Theory reading group
The Theory reading group meetings run for 3 hours plus minus, during
which time we have the possibility to first get an overview of some
interesting research results and then study them in some depth.
Meetings in the Theory reading group are announced in the
KTH CSC calendar
and the
Stockholm
Mathematics Center calendar,
as well as on the
TCS seminar mailing list. If you want to receive the announcements
but are not (yet) on the TCS mailing list, just contact
Jakob Nordström.
We usually start at noon with a light lunch, and at 12:10 pm there is a
presentation which ends at 1 pm sharp. This should be an accessible
talk, not requiring any particular prerequisites. Then there is a
short break. Up to this point the whole exercise is usually pretty
much indistinguishable from a regular (lunch) seminar.
After the break, we return for ca two hours of more technical
discussions. Now we open up the hood, look at the formal definitions,
and see the proofs of some of the key ingredients in the results
including all (or at least some of) the gory technical details glossed
over in a polished seminar presentation. If the first part of the
seminar was with slides, we now switch to the board, and there is
usually a lot of interaction (making this into more of a discussion
than a presentation at times).
However, for those who feel that the first onehour regular seminar
was enough for today, it is perfectly fine to just discretely drop out
during the break. No excuses needed; no questions asked.
In addition to the Theory reading group we also have a series of less
formal (and perhaps slightly less listenerfriendly)
Complexity meetings.
These meetings are not announced publicly in any calendars or via the TCS seminars mailing list, but there is a
special Complexity meetings mailing list
that you can join if you are interested. See the
Complexity meetings webpage
for more information.
Upcoming seminars
It seems like the Theory reading group might have gone on summer
vacation, but it should restart some time in AugustSeptember.
Theory reading group meetings spring 2017

Monday May 29 at 12:00 in 4523, Lindstedtsvägen 5
Unified and optimal lower bounds for monotone computation
(Robert Robere, University of Toronto)
A classic counting argument due to Claude Shannon shows that almost
all boolean functions have high complexity — more formally, all but
an exponentially small fraction of boolean functions on n bits require
strongly exponential (i.e. 2^Ω(n)) size circuits. On the other
hand, the best lower bounds on circuit size that we have proven for
any explicit function is on the order of 5n  o(1), which is not even
superlinear! The state of affairs is not much better for boolean
formulas, for which we merely have cubic lower bounds. Even for monotone
circuits and formulas, which are much more understood, the best lower
bounds for explicit functions are weakly
exponential (i.e. of the form 2^{n^c} where c < 1).
In this talk, we discuss some recent work in which we have nearly
matched Shannon's lower bound for computing an explicit function by
monotone circuit models. In particular, we prove that a monotone
variant of the SAT problem requires monotone formulas of size 2^{c*n}
for some universal constant c > 0. Our lower bounds are tight (up to
the constant c) for any monotone boolean function, and thus represent
the first example of any explicit monotone boolean function with
strongly exponential size lower bounds. Furthermore, our techniques
are very general and apply to a wide variety of monotone circuits
models.
This work is joint with Toniann Pitassi.

Wednesday May 24 at 12:00 in 4523, Lindstedtsvägen 5
"Secure" execution platforms for application software —
some definitions of "secure"
(Frank Piessens, Katholieke Universiteit Leuven)
Software applications run on top of execution platforms consisting of
hardware (processors, devices, communication networks, and so forth) as
well as software (operating systems, compilers, virtual machines,
language runtimes, databases, and so forth). It is obviously desirable
that these platforms are "secure", but it is less obvious what "secure"
means in this context. It is an empirical fact that attacks against
application software often rely at least to some extent on aspects of
the execution platform, but often the possibility of an attack is a
joint responsibility between application code vulnerabilities and
execution platform characteristics. For instance, a memory safety
vulnerability is only a security concern if the compiler or operating
system do not implement adequate countermeasures against exploitation
of such vulnerabilities. As another example, an attacker sniffing the
network is less of a concern if either the application, or the platform
provide adequate cryptographic protection.
So what exactly should we require of a platform for it to be secure?
This seminar will discuss the problem of platform security by proposing
and analyzing a number of different formal definitions for platform
security, building on classical concepts from programming language
theory.

Monday May 22 at 10:00
(note the time!)
in 1537, Lindstedtsvägen 3
Part I: Futureproof software +
Part II: How to test the universe?  Efficient testing of software product lines
(Ina Schaefer,
Technische Universität Braunschweig)
Part I: Modern software systems are extremely longlived and evolve
over time in order to adapt to changing user requirements,
application contexts or technological changes. This leads to
challenges for software design, implementation, quality assurance
and reengineering in order to make those software systems
futureproof. In this talk, I will present an overview of ongoing
research in my group at the Institute of Software Engineering and
Automotive Informatics at TU Braunschweig addressing the challenges
of longliving and evolving software systems. I will focus on two
areas of our work:

applying machine learning techniques for
efficient systemlevel regression testing and

similaritybased
variability model mining in order to transform grown software
systems into wellstructured software product line.
Part II: Software product line engineering has gained considerable
momentum in recent years, both in industry and in academia. A
software product line is a family of software products that share a
common set of features. Software product lines challenge traditional
analysis, test and verification techniques in their quest of
ensuring correctness and reliability of software. Simply creating
and testing all products of a product line is usually not feasible,
due to the potentially exponential number of valid feature
combinations. In this talk, I present different strategies for
software product line testing and focus on our recent work on
incremental test case selection and test case prioritization of
efficient product line testing.

Monday May 8 at 12:00
in 4523, Lindstedtsvägen 5
Informationtheoretic thresholds
(Amin CojaOghlan, GoetheUniversität)
Vindicating a sophisticated but nonrigorous physics approach called
the cavity method, we prove a formula for the mutual information in
statistical inference problems induced by random graphs. This general
result implies the conjecture on the informationtheoretic threshold
in the disassortative stochastic block model [Decelle et al.:
Phys. Rev. E (2011)] and allows us to pinpoint the exact condensation
phase transition in random constraint satisfaction problems such as
random graph coloring, thereby proving a conjecture from [Krzakala et
al.: PNAS (2007)]. As a further application we establish the formula
for the mutual information in LowDensity Generator Matrix codes as
conjectured in [Montanari: IEEE Transactions on Information Theory
(2005)]. Joint work with Florent Krzakala, Will Perkins and Lenka
Zdeborova.

Monday Apr 24 at 12:00 in 4523, Lindstedtsvägen 5
Dynamic (minimum) spanning forest with worstcase update time
(Thatchaphol Saranurak, Theory Group, KTH)
We will discuss recent algorithms for dynamically maintaining a (minimum)
spanning forest of a graph undergoing edge insertions and deletions. This
problem played a central role in the study of dynamic graph algorithms.
We provide the "first polynomial improvement" over the longstanding
O(sqrt{n}) bound of [Frederickson STOC'83, Eppstein et. al FOCS'92] for
dynamic spanning forest problem. Independently, WulffNilsen shows the
first polynomial improvement for dynamic minimum spanning forest problems.
Both works will be presented in STOC'17.
The new crucial techniques are about expanders:
 an algorithm for
decomposing a graph into a collection of expanders in nearlinear time, and
 an algorithm for "repairing" the expansion property of an expander after
deleting some edges of it.
These techniques can be of independent interest.

Monday April 10 at 12:00 in 4523, Lindstedtsvägen 5
Complexity results in automated planning
(Meysam Aghighi, Linköping University)
Planning is the problem of finding a sequence of actions that achieve
a specific goal from an initial state. This problem is computationally
hard in the general case. Propositional planning is PSPACEcomplete
and firstorder planning is undecidable.
Costoptimal planning (COP) is a type of planning where each action
has a cost and the goal is to find a satisfying plan with minimal
cost. We show how under different restrictions, COP varies from being
polynomialtime solvable to PSPACEcomplete. Netbenefit planning
(NBP) is a type of planning where the goal is to find a plan that
maximizes the achieved utility minus the cost spent. We present a
method for efficiently compiling NBP into the ordinary plan existence
problem.
We also show the effect of numeric domains for action costs in COP
using parameterized complexity. Planning is PSPACEcomplete regardless
of whether action costs are positive or nonnegative, integer or
rational. We distinguish between these cases and show that COP is
W[2]complete for positive integer costs, but it is paraNPhard if
the costs are nonnegative integers or positive rationals.

Monday Mar 27 at 12:00 in 4523, Lindstedtsvägen 5
Complexity theory beyond deterministic exponential time and applications
(Igor Carboni Oliveira, Charles University)
In the first part of the talk, I'll survey a variety of connections between
nonuniform circuit lower bounds, nontrivial proof systems, nontrivial
learning algorithms, and variants of natural properties. In particular, I
plan to discuss how one can prove new unconditional lower bounds for
standard complexity classes such as REXP, ZPEXP, BPEXP, and NEXP by
designing algorithms with a nontrivial running time.
After the break, we will focus on a recent application of some of the
complexitytheoretic techniques behind these results. I'll describe in more
detail recent progress on the problem of efficiently generating large
canonical prime numbers, via pseudodeterministic pseudorandomness. If there
is enough time and interest, I'll also discuss some nonconstructive
aspects of the result, and connections to derandomization.
Based on joint work with Rahul Santhanam.

Thu Mar 16 at 13:15
(note the time!)
in 4523, Lindstedtsvägen 5
Daglike communication and its applications
(Dmitry Sokolov, St. Petersburg Department of V. A. Steklov Institute of Mathematics)
In this talk we consider a generalization of communication protocols to
daglike case (instead of classical one where protocols correspond to
trees). We prove an analogue of KarchmerWigderson Theorem for this model
and boolean circuits. We also consider a relation between this model and
communication PLS games proposed by Razborov in 1995 and simplify the proof
of Razborov's analogue of KarchmerWigderson Theorem for PLS games.
We also consider a daglike analogue of realvalued communication protocols
and adapt a lower bound technique for monotone real circuits to prove a
lower bound for these protocols. We use realvalued daglike communication
protocols to generalize ideas of "feasible interpolation" technique,
which helps us to prove a lower bound on the Cutting Planes proof system
(CP) and adapt it to "random CP" by using recent result of
Krajíček.
Our notion of daglike communication games allows us to use a RazMcKenzie
transformation from Goos and Pitassi paper, which yields a lower bound on
the real monotone circuit size for the CSPSAT problem.

Monday Mar 13 at 12:00 in 4523, Lindstedtsvägen 5
A nearly tight sumofsquares lower bound for the planted clique problem
(Aaron Potechin, Institute for Advanced Study)
The sum of squares (SoS) hierarchy, a method for deciding the
feasibility of polynomial equations over the reals, is an exciting frontier
of complexity theory. SoS is exciting because it can be applied to a wide
variety of combinatorial optimization problems and is in fact conjectured
to be optimal for many important problems. However, its performance is only
partially understood.
In this talk, I will describe recent work proving sum of squares lower
bounds for the planted clique problem, a classic average case problem. In
the first part of the talk, I will describe the sum of squares hierarchy,
planted clique, and pseudocalibration, a key idea for the lower bound. In
the second part of the talk, I will describe the main technical ideas
needed in the proof.

Friday Mar 10 at 12:00 in 4523, Lindstedtsvägen 5
Finding little graphs inside big graphs (in parallel)
(Ciaran McCreesh, University of Glasgow)
I'll give a high level overview of the best practical algorithms
for three families of subgraph problems: maximum clique (finding
the largest group of people, all of whom are friends with
everyone else in the group), subgraph isomorphism (finding a
pattern graph inside a target graph), and maximum common
subgraph (finding what two graphs have in common). These
problems are all NPhard, but nonetheless we can solve some
instances involving graphs with tens of thousands of vertices,
and millions of edges. I'll discuss the kinds of graphs that
make subgraph problems easy to solve in practice, and then
explain how to generate "really hard" instances (and why this
matters). Finally, I'll present an overview of my research on
parallelising these algorithms to exploit multicore hardware:
with the right worksplitting mechanisms, we can get strong,
consistent, reproducible speedups up to at least the 64 core
mark. This may come as a surprise to those familiar with
parallel SAT, CP or MIP solvers, so I'll explain what makes
these subgraph algorithms different.
After the break, we will have an indepth look at a stateoftheart maximum clique algorithm due to Tomita et al. This algorithm is easy to understand from a correctness perspective, but it features several design choices which are hard to justify theoretically. We'll discuss three questions, and I'll show some empirical work which suggests that they share a common answer:

How exactly does this algorithm behave on random graphs, and why?

Why is it so much better for the main loop in the algorithm to iterate
backwards rather than forwards?

Why does introducing parallelism via randomised workstealing behave
so erratically, and how can we do better?

Monday Mar 6 at 13:15 (note the time!)
in 4523, Lindstedtsvägen 5
SATbased learning to verify liveness of randomised parameterised systems
(Philipp Rümmer, Uppsala University)
We consider the problem of verifying liveness for systems with a
finite, but unbounded, number of processes, commonly known as
parameterised systems. Typical examples of such systems include
distributed protocols (e.g. for the dining philosopher
problem). Unlike the case of verifying safety, proving liveness is
still considered extremely challenging, especially in the presence of
randomness in the system. We introduce an automatic method of proving
liveness for randomised parameterised systems under arbitrary
schedulers. Viewing liveness as a twoplayer reachability game
(between Scheduler and Process), our method is a CEGAR approach that
synthesises a progress relation for Process that can be symbolically
represented as a finitestate automaton. The method constructs a
progress relation by means of a suitable Boolean encoding and
incremental SAT solving. Our experiments show that our algorithm is
able to prove liveness automatically for wellknown randomised
distributed protocols, including LehmannRabin Randomised Dining
Philosopher Protocol and randomised selfstabilising protocols (such
as the IsraeliJalfon Protocol). To the best of our knowledge, this is
the first fullyautomatic method that can prove liveness for
randomised protocols.
Joint work with Anthony W. Lin.

Monday Jan 23 at 12:00 in 4523, Lindstedtsvägen 5
Cumulative space in blackwhite pebbling and resolution
(Susanna F. de Rezende, Theory Group, KTH)
The space usage of a computation is usually defined as the maximum memory it
requires, and this is essentially everything one would want to know if the
computational model is static. But what if we are computing in an
environment where we can allocate memory dynamically? Then it makes a big
difference whether we only have one single spike of high memory consumption
or whether we consistently need large amounts of memory throughout the whole
computation.
In this talk we will explore a complexity measure, namely cumulative
space, which allows us to measure this difference by accounting for the sum
of memory usage over all time steps rather than just the maximum. It was
introduced for pebble games by [Alwen and Serbinenko '15] and used to
propose a more robust definition of cryptographic functions that are hard to
invert (memoryhard functions).
We will present some results regarding the cumulative space of another
pebble game, namely the blackwhite pebble game, and relate this to the
cumulative space complexity of refuting formulas in the resolution proof
system.
This is joint work with Joël Alwen, Jakob Nordström
and Marc Vinyals.

Wednesday Jan 18 at 12:00 in 1537, Lindstedtsvägen 3
The QRAT proof system
(Martina Seidl, Johannes Kepler University Linz)
We present QRAT, a novel proof system for quantified Boolean formulas.
With QRAT it is now possible to produce certificates for all reasoning
techniques implemented in stateoftheart QBF preprocessors. Such
certificates are not only useful to efficiently validate the
correctness of a result, but they also allow the extraction of Skolem
functions, i.e., winning strategies for satisfiable formula instances.
In this talk, we first introduce the rules of QRAT, then we show how
they can be used to express some powerful reasoning techniques
implemented in modern preprocessors, and finally we outline how to
extract a Skolem function from a QRAT proof.
Theory reading group meetings autumn 2016
During the autumn of 2016 we kept ourselves quite busy with the course
DD2442
Seminars on Theoretical Computer Science: Proof Complexity,
which served pretty much as a long series of proof complexity seminars.

Monday Nov 28 at 12:00 in 4523, Lindstedtsvägen 5
Space in proof complexity
(Marc Vinyals, Theory Group, KTH)
Space in proof complexity measures the amount of memory that a
verifier needs to check a proof, which imposes a lower bound on the
memory a SAT solver needs to generate a proof. Space is very well
understood for the most common proof system, resolution, but less so
in other proof systems. In this talk we will survey some recent
results about space in other settings: stronger proof systems such as
polynomial calculus and cutting planes on the one hand, and a weaker
"CDCL" proof system that is closer to the actual capabilities of
SATsolving algorithms on the other hand. We will even explore
alternative definitions of space. The proof techniques will lead us to
discussing adjacent areas of computational complexity such as pebble
games and communication complexity.

Monday Nov 21 at 12:00 in 4523, Lindstedtsvägen 5
Strong size lower bounds in regular resolution via games
(Ilario Bonacina, Theory Group, KTH)
The Strong Exponential Time Hypothesis (SETH) says that solving the
SATISFIABILITY problem on formulas that are kCNFs in n variables
require running time 2^(n(1  c_k)) where c_k goes to 0 as k goes to
infinity. Beck and Impagliazzo (2013) proved that regular resolution
cannot disprove SETH, that is they prove that there are unsatisfiable
kCNF formulas in n variables such that each regular resolution
refutation of those has size at least 2^(n(1  c_k)) where c_k goes to
0 as k goes to infinity. We give a different/simpler proof of such
lower bound based on the known characterizations of width and size in
resolution and our technique indeed works for a proof system stronger
than regular resolution. The problem of finding kCNF formulas for
which we can prove such strong size lower bounds in general resolution
is still open.
This is a joint work with Navid Talebanfard.

Monday Nov 7 at 12:00 in 4523, Lindstedtsvägen 5
Graphbased pseudoindustrial random SAT instance generators
(Jesús Giráldez Crú, Theory Group, KTH)
The Boolean Satisfiability problem (SAT) is the canonical NPcomplete
problem. However, ConflictDriven Clause Learning (CDCL) SAT solvers
are nowadays able to efficiently solve many industrial, or realworld,
SAT instances — as hardware and software verification, cryptography,
planning or scheduling, among others. On the other hand, relatively
small random SAT instances are still too hard. The common intuition to
explain the success of CDCL solving industrial instances is the
existence of some hidden structure in them (whereas random formulas
would not show such structure). In some works, this structure is
studied representing SAT instances as graphs and analyzing some graph
features, showing that these features are shared by the majority of
existing industrial SAT instances. Some examples are the scalefree
structure or the community structure.
Unfortunately, the process of development and testing of new SAT
solving techniques (specialized in industrial problems) is conditioned
to the reduced number of existing industrial benchmarks. Therefore,
the generation of random SAT instances that captures realistically
some computational properties of such industrial instances is an
interesting open problem.
In this talk, we review some models of pseudoindustrial random SAT
instances generation. They are the scalefree model and the community
attachment model, which are related to some wellknown concepts in
realworld complex networks: preferential attachment and high
modularity, respectively. In the scalefree model, the number of
variable occurrences k follows a powerlaw distribution P(k) \propto
k^{\alpha}. With the community attachment model, it is possible to
generate SAT instances with clear community structure (i.e., high
modularity). These models will be reviewed from the perspectives of
both graphs and SAT instances. Finally, we discuss some models for
generating complex networks — not adapted to SAT instances yet
— that may reduce the limitations of the previous models.

Monday Oct 17 at 12:00 in 4523, Lindstedtsvägen 5
Simulation theorem and forklift
(Sagnik Mukhopadhyay, Tata Institute of Fundamental Research, Mumbai)
Recently, proving theorems of the form that the communication
complexity of a composed function f \circ g is essentially of the
order of the decision tree complexity of f times the communication
complexity of g has received a lot of attention. In particular,
GoosPitassiWatson (2015) simplified the proof of such a theorem for
deterministic complexity due to RazMcKenzie (1997) that worked only
when g is the Indexing function. They used this theorem to settle a
longstanding open problem in communication complexity. The
RazMcKenzie theorem needs the size of the Indexing gadget to be at
least p^20, where p is the number of instances of Index.
We identify a simple sufficient condition for g to be satisfied to
prove such deterministic simulation theorems. Using this, we show that
CC(f \circ IP) = Omega(DT(f). n), provided n = Omega(log p), where IP
is the innerproduct function. This gives an exponential improvement
over the gadget size of RazMcKenzie.
We also prove a tight lower bound for randomized communication
complexity of OrdSearch \circ IP in terms of randomized decision tree
complexity of the function OrdSearch, which is Omega(log p). Proving a
randomized simulation theorem remains elusive and we will discuss the
hurdles that are needed to be faced and overcome.
This is a joint work with Arkadev Chattopadhyay (TIFR Mumbai), Michal
Koucky and Bruno Loff (Charles University, Prague).
Theory reading group meetings spring 2016

Monday May 2 at 12:00 in 4523, Lindstedtsvägen 5
Dynamic primaldual algorithms for vertex cover and matching
(Sayan Bhattacharya, Institute of Mathematical Sciences Chennai)
Consider a scenario where we are given an input graph G = (V, E) with n nodes
and m edges. The nodeset of the graph remains unchanged over time, but the
edgeset is dynamic. Specifically, at each timestep an adversary either inserts
an edge into the graph, or deletes an already existing edge from the graph. The
problem is to maintain an approximately maximum matching (resp. minimum vertex
cover) in this dynamic setting.
We present a clean primaldual algorithm for this problem that has O(log
n/\epsilon^2) amortized update time. The approximation ratio of the algorithm is
(2+\epsilon) for minimum vertex cover and (3+\epsilon) for maximum matching. We
also show how to extend this result to the dynamic bmatching and setcover
problems. This is the first application of the primaldual framework in a
dynamic setting.
Joint work with M. Henzinger and G. F. Italiano (based on papers that appeared
in SODA 2015 and ICALP 2015).

Monday April 25 at 12:00 in 4523, Lindstedtsvägen 5
Memoryhard functions and parallel graph pebbling
(Joël Alwen, Institute of Science and Technology Austria)
There is growing interest in the security community in MemoryHard Functions
(MHFs). These require moderate amounts of memory to compute on a generalpurpose
singleprocessor (i.e. sequential) machine, but cannot be repeatedly computed
with significantly less memory amortized per instance on dedicated custom
hardware (i.e. a parallel generalized circuit). For example, such functions
are used for passwordhashing and keyderivation to prevent bruteforce attacks
being costeffectively implemented on custom circuits and in proofsofwork for
more egalitarian decentralized cryptocurrencies.
In (STOC 2015) Alwen and Serbinenko showed that, in order to construct a secure
MHF it suffices to find a constant indegree directed acyclic graph with a high
cumulative pebbling complexity in a simple game of parallel pebbling. Conversely
a wide class of candidate MHF constructions from the literature are given by
fixing some particular (constant indegree) DAG and showing an efficient way to
pebble these DAGs immediately leads to a break of the MHF construction (i.e. a
method of computing the MHF with low parallel memory complexity).
The first part of this talk will be aimed at providing an overview of this area
of research. In particular will cover the following:

The motivation for and definition of MHFs.
 The close connection to a certain parallel pebbling game over DAGs and new
pebbling complexity measure called the cumulative complexity (CC) of the DAG.
 What won't work: line graphs, bitreversal graphs, superconcentrators and
stacks of superconcentrators.
 A powerful parallel pebbling algorithm with low CC. In particular we show how
this gives us a nontrivial general lowerbound on the CC of any DAG of a fixed
size.
 A method for lowerbounding the CC of a DAG using a wellstudied
combinatorial property of the DAG called its depthrobustness.
 Finally we conclude with two strong positive results. Namely a pair of
constant indegree DAGs enjoying very high CC. Indeed the second has maximal CC
for any constant indegree DAG of equal size. Moreover it can be sequentially
pebbled with this same CC. Thus we obtain a provably secure MHF.
To demonstrate the power of these tools we will also briefly describe their
implications for several of the most important MHF candidate constructions from
the literature including the winner and several of the finalists of the recent
multiyear international Password Hashing Competition. For each candidate we
will see an attack strongly invalidating the conjectured security of the
construction. We will also see a (weak) security proof for the construction
showing that the attack is (essentially) optimal.
The second part of this talk will focus on some of the most important proof
techniques underlying these results. In particular we will cover the following:
 The "metanode" technique for analysing the CC of random DAGs.
 A method for indegree reduction in graphs.
 Lowerbounding CC using depthrobustness.
 Using the "dispersal" property of a graph to lowerbound its CC.
 Stacks of depthrobust graphs with maximal parallel CC which can nevertheless
be sequentially pebbled with the same CC.

Monday April 18 at 12:00 in 4523, Lindstedtsvägen 5
Experimenting with CDCL SAT solvers and glue clauses
(Laurent Simon, Université de Bordeaux)
Trying to tackle in practice NPComplete problems was believed hopeless 20 years
ago. However, with the introduction of Conflict Driven Clause Learning
algorithms (CDCL for short) in the late 90's, we observed one of the most
fascinating victories against hard problems. However, despite these impressive
results, the underlying reasons for these successes are just partially known. We
will begin this talk by presenting a set of experiments showing why SAT solvers
are so hard to study in practice.
In a second part of the talk, we will focus on one of the many ingredients of
SAT solvers: the concept of glue clauses and Literal Bock Distance. This measure
for the quality of learnt clauses was introduced in 2009 and is now used in most
of CDCL solvers. However, despite its interest, this measure is not fully
understood. We will present the concept of glue clauses, as it was stated five
years ago, and develop new insights in what may explain its performance, for
instance by trying to find correlations between blocks as stated in the LBD
measure and communities.

Monday April 4 at 12:00 in 4523, Lindstedtsvägen 5
Unconditional lower bounds for data structures
(Kasper Green Larsen, Aarhus University)
In the first part of this talk, we will introduce and motivate the
cell probe model for proving data structure lower bounds. We will then
survey some of the recent techniques for proving lower bounds in this
model, with an emphasis on the results obtained by the speaker and
coauthors. The talk will highlight the current limitations of our
techniques and we will also briefly discuss work by the speaker on
lower bounds in more restricted models of computation.
The second part of the talk is more technical and will be based on a
FOCS'15 paper joint with Raphal Clifford (Bristol) and Allan Grønlund
(Aarhus). The main focus here is a new type of threshold lower bound
proved for the wellstudied Multiphase Problem. The Multiphase
Problem, presented by Patrascu at STOC'10, was one of the problems
that really sparked the recently very popular discipline of proving
conditional lower bounds. Our focus is on proving unconditional lower
bounds for the Multiphase Problem in the regime of parameters where we
can actually make useful statements. More specifically, we show that
any data structure for the Multiphase Problem which insist on having a
very fast query time of o(lgn/lglgn) must have n^{1o(1)} update
time. This is a whole new type of lower bound as previous techniques
could only prove n^{eps} update time lower bounds, even when insisting
on O(1) query time. We will also briefly touch on new lower bounds we
prove for Matrixvector multiplication.

Monday March 21 at 12:00 in 4523, Lindstedtsvägen 5
Deterministic communication vs. partition number
(Marc Vinyals, Theory Group, KTH)
Alice is given a clique in a graph and Bob an independent set in the
same graph. How much communication do they need to decide if these two
sets of vertices intersect? This seemingly innocent question is
connected to deep topics in communication complexity and analysis of
Boolean functions.
In a breakthrough paper in FOCS 2015,
Göös, Pitassi and Watson solved
this and other problems by proving lower bounds in query complexity,
and then giving an explicit way of amplifying query complexity lower
bounds to communication complexity lower bounds. This solved a problem
that had been open since 1979, and the paper has already generated a
long (and growing) list of followup works that have made progress on
other longstanding open problems in different areas of communication
complexity and query complexity.
In this seminar, we will go over the GPW paper. During the first hour
we will review the new results, and after the break we will present a
detailed proof of their main theorem.

Wednesday March 16 at 12:00 in 4523, Lindstedtsvägen 5
Title: Structural restrictions of CNF formulas: applications and limitations
(Florent Capelli, Université Paris Diderot)
It is wellknown that clauses restrictions of CNF formulas
such as 2SAT or HornSAT are easy instances of the problem SAT. It is
however not the case for harder problems such as #SAT, the problem of
counting the satisfying assignments of a CNF formula: #2SAT is already
as hard as the general case. Fortunately, restrictions on the way the
variables interact with the clauses have been a successful approach to
find large classes of formulas for which #SAT was doable in polynomial time.
In the first part of this lunch seminar, I will give a broad picture
of what can be done with these structural restrictions of CNF formulas.
I will first present how such restrictions are defined and give an
overview of the tractability results they enable for #SAT. I will then
leverage these results to the broader problem of knowledge compilation,
that is, the problem of finding a good data structure representing F
that supports queries such as model counting or enumeration in
polynomial time. This naturally raises the questions of finding hard
instances for such algorithmic techniques. We reformulate these
questions as proving lower bounds in knowledge compilation and answer
this by giving new exponential lower bounds on the compilation of some
family of CNF formulas.
In the second and more technical part of the talk, I will either
present the algorithmic techniques in more details or give a complete
proof of one of the lower bounds mentioned above depending on what the
audience prefers. Most of the results presented in this talk were
conceived in collaboration with Simone Bova, Stefan Mengel and Friedrich
Slivovsky.

Monday March 14 at 12:00 in 4523, Lindstedtsvägen 5
Verification of bitvector arithmetic using finite integer algebras
(Priyank Kalla, University of Utah)
Finiteprecision integer arithmetic plays an important role in many hardware and
software systems, minimizing resource usage while maintaining necessary
precision. However, operating on these bitvector (BV) datatypes can introduce
subtle, unforeseen errors, causing incorrect function or even security
vulnerabilities. With the widespread use of finiteprecision arithmetic from
multimedia DSP to embedded automotive control, it is imperative to devise new
techniques to efficiently model and verify such systems at higher levels of
abstractions  raising the abstraction from bits to words, yet maintaining
precision.
A bitvector of size "m" represents integer values reduced "(mod 2^m)".
Therefore, BVarithmetic can be modeled as a system of polynomial functions over
Z_{2^m}; and numbertheoretic and algebraic techniques can be devised for
verification. In this talk, I will describe decision procedures for verification
of bitvector arithmetic that lie at the crossroads of number theory and
commutative algebra  such as canonical simplification of polynomial
functions, Newton's padic iterations, etc. We will look at the challenge of
making such techniques practical, and also discuss their potential for
integration with SMTsolvers.

Tuesday March 8 at 12:00 in 4523, Lindstedtsvägen 5
SATenabled verification of state transition systems
(Karem Sakallah, University of Michigan and Qatar Computing Research Institute)
Modern conflictdriven clauselearning (CDCL) SAT solvers, introduced twenty
years ago, have had a profound impact in many domains by enabling the solution
of largescale problems that were thought to be out of reach. It is now routine
for modern SAT and SAT modulo Theories (SMT) solvers to tackle decision problems
containing millions of variables and constraints. Verification of complex
hardware and software systems is now increasingly facilitated by the automated
reasoning capabilities of modern SAT technology.
In this seminar I argue that the CDCL paradigm is now sufficiently mature and
attempts to improve it further can only yield incremental gains in performance
and capacity. Instead, I propose to combine it with two equally powerful
concepts to allow for scalable reasoning about exponentiallylarge state
transition systems. The first concept, pioneered by the IC3 and later PDR
incremental induction reachability solvers, culminates a decadesold quest for
solving the socalled state explosion problem in model checking. The second
concept, CounterExampleGuided Abstraction Refinement (CEGAR for short),
provides an adaptive computational framework for managing complexity by a)
judicious elimination of irrelevant details (abstraction/overapproximation)
and by b) automatically filtering any false positives/spurious counterexamples
(refinement).
After briefly describing the salient features of these two concepts I will
illustrate their use, along with an underlying SAT/SMT engine, on two example
applications of state transition systems: sequential hardware verification and
detection of crosssite scripting (XSS) in PHP web servers. In both cases the
goal is to show that all states reachable from a good initial state satisfy a
given safety property or to produce a counterexample trace demonstrating
violation of the property.
Theory reading group meetings autumn 2015

Monday December 14 at 12:00 in 4523, Lindstedtsvägen 5
Linear temporal logic satisfiability checking
(Kristin Yvonne Rozier, University of Cincinnati)
Formal verification techniques are growing increasingly vital for the
development of safetycritical software and hardware. Techniques such
as requirementsbased design and model checking have been successfully
used to verify systems for air traffic control, airplane separation
assurance, autopilots, logic designs, medical devices, and other
functions that ensure human safety. Formal behavioral specifications
written early in the systemdesign process and communicated across all
design phases increase the efficiency, consistency, and quality of the
system under development. We argue that to prevent introducing design
or verification errors, it is crucial to test specifications for
satisfiability.
In 2007, we established LTL satisfiability checking as a sanity check:
each system requirement, its negation, and the set of all requirements
should be checked for satisfiability before being utilized for other
tasks, such as propertybased system design or system verification via
model checking. We demonstrated that LTL satisfiability checking
reduces to model checking; an extensive experimental evaluation proved
that for LTL satisfiability checking, the symbolic approach is
superior to the explicit approach. However, the performance of the
symbolic approach critically depends on the encoding of the
formula. Since 1994, there had been essentially no new progress in
encoding LTL formulas as symbolic automata for BDDbased analysis. We
introduced a set of 30 symbolic automata encodings, demonstrating that
a portfolio approach utilizing these encodings translates to
significant, sometimes exponential, improvement over the standard
encoding for symbolic LTL satisfiability checking. In recent years,
LTL satisfiability checking has taken off, with others inventing
exciting new methods to scale with increasingly complex systems. We
revisit the benchmarks for LTL satisfiability checking that have
become the de facto industry standard and examine the encoding methods
that have led to leaps in performance. We highlight the past and
present, and look to the future of LTL satisfiability checking, a
sanity check that now has an established place in the development
cycles of safetycritical systems.

Monday November 30 at 12:00 in 4523, Lindstedtsvägen 5
Model checking, SAT and bitvectors + Evaluating CDCL restart schemes
(Armin Biere, Johannes Kepler University Linz)
The lunch seminar part is titled "Model checking, SAT and bitvectors"
with abstract as follows.
Both SAT solving and Model Checking are considered to have saved the
reputation of formal methods. We will highlight how their recent
history is actually closely related. We further discuss important
developments in both domains, mostly from the historical and practical
point of view, but then will delve into the complexity of deciding
bitvector logic. This logic is the most important theory for
bitprecise reasoning with SMT solvers and has many practical
applications in testing and verification both of Hardware and
Software. Related to solving certain bitvector problems is the
challenge to make bitlevel arithmetic reasoning work.
After the break, there will be a more technical presentation on
evaluating restart schemes for CDCL SAT solvers, which is based on
joint work with Andreas Frölich.
Modern CDCL (conflictdriven clause learning) SAT solvers are used for
many practical applications. One of the key ingredients of
stateoftheart CDCL solvers are efficient restart schemes. The main
contribution of this work is an extensive empirical evaluation of
various restart strategies. We show that optimal static restart
intervals are not only correlated with the satisfiability status of a
certain instance, but also with the more specific problem class of the
given benchmark. We further compare uniform restart intervals with the
performance of nonuniform restart schemes, such as Luby
restarts. Finally, we revisit the dynamic restart strategy used in
Glucose and propose a new variant thereof, which is based on the
concept of exponential moving averages. The resulting implementation
in Lingeling improves stateoftheart performance in SAT solving.

Monday November 23 at 12:00 in 4523, Lindstedtsvägen 5
Lower bounds: from circuits to QBF proof systems
(Ilario Bonacina, Theory Group, KTH)
A general and longstanding belief in the proof complexity community
asserts that there is a close connection between progress in lower
bounds for Boolean circuits and progress in proof size lower bounds
for strong propositional proof systems. Although there are famous
examples where a transfer from ideas and techniques from circuit
complexity to proof complexity has been effective, a formal connection
between the two areas has never been established so far. Here we
provide such a formal relation between lower bounds for circuit
classes and lower bounds for Frege systems for quantified Boolean
formulas (QBF). Using the full spectrum of the stateoftheart
circuit complexity lower bounds we will prove lower bounds for very
strong QBF proof systems (e.g. for what we called AC0[p]FREGE +
\forall red). Such lower bounds corresponds, in the propositional
case, to major open problems in proof complexity.
This talk is based on the joint work with Olaf Beyersdorff and Leroy
Chew (ECCC TR15133 and ITCS16, to appear).

Monday November 9 at 12:00 in 4523, Lindstedtsvägen 5
Oblivious and online matching via continuous linear programming
(Fei Chen, Theory Group, KTH)
Variants of the maximum matching problem have wide applications in the
real world. Motivated by a kidney exchange program, where kidney
transfer is expected to be performed right after patients and donors
pass the compatibility tests, the oblivious matching problem was
proposed allowing greedy matching algorithms only. Motivated by online
advertising, where for each keyword arriving online the advertising
system should immediately decide which ad to display to maximize the
profit, the online matching setting was proposed that requires the
algorithm to maintain a matching in an online fashion.
We study the oblivious and online matching problems. For oblivious
matching, we prove that the Ranking algorithm has a performance ratio
of at least 0.523 on arbitrary graphs. For edgeweighted online
bipartite bmatching, we give a procedure to construct an
asymptotically optimal algorithm. The analysis of both problems relies
on linear programming framework. We use a continuous linear
programming approach to analyze the limiting behavior of normal
LPs. In particular, our results for online bipartite bmatching are
based on a generalized secretary problem. The continuous LP gives a
clear perspective on the secretary problem from which we are able to
make a connection between secretary and online matching.

Monday October 12 at 12:00 in 4523, Lindstedtsvägen 5
Fast and powerful hashing using tabulation
+ Deterministic edge connectivity in nearlinear time
(Mikkel Thorup, University of Copenhagen)
The first part of the presentation, titled "Fast and powerful hashing using
tabulation," is intended for a general audience.
Randomized algorithms are often enjoyed for their simplicity, but the hash
functions employed to yield the desired probabilistic guarantees are often too
complicated to be practical. Here we survey recent results on how simple hashing
schemes based on tabulation provide unexpectedly strong guarantees.
Simple tabulation hashing dates back to Zobrist [1970]. Keys are
viewed as consisting of c characters and we have precomputed character tables
h_1,...,h_c mapping characters to random hash values. A key x=(x_1,...,x_c) is
hashed to h_1[x_1] ⊕ h_2[x_2].....⊕ h_c[x_c]. This schemes is very
fast with character tables in cache. While simple tabulation is not even
4independent, it does provide many of the guarantees that are normally obtained
via higher independence, e.g., linear probing and Cuckoo hashing.
Next we consider twisted tabulation where one input character is
"twisted" in a simple way. The resulting hash function has powerful
distributional properties: ChernoffHoeffding type tail bounds and a very small
bias for minwise hashing. This is also yields an extremely fast pseudorandom
number generator that is provably good for many classic randomized algorithms
and datastructures.
Finally, we consider double tabulation where we compose two simple
tabulation functions, applying one to the output of the other, and show that
this yields very high independence in the classic framework of Carter and
Wegman [1977]. In fact, with high probability, for a given set of size
proportional to that of the space consumed, double tabulation gives fullyrandom
hashing. We also mention some more elaborate tabulation schemes getting
nearoptimal independence for given time and space.
While these tabulation schemes are all easy to implement and use, their analysis
is not.
After the break, there will be a more technical presentation titled
"Deterministic edge connectivity in nearlinear time," based on joint work with
Kenichi Kawarabayashi.
We present a deterministic algorithm that computes the edgeconnectivity of a
graph in nearlinear time. This is for a simple undirected unweighted graph G
with n vertices and m edges. This is the first o(mn) time deterministic
algorithm for the problem. Our algorithm is easily extended to find a concrete
minimum edgecut. In fact, we can construct the classic cactus representation of
all minimum cuts in nearlinear time.
The previous fastest deterministic algorithm by Gabow from STOC'91 took ~O(m+k^2
n), where k is the edge connectivity, but k could be Omega(n).
At STOC'96 Karger presented a randomized near linear time Monte Carlo algorithm
for the minimum cut problem. As he points out, there is no better way of
certifying the minimality of the returned cut than to use Gabow's slower
deterministic algorithm and compare sizes.
Our main technical contribution is a nearlinear time algorithm that contract
vertex sets of a simple input graph G with minimum degree d, producing a
multigraph with ~O(m/d) edges which preserves all minimum cuts of G with at
least 2 vertices on each side.
In our deterministic nearlinear time algorithm, we will decompose the problem
via lowconductance cuts found using PageRank á la Brin and Page (1998), as
analyzed by Andersson, Chung, and Lang at FOCS'06. Normally such algorithms for
lowconductance cuts are randomized Monte Carlo algorithms, because they rely on
guessing a good start vertex. However, in our case, we have so much structure
that no guessing is needed.

Wednesday October 7 at 13:15 in 4523, Lindstedtsvägen 5
Sizespace tradeoffs in proof complexity
(Susanna F. de Rezende, Theory Group, KTH)
The study of proof size, proof size, and sizespace tradeoffs has
recently been an active line of research in proof complexity. This
study was originally motivated by concerns in applied SAT solving, but
has also led to questions of intrinsic interest to proof complexity.
The resolution proof system underlying current stateoftheart SAT
solvers is now fairly wellunderstood in this regard, but for stronger
proof systems many open problems remain. In this talk, we will give an
overview of what is known and then present our current research aimed
at obtaining sizespace tradeoffs for the cutting planes proof
system.

Monday October 5 at 12:00 in 4523, Lindstedtsvägen 5
An averagecase depth hierarchy theorem for Boolean circuits +
Circuit complexity of small distance connectivity
(LiYang Tan, Toyota Technological Institute at Chicago)
In the first hour I will speak about recent work with Ben Rossman and Rocco
Servedio. We prove an averagecase depth hierarchy theorem for Boolean
circuits over the standard basis of AND, OR, and NOT gates. Our hierarchy
theorem says that for every
d ≥ 2, there is an explicit nvariable Boolean
function f, computed by a linearsize
depthd formula, which is such that
any depth(d1)
circuit that agrees with f on (1/2 + o_{n}(1)) fraction of
all inputs must have size
exp(n^{Ω(1/d)}). This answers an open
question posed by Håstad in his Ph.D. thesis, and has applications in
structural complexity and the analysis of Boolean functions. A key
ingredient in our proof is a notion of random projections which generalize
random restrictions.
After the break, I'd be happy to present the technical details and/or speak
about related subsequent work with Xi Chen, Igor Oliveira, and Rocco
Servedio on the stconnectivity problem:
We show that any depthd circuit for
determining whether an nnode graph
has an stot path of length at most k must have size
n^^{Ω(k^{1/d}/d)}.
The previous best circuit size lower bounds for this
problem were
n^{k^(exp(O(d))} [Beame, Impagliazzo, Pitassi 1998] and
n^{Ω((log k)/d)} [Rossman 2014].
Our lower bound is quite close to
optimal, since a simple construction gives depthd circuits of size
n^{O(k^(2/d))}
for this problem. Our proof is by reduction to a new lower
bound on the size of smalldepth circuits computing a skewed variant of the
"Sipser functions" that have played an important role in classical circuit
lower bounds of Sipser, Yao, and Håstad. A key ingredient in our proof is
the use of random projections, an extension of random restrictions which
allow us to obtain sharper quantitative bounds while employing simpler
arguments, both conceptually and technically, than in the previous works.

Monday September 14 at 12:00 in 4523, Lindstedtsvägen 5
Conflictdriven clause learning and pseudoBoolean SAT solving
(Jan Elffers, Theory Group, KTH)
Conflictdriven clause learning (CDCL) is the most popular method to
solve the Boolean satisfiability (SAT) problem in practice. This
approach is based on backtracking search and uses clause learning to
avoid solving the same subproblem multiple times. I will present the
core algorithm and a number of extensions and optimizations that are
incorporated in modern SAT solvers. I will also present possible
directions for future research aimed at improving the understanding of
this method.
The pseudoBoolean SAT problem is a generalization of SAT with linear
constraints instead of disjunctive clauses. This area is much less
well developed. One approach is to use an extension of CDCL with a
modified implementation of clause learning to handle linear
constraints. I will present this approach as well, and I will go
through an example execution of the method on the Pigeonhole
Principle. I will also outline some interesting research questions
regarding pseudoBoolean SAT solving.
Theory reading group meetings spring 2015

Monday June 22 at 12:00 in 4523, Lindstedtsvägen 5
Hardness of dynamic problems and the geometry of binary search trees
(Thatchaphol Saranurak, Theory Group, KTH)
This talk consists of two separated parts; both about
dynamic data structures.
The first part is about proving hardness of dynamic graph problems. In
dynamic graph problems, one wants to maintain some properties (e.g.
connectivity, distances between nodes, maximum matching) of a given graph
where edges of a graph can be inserted and deleted. While there are several
techniques for proving polylogarithmic lower bounds for the time required
for the update, these techniques currently do not give polynomial
(n^^{ε})
lower bounds. Then, one way to show hardness of the problems is
to assume some conjectures (e.g. SETH, 3SUM) and prove that an algorithm
with fast update time would contradict the conjecture. I will survey the
active development of these techniques for proving hardness based on
conjectures. Then, I will talk about our new conjecture called the Online
MatrixVector Multiplication, which very well tightly captures the hardness
of many dynamic problems. This is joint work with Monika Henzinger,
Sebastian Krinninger, and Danupon Nanongkai.
The second part is about the geometry of binary search trees. First, I will
talk about a Demaine et al. paper which shows how the "execution log" of a
binary search tree algorithm can be represented and characterized by a
point set in a plane with a simple property. This characterization suggests
a natural algorithm called Greedy. Next, I will talk about our work which
shows that, using forbiddenpattern theory, Greedy almost solves
30yearold Traversal Conjecture up to a function depending on alpha(n)
(inverseackerman function). This is based on a joint work with Parinya
Chalermsook, Mayank Goswami, Laszlo Kozma, and Kurt Mehlhorn.

Tuesday June 2 at 12:00 in 4423, Lindstedtsvägen 5
From graphs to matrices, and back: bridging the combinatorial and
the continuous
(Aleksander Mądry, MIT)
Over the last several years there was an emergence of new type of fast
algorithms for various fundamental graph problems. A key primitive
employed in these algorithms is electrical flow computation, which
corresponds to solving a Laplacian linear system.
In this talk, I will discuss how this approach enabled improvement over
longstanding bounds for the maximum flow problem. This progress will
also illustrate a broader emerging theme of employing optimization
methods as an algorithmic bridge between our combinatorial and
continuous understanding of graphs.
Additionally, I will briefly outline how this line of work brings a new
perspective on some of the core continuous optimization primitives —
most notably, interiorpoint methods.

Monday May 18 at 12:00 in 4523, Lindstedtsvägen 5
Symbol elimination for program analysis
(Laura Kovács, Chalmers University of Technology)
Automatic understanding of the intended meaning of computer
programs is a very hard problem, requiring intelligence and
reasoning. In this talk we describe applications of our symbol
elimination methods in automated program analysis.
Symbol elimination uses firstorder theorem proving techniques
in conjunction with symbolic computation methods, and derives
nontrivial program properties, such as loop invariants and
loop bounds, in a fully automatic way.
Moreover, symbol elimination can be used as an alternative
to interpolation for software verification.

Thursday May 7 at 12:00 in 4523, Lindstedtsvägen 5
The notion of structure in realworld SAT solving
(Jesú Giráldez Crú,
Artificial Intelligence Research Institute IIIACSIC, Barcelona)
Modern SAT solvers have experienced a remarkable progress on solving
industrial (or realworld) SAT instances. Most of the techniques have
been developed after an intensive experimental testing process. The
common wisdom in the SAT community is that the success of these
techniques is because they exploit some kind of hidden structure that
industrial formulas actually have. Recently, there have been some
attempts to analyze this structure in terms of complex networks, with
the longterm aim of explaining the success of these SAT solving
techniques, and possibly improving them.
In this talk, we analyze some structural properties of SAT instances,
viewed as graphs. Namely, the scalefree structure, the community
structure and the selfsimilar structure. In addition, we explore how
these features are affected during the SAT solving search by effects
of variable instantiation or clause learning. Finally, we present some
applications in SAT based on these notions of structure.

Monday May 4 at 12:00 in 4523, Lindstedtsvägen 5
The parity of set systems under random restrictions with applications
to exponential time problems
(Thore Husfeldt, Lund University and IT University of Copenhagen)
We reduce the problem of detecting the existence of an object to the
problem of computing the parity of the number of objects in
question. In particular, when given any nonempty set system, we prove
that randomly restricting elements of its ground set makes the size of
the restricted set system an odd number with significant
probability. When compared to previously known reductions of this
type, ours excel in their simplicity: For graph problems, restricting
elements of the ground set usually corresponds to simple deletion and
contraction operations, which can be encoded efficiently in most
problems. We find three applications of our reductions:

An exponentialtime algorithm: We show how to decide Hamiltonicity in
directed nvertex graphs with running time 1.9999^{n} provided that the
graph has at most 1.0385^{n} Hamiltonian cycles. We do so by reducing to
the algorithm of Björklund and Husfeldt (FOCS 2013) that computes the
parity of the number of Hamiltonian cycles in time 1.619^{n}.

A new result in the framework of Cygan et al. (CCC 2012) for analyzing
the complexity of NPhard problems under the Strong Exponential Time
Hypothesis: If the parity of the number of Set Covers can be
determined in time 1.9999^{n}, then Set Cover can be decided in the same
time.

A structural result in parameterized complexity: We define the
parameterized complexity class ⊕W[1] and prove that it is at least as
hard as W[1] under randomized fptreductions with bounded onesided
error; this is analogous to the classical result NP ⊆ RP ⊕ P by Toda
(SICOMP 1991).
This is joint work with Andreas Björklund and Holger Dell.

Wednesday April 8 at 12:00 in 4523
Space for random CNFs in proof complexity
(Ilario Bonacina, Sapienza – Università di Roma)
The aim of this talk is to give an overview of some recent results
about space lower bounds in proof complexity. In particular, for
random 3CNFs we will present a (reasonably complete) proof of an
optimal monomial space lower bound for Polynomial Calculus with
Resolution (PCR) and an optimal total space lower bound in Resolution.
We will see how to apply a fairly general combinatorial framework for
proving space lower bounds in Resolution and PCR and, more in detail,
the difficulties we had to overcame for the particular case of
random 3CNFs.
A crucial point is a result independent from proof complexity: a
variation of Hall's Theorem for matchings. We show that in bipartite
graphs G with bipartition (L,R) and leftdegree at most 3,
L can be
covered by certain families of disjoint paths, called VWmatchings,
provided that L expands in R by a factor of
2ε for ε > 1/23.
This talk is mainly based on a joint work with Patrick Bennett, Nicola
Galesi, Tony Huynh, Mike Molloy and Paul Wollan.

Monday March 23 at 12:00 in 4523, Lindstedtsvägen 5
An ultimate tradeoff in propositional proof complexity
(Alexander Razborov, University of Chicago)
We exhibit an unusually strong tradeoff between resolution proof
width and treelike proof size. Namely, we show that for any parameter k =
k(n) there are unsatisfiable kCNFs that possess refutations
of width O(k), but such that any treelike refutation of width
n^{1ε} / k
must necessarily have double exponential size exp(n^{Ω(k)}).
Conceptually, this means that there exist contradictions that allow narrow
refutations, but in order to keep the size of such a refutation even within
a single exponent, it must necessarily use a high degree of parallelism.
Viewed differently, every treelike narrow refutation is exponentially
worse not only than wide refutations of the same contradiction, but
of any other contradiction with the same number of variables. This
seems to significantly deviate from the established pattern of most, if
not all, tradeoff results in complexity theory.
Our construction and proof methods combine, in a nontrivial way,
two previously known techniques: the hardness escalation method
based on substitution formulas and expansion. This combination results
in a hardness compression approach that strives to preserve hardness
of a contradiction while significantly decreasing the number of its variables.

Thursday March 12 at 12:00 in 4523, Lindstedtsvägen 5
Limitations of algebraic approaches to graph isomorphism testing
(Christoph Berkholz, RWTH Aachen and Theory Group, KTH)
We investigate the power of algebraic techniques to test whether two
given graphs are isomorphic.
In the algebraic setting, one encodes the graphs into a system of
polynomial equations that is satisfiable if the graphs are isomorphic.
Afterwards, one tries to test satisfiability of this system using, for
example, the Gröbner basis algorithm.
In some cases this can be done in polynomial time, in particular, if the
equations admit a constant degree refutation in algebraic proof systems
such as Nullstellensatz or Polynomial Calculus.
We compare this approach to other known relaxations and show that the
kdimensional WeisfeilerLehman Algorithm can be characterized
in terms of algebraic proof system over the reals that lies between
degreek Nullstellensatz and degreek Polynomial Calculus.
Furthermore, we prove a linear lower bound on the Polynomial Calculus
degree of CaiFürerImmerman graphs, which holds over any field of
characteristic different from two.
This is joint work with Martin Grohe.

Monday January 26 at 12:00 in 4523, Lindstedtsvägen 5
Distributed graph algorithms and lower bounds
(Danupon Nanongkai, Theory Group, KTH)
This talk will focus on distributed approximation algorithms for
solving basic graph problems such as shortest paths, minimum spanning
trees, and minimum cut. I will cover:

Survey of the recent progress.

Open problems and some related conjectures.

A basic technique for proving lower bounds by a reduction from twoparty
communication complexity based on [1].
If time permits, I will touch some algorithmic techniques as well (e.g.
[2,3]). No background in distributed algorithms will be assumed in this
talk.
[1] Atish Das Sarma, Stephan Holzer, Liah Kor, Amos Korman, Danupon
Nanongkai, Gopal Pandurangan, David Peleg, Roger Wattenhofer: Distributed
Verification and Hardness of Distributed Approximation. SIAM J. Comput.
41(5): 12351265 (2012)
[2] Danupon Nanongkai: Distributed Approximation Algorithms for Weighted
Shortest Paths. STOC 2014: 565573
[3] Danupon Nanongkai, HsinHao Su: AlmostTight Distributed Minimum Cut
Algorithms. DISC 2014: 439453
Theory reading group meetings autumn 2014

Monday December 8 at 12:00 in 4523, Lindstedtsvägen 5
(2+ε)SAT is NPhard
(Per Austrin, Theory Group, KTH)
We prove the following hardness result for a natural promise variant of
the classical CNF satisfiability problem: given a CNF formula where each
clause has width w and the guarantee that there exists an assignment
satisfying at least g = w/2  1 literals in each clause, it is NPhard to
find a satisfying assignment to the formula (that sets at least one
literal to true in each clause). On the other hand, when g = w/2, it is
easy to find a satisfying assignment via simple generalizations of the
algorithms for 2SAT.
We also generalize this to prove strong NPhardness for discrepancy
problems with small size sets.
The talk will be elementary and (almost) selfcontained.
Joint work with Venkatesan Guruswami and Johan Håstad.

Monday November 3 at 12:00 in 4523, Lindstedtsvägen 5
Polynomial identity testing of readonce oblivious algebraic
branching programs
(Michael Forbes,
Simons Institute for the Theory of Computing at UC Berkeley)
Polynomial Identity Testing (PIT) is the problem of identifying
whether a given algebraic circuit computes the identically zero
polynomial. It is wellknown that this problem can be solved with a small
probability of error by testing whether the circuit evaluates to zero on
random evaluation points. Recently, there has been much interest in
solving this problem deterministically, because it has close connections
with circuit lower bounds, and this problem is now one of the frontiers of
the area of pseudorandomness.
In this talk we will discuss recent progress in this area, in particular
focusing on a model of algebraic computation known as the "readonce
oblivious algebraic branching program" which has been the focus of most
PIT research for the past several years. Developing PIT algorithms for
this class is a natural algebraic analogue of derandomizing smallspace
computation (the RL vs L question), and this class of computation
naturally has a linearalgebraic flavor. I will discuss deterministic
algorithms for determining if computations in this model compute the zero
polynomial, and will expose the linear algebraic nature of this question.
In particular, I will construct a natural pseudorandom object from linear
algebra called a "rank extractor" and show how it yields the desired PIT
algorithms.
Based on joint works with Amir Shpilka and Ramprasad Saptharishi.

Monday October 27 at 12:00 in 1537, Lindstedtsvägen 3
Easy generation and efficient validation of proofs for SAT and QBF
(Marijn Heule,
University of Texas at Austin)
Several proof systems have been proposed to verify results produced
by satisfiability (SAT) and quantified Boolean formula (QBF) solvers.
However, existing proof systems are not very suitable for validation
purposes: It is either hard to express the actions of solvers in
those systems or the resulting proofs are expensive to validate. We
present two new proof systems (one for SAT and one for QBF) which
facilitate validation of results in a time similar to proof discovery
time. Proofs for SAT solvers can be produced by making only minor
changes to existing conflictdriven clauselearning solvers and their
preprocessors. For QBF, we show that all preprocessing techniques can
be easily expressed using the rules of our proof system and that the
corresponding proofs can be validated efficiently.

Monday October 20 at 12:00 in 4523, Lindstedtsvägen 5
Lifting locally consistent partial solutions to a global solution
(Irit Dinur, Weizmann Institute of Science)
We are given a collection of (alleged) partial views of a function. We are
promised "local consistency", i.e., that the partial views agree on their
intersection with probability p.
The question is whether the partial views
can be lifted to a global function f, i.e. whether a p'
fraction of the partial views agree with f (aka "global consistency").
This scenario captures "low degree tests" and "direct product tests", both
studied for constructions of PCPs. We describe other possible settings
where a lifting theorem may hold.
We will relate this to questions on proving a "derandomized" parallel
repetition theorem, and the sliding scale conjecture.

Wednesday October 1 at 12:00 in 4523, Lindstedtsvägen 5
Parallel repetition from fortification
(Dana Moshkovitz, MIT)
The Parallel Repetition Theorem upperbounds the value of a repeated
(tensored) two prover game in terms of the value of the base game and
the number of repetitions. In this work we give a simple
transformation on games — "fortification" — and show that for
fortified games, the value of the repeated game decreases perfectly
exponentially with the number of repetitions, up to an arbitrarily
small additive error. Our proof is combinatorial and short.
As corollaries, we obtain:

Starting from a PCP Theorem with soundness error bounded away from 1,
we get a PCP with arbitrarily small constant soundness error. In
particular, starting with the combinatorial PCP of Dinur, we get a
combinatorial PCP with low error. The latter can be used for hardness
of approximation as in the work of Håstad.
 Starting from the work of the author and Raz, we get a projection
PCP theorem with the smallest soundness error known today. The theorem
yields nearly a quadratic improvement in the size compared to previous
work.
Theory reading group meetings spring 2014

Monday June 23 at 12:00 in 4523, Lindstedtsvägen 5
Indistinguishability obfuscation from semanticallysecure multilinear
encodings
(Rafael Pass, Cornell University and Cornell NYC Tech)
The goal of program obfuscation is to "scramble" a computer program,
hiding its implementation details while preserving functionality.
Unfortunately, the "dream" notion of security, guaranteeing that
obfuscated code does not reveal any information beyond blackbox access to
the original program, has run into strong impossibility results, and is
known to be unachievable for general programs (Barak et al, JACM 2012).
Recently, the first plausible candidate for generalpurpose obfuscation
was presented (Garg et al, FOCS 2013) for a relaxed notion of security,
referred to as indistinguishability obfuscation; this notion, which
requires that obfuscations of functionally equivalent programs are
indistinguishable, still suffices for many important applications of
program obfuscation.
We present a new hardness assumption—the existence of semantically
secure multilinear encodings—which generalizes a multilinear DDH
assumption and demonstrate the existence of indistinguishability
obfuscation for all polynomialsize circuits under this assumption (and
the LWE assumption). We rely on the beautiful candidate obfuscation
constructions of Garg et al (FOCS'13), Brakerski and Rothblum (TCC'14) and
Barak et al (EuroCrypt'14) that were proven secure only in idealized
generic multilinear encoding models, and develop new techniques for
demonstrating security in the standard model, based on semantic security
of multilinear encodings (which trivially holds in the generic multilinear
encoding model).
Joint work with Karn Seth and Sidharth Telang.

Monday June 16 at 12:00 in 4523, Lindstedtsvägen 5
Formulas vs. circuits for small distance connectivity
(Benjamin Rossman, National Institute of Informatics, Tokyo)
Are polysize boolean circuits strictly more powerful than polysize
boolean formulas? This question (also known as NC^1 vs. P) is one of
the central open problems in complexity theory. We can also consider
versions of this question for restricted classes of circuits. In the
monotone setting, a superpolynomial separation of formulas
vs. circuits was shown by Karchmer and Wigderson (1988). In recent
work, we give the first superpolynomial separation in the
(nonmonotone) boundeddepth setting.
Our main result is a lower bound showing that depthd formulas
solving the Distancek stConnectivity problem require size n^(log k)
for all k <= loglog n and d <= log n/(loglog n)^O(1). In contrast,
this problem has circuits of size n^Ω(1) and depth O(log k) by the
"recursive doubling" method of Savitch. As a corollary, it follows
that depthd circuits of size S cannot be simulated by depthd
formulas of size S^o(d) for all d <= logloglog S (previously this was
not known for any unbounded d > \infty).
The first part of the talk will be a gentle introduction to the
question of formulas vs. circuits and the Distancek stConnectivity
problem. After the break, I will give an overview of the new proof
technique.

Wednesday May 14 at 12:00 in 1537, Lindstedtsvägen 3 (joint with the Combinatorics Group)
Separating path systems
(Victor FalgasRavry, Umeå University)
Let
G
be a graph on
n
vertices. A family
F
of paths in
G
constitutes a separating
system of
G
if for ever pair of distinct edges
e,f
in
E(G)
there exists a path
p
in
F
which contains exactly one of
e
and
f.
How small a separating path system can we
find?
This question arises naturally in the context of network design. The graph
G
represents a communication network in which one link is defective; to identify this
link, we can send messages between nodes along predetermined paths. If a message
does not reach its destination, then we deduce that the defective link lies on the
corresponding path. A minimal separating path system thus allows us to determine
precisely which link is defective using a minimal number of messages.
We show that for asymptotically almost all
nvertex graphs, we can find a separating
system containing at most
48n
paths. In addition we prove some exact extremal
results in the case where
G
is a tree.
This is joint work with Teeradej Kittipassorn, Daniel Korandi, Shoham Letzter and
Bhargav Narayanan. Similar results have recently and independently been obtained by
Balogh, Csaba, Martin and Pluhar.

Monday May 12 at 12:00 in 4523, Lindstedtsvägen 5 (joint with the Combinatorics Group)
Models for wireless algorithms
(Magnús M. Halldórsson, Reykjavik University)
The design and analysis of algorithms requires appropriate models —
models that capture reality, yet are algorithmically usable; general,
yet analyzable. The wireless setting has proved most challenging in
this regard.
We survey some of the recent progress on fundamental problems
in the SINR (or physical) model, including link capacity and scheduling,
aggregation, and the relative value of power control.
The basic SINR model, however, still makes unrealistic assumptions
that hold only in idealistic situations. We outline how to allow for
arbitrary static environments while maintaining comparable performance
guarantees with what holds in the basic SINR model. We might
therefore be approaching an algorithmic model that captures reality
with high fidelity while maintaining generality and analytic
feasibility.

Thursday April 24 at 12:00 in 4523, Lindstedtsvägen 5
Lower bounds from the strong exponential time hypothesis
(Janne H. Korhonen, University of Helsinki)
Exact exponential algorithmics considers the development of faster, still
exponentialtime, algorithms for problems that are believed to lie outside
polynomial time. There have been notable successes in the recent years, such as the
Björklund's 1.657^n time Hamiltonian path algorithm; however, for the central
problem of CNFSAT, no such strictly exponential improvement has been obtained. That
is, while o(2^n) algorithms are known, there is no known algorithm with running time
c^n poly(n,m) for some c < 2.
Following the influential work of Impagliazzo, Paturi and Zane, connections between
the exponential complexity of CNFSAT and other problems have been investigated
extensively. The basic idea is that if we assume CNFSAT in fact cannot be solved in
time c^n poly(n,m) for any c < 2  this assumption is known as the strong
exponential time hypothesis  we can then derive conditional lower bounds for other
problems. These results can then be viewed as hardness results or new attacks on the
complexity of CNFSAT, depending on one's view on the strong exponential time
hypothesis.
In this talk, we will survey these connections between the strong exponential time
hypothesis and other problems. In particular, we will focus on the perhaps somewhat
unexpected conditional lower bounds for polynomialtime problems, and the basic
strategies used in the proofs of these results.

Tuesday April 22 at 12:00 in 4523, Lindstedtsvägen 5
BiLipschitz bijection between the Boolean cube and the Hamming ball
(Igor Shinkar, Weizmann Institute of Science)
We construct a biLipschitz bijection from the Boolean cube to the Hamming ball of
equal volume. More precisely, we show that for all even n there exists an explicit
bijection f from the ndimensional Boolean cube to the Hamming ball of equal volume
embedded in (n+1)dimensional Boolean cube, such that for all x and y it holds that
distance(x,y) / 5 <= distance(f(x),f(y)) <= 4 distance(x,y) , where distance(,)
denotes the Hamming distance. In particular, this implies that the Hamming ball is
biLipschitz transitive.
This result gives a strong negative answer to an open problem of Lovett and Viola
(CC 2012), who raised the question in the context of sampling distributions in
lowlevel complexity classes. The conceptual implication is that the problem of
proving lower bounds in the context of sampling distributions will require some new
ideas beyond the sensitivitybased structural results of Boppana (IPL 97).
We also study the mapping f further and show that it (and its inverse) are
computable in DLOGTIMEuniform TC0, but not in AC0. Moreover, we prove that f is
"approximately local" in the sense that all but the last output bit of f are
essentially determined by a single input bit.
Joint work with Itai Benjamini and Gil Cohen.

Monday April 14 at 12:00 in 4523, Lindstedtsvägen 5 (joint with the Combinatorics Group)
First order convergence of graphs
(Daniel Král', University of Warwick)
Nesetril and Ossona de Mendez introduced the notion of
the first order (FO) convergence of graphs. This notion
can be viewed as a unified notion of the existing notions
of convergence of dense and sparse graphs. In particular,
every FO convergent sequence of graphs is convergent
in the sense of left convergence of dense graphs, and
every FO convergent sequence of sparse graphs is
convergent in the BenjaminiSchramm sense.
During the talk, we will provide a brief introduction
to the theory of graph limits and its applications
in extremal combinatorics and theoretical computer
science. We will then present our results on the FO
convergence of trees, the FO convergence of matroids and
we will also discuss the relation of the limit object
to the convergent sequence. In particular, our results
yield answers to several problems raised by Nesetril and
Ossona de Mendez.
The presented results were obtained jointly
with Demetres Christofides, Frantisek Kardos,
Martin Kupec, Anita Liebenau, Lukas Mach and
Vojtech Tuma.

Monday April 7 at 12:00 in 3721, Lindstedtsvägen 25 (joint with the Combinatorics Group)
A relative Szemerédi theorem
(David Conlon, University of Oxford)
The celebrated GreenTao theorem states that there are arbitrarily
long arithmetic progressions in the primes. One of the main
ingredients in their proof is a relative Szemerédi theorem which says
that any subset of a pseudorandom set of integers of positive relative
density contains long arithmetic progressions.
In this talk, we will discuss a simple proof of a strengthening of the
relative Szemerédi theorem, showing that a much weaker
pseudorandomness condition is sufficient. Our strengthened version can
be applied to give the first relative Szemerédi theorem for kterm
arithmetic progressions in pseudorandom subsets of Z_N of density
N^{c_k}.
The key component in our proof is an extension of the regularity
method to sparse pseudorandom hypergraphs, which we believe to be
interesting in its own right. From this we derive a relative extension
of the hypergraph removal lemma. This is a strengthening of an earlier
theorem used by Tao in his proof that the Gaussian primes contain
arbitrarily shaped constellations and, by standard arguments, allows
us to deduce the relative Szemerédi theorem.
This is joint work with Jacob Fox and Yufei Zhao.

Wednesday April 2 at 11:15
in 4523, Lindstedtsvägen 5
(joint with the Combinatorics Group)
Induced matchings, arithmetic progressions and communication
(Benjamin Sudakov, ETH Zürich)
Extremal Combinatorics is one of the central branches of discrete
mathematics which deals with the problem of estimating the maximum
possible size of a combinatorial structure which satisfies certain
restrictions. Often, such problems have also applications to other
areas including Theoretical Computer Science, Additive Number Theory
and Information Theory. In this talk we will illustrate this fact by
several closely related examples focusing on a recent work with Alon
and Moitra.

Monday March 31 at 12:00 in 3721, Lindstedsvägen 25 (joint with the Combinatorics Group)
The graph regularity method
(Jacob Fox, MIT)
Szemerédi's regularity lemma is one of the most powerful tools in graph
theory, with many applications in combinatorics, number theory, discrete
geometry, and theoretical computer science. Roughly speaking, it says that
every large graph can be partitioned into a small number of parts such
that the bipartite subgraph between almost all pairs of parts is
randomlike. Several variants of the regularity lemma have since been
established with many further applications. In this talk, I will survey
recent progress in understanding the quantitative aspects of these lemmas
and their applications.

Monday March 24 at 12:00 in 4523, Lindstedsvägen 5
Improved inapproximability results for hypergraph coloring
(Prahladh Harsha, Tata Institute of Fundamental Research, Mumbai)
Despite the tremendous progress in understanding the approximability
of several problems, the status of approximate coloring of constant
colorable (hyper)graphs is not resolved and in fact, there is an
exponential (if not doubly exponential) gap between the best known
approximation algorithms and inapproximability results. The best known
approximation algorithms which are a combination of combinatorial and
semidefinite programming methods, require at least n^{δ} colors to
color a 2 colorable 4uniform hypergraph for some constant
δ ∈ (0,1).
On the contrary, till recently, the best known hardness results
could rule out at best coloring a 2colorable hypergraph with polylog
n colors (if not polyloglog n colors in some cases).
Recently, with the discovery of the lowdegree polynomial long code
(aka short code of Barak et al [FOCS 2012]), there has been a
superpolynomial (and in some cases, exponential) improvement in the
inapproximability results. In particular, we prove quasiNPhardness
of
the following problems on nvertex hypergraphs:

Coloring a 2colorable 8uniform hypergraph with 2^{2^{\sqrt{log log
n}} colors.

Coloring a 4colorable 4uniform hypergraph with 2^{2^{\sqrt{log log
n}} colors

Coloring a 3colorable 3uniform hypergraph with (log n)^{1/log log
log n} colors.
These results are obtained using the lowdegree polynomial code and
the techniques proposed by Dinur and Guruswami [FOCS 2013] to
incorporate this code for inapproximability results.
In this talk, I'll explain the bottleneck in obtaining improved
coloring inapproximability results using the long code and how
derandomizations of the long code (the short code in our setting) can
be used to improve the inapproximability factors.
Joint work with V. Guruswami, J. Håstad, S. Srinivasan, G. Varma.

Monday March 10 at 12:00 in 3721, Lindstedsvägen 25 (joint with the Combinatorics Group)
Random Cayley graphs
(Demetres Christofides, UCLan Cyprus)
In this talk we will define various models of random Cayley graphs. We
will then mention some results and partial results about random Cayley
graphs and compare them with the corresponding results for the
classical random graphs. We will also recall some of the proofs of the
results in the classical case and pinpoint at which places these
proofs fail to work in the Cayley case. Finally we will briefly
discuss how some of these results in the Cayley case can be obtained.
In the second part of the talk we will concentrate on a specific
result for classical random graphs and its corresponding extension for
random Cayley graphs: It is known that there is a threshold p(n)
such that G(n,p) has diameter greater than 2 if p is slightly
below the threshold and diameter equal to 2 if p is slightly above
the threshold. In this part of the talk we will discuss how this
result extends to random Cayley graphs.

Monday February 10 at 12:00 in 4523, Lindstedsvägen 5 (joint with the Combinatorics Group)
Smoothed analysis on connected graphs
(Michael Krivelevich, Tel Aviv University)
The main paradigm of smoothed analysis on graphs suggests that
for any large graph G in a certain class of graphs, perturbing
slightly the edges of G at random (usually adding few random edges
to G) typically results in a graph having much nicer properties.
In this talk we discuss smoothed analysis on trees, or
equivalently on connected graphs. A connected graph G on n vertices
can be a very bad expander, can have very large diameter, very high
mixing time, and possibly has no long paths. The situation changes
dramatically when \epsilon n random edges are added on top of G,
the so obtained graph G* has with high probability the following
properties:
 its edge expansion is at least c/log n;
 its diameter is O(log n);
 its vertex expansion is at least c/log n;
 it has a linearly long path;
 its mixing time is O(log^2n)
(the last three items assuming the base graph G has bounded
degrees).
All of the above estimates are asymptotically tight.
Joint work with Daniel Reichman (Weizmann Institute)
and Wojciech Samotij (Tel Aviv/Cambridge).
Theory reading group meetings autumn 2013

Monday December 16 at 12:00 in 4523
Approximate groups and almostperiodicity
(Olof Sisask, Department of Mathematics, KTH)
One can characterise the finite subgroups of an abelian group — or
rather cosets of subgroups — as those nonempty subsets A of the
group for which the sumset A+A = {a+b : a,b in A} has the same size as
A itself. What if one relaxes this condition to A+A < KA for some
fixed number K — must such sets look a bit like subgroups?
This kind of question forms the backdrop for a fundamental result of
additive combinatorics, known as Freiman's theorem, that gives a
partial classification of such 'approximate subgroups'. There has been
great progress in the past few years on obtaining stronger
descriptions of such sets, but there are still important unsolved
problems in this direction, the most famous probably being the
Polynomial FreimanRuzsa conjecture. A positive resolution of this
conjecture would provide very interesting and strong classifications
of some fundamental approximate algebraic objects, such as approximate
subgroups and approximate homomorphisms, and has been shown to have
nice applications in other areas — among them theoretical computer
science. The plan for this talk is to describe some of the background
of the Polynomial FreimanRuzsa conjecture and some of the underlying
theory, and possibly to mention some of the application areas in TCS.

Monday December 9 at 12:00 in 4523
From degree to space and back: A journey through algebraic proofs
(Mladen Mikša, Theory Group, KTH)
Have you ever run a SAT solver on your favourite problem just to
realize too late that going cheap on RAM was a bad idea and that the
solver won't be able to find a solution? Attending this talk might
help you understand better your SAT solver's needs.
In order to theoretically study a SAT solver, we model the reasoning
that goes on inside the solver by a corresponding proof system and map
different resources of the SAT solver to corresponding complexity
measures of the proof system. Thus, we map the running time of a SAT
solver to the size of the proof in the proof system, while the memory
consumption is mapped to space. In the talk, we focus on the space
measure in the algebraic proof system polynomial calculus, which is
still poorly understood compared to resolution (the basis of most
stateoftheart SAT solvers).
One of the most important results for our understanding of space in
resolution was given by Atserias and Dalmau in 2003, in which they
showed a relation between space and resolution width, another very
useful complexity measure. As we can generalize resolution width to
polynomial calculus degree, a natural question that arises is whether
a similar relation to the one given by Atserias and Dalmau holds for
polynomial calculus space and degree? In this talk we discuss some
partial results towards answering that question.
This talk is based on the paper Towards an Understanding of
Polynomial Calculus: New Separations and Lower Bounds, which is a
joint work with Yuval Filmus, Massimo Lauria, Jakob Nordström, and
Marc Vinyals, and which was presented at ICALP 2013.

Monday December 2 at 12:00 in 4523
Pebble games and complexity
(Siu Man Chan, Princeton University)
We will discuss space and parallel complexity, ranging from some
classical results which motivated the study, to some recent results
concerning combinatorial lower bounds in restricted settings. The
recurring theme is some pebble games. We will highlight some of their
connections to Boolean complexity and proof complexity.

Monday November 25 at 12:00 in 4523
Integrating cutting planes in a modern SAT solver
(Daniel Le Berre, Université d'Artois)
SAT solvers used in a daily basis in hardware or software verification
are based on the so called "conflict driven clause learning (CDCL)"
architecture. Such architecture is based on a proof system equivalent
to full resolution. Resolution in that context is used to derive new
clauses during conflict analysis. SAT solvers can easily be extended
to deal with cardinality constraints and PseudoBoolean constraints
while keep a resolutionbased proof system. A major focus has been
to study the translation of those constraints into CNF to reuse
without modifications the current SAT solvers. Another approach is to
extend the CDCL architecture to use cutting planes instead of
Resolution on cardinality or pseudo Boolean constraints. The
presentation will focus on the design of such a kind of solver, from the
specific properties of PseudoBoolean constraints to the design of a
cutting planes based conflict analysis procedure. Some experimental
results based on the implementation of such procedure available in
Sat4j will conclude the talk.

Wednesday November 13 at 12:00 in 1537
Turanproblem for hypergraphs
(Klas Markström, Umeå University)
The classical Turanproblem asks the following question. Given a graph
H, what is the maximum number of edges in a graph on n vertices which
does not contain a copy of H? For ordinary graphs a results of Erdös,
Stone and Simonovits gives an asymptotic solution to this
problem. However the asymptotics for bipartite graphs H and graphs H
which do not have constant size still present problems. The latter
connects to the well known triangle removal lemma.
A 3graph, or 3uniform hypergraph, consists of a vertex set and a set
of edges, which are vertex sets of size 3. Unlike the Turanproblem
for graphs, the Turanproblem for 3graphs is still far from
understood, for example we do not know the correct asymptotics for any
complete 3graph.
I will survey some methods and problems in this area, discussing how
lower and upper bounds have been proven.

Monday October 14 at 12:00 in 4523
Lower bounds (slightly) beyond resolution
(Marc Vinyals, Theory Group, KTH)
One of the goals of proof complexity is to show lower bounds for
stronger and stronger proof systems. For the wellknown resolution
proof systems exponential lower bounds are known for many formula
families, but for stronger proof systems many of these formulas
quickly become either provably easy or very hard to analyze.
In this talk, we will look at kDNF resolution, an extension of
resolution that works with kDNF formulas instead of clauses, which is
exponentially more powerful than resolution yet weak enough so that
one can prove interesting lower bounds and see how the hardness of
different formulas change. We will show exponential lower bounds for
the weak pigeonhole principle and for random kCNF formulas as well as
separations between kDNF and (k+1)DNF resolution for increasing
k. The main technical tool is a version of the switching lemma that
works for restrictions that set only a small fraction of the variables
and is applicable to DNF formulas with small terms.
This presentation is based on the paper
Segerlind, Buss, Impagliazzo:
A Switching Lemma for Small Restrictions and Lower Bounds for kDNF Resolution
(dx.doi.org/10.1137/S0097539703428555).

Monday September 30 at 12:00 in 4523
The complexity of proving that a graph is Ramsey
(Massimo Lauria, Theory Group, KTH)
We say that a graph with n vertices is cRamsey if it does not contain
either a clique or an independent set of size c*log(n). We define a
CNF formula which expresses this property for a graph G. We show a
superpolynomial lower bound on the length of resolution proofs that G
is cRamsey, for every graph G. Our proof makes use of the fact that
every Ramsey graph must contain a large subgraph with some of the
statistical properties of the random graph.
Joint work with Pavel Pudlák, Vojtech Rödl and Neil Thapen. The paper appeared at ICALP 2013.

Monday September 9 at 12:00 in 1537
Averagecase complexity of lattice problems
(Rajsekar Manokaran, Theory Group, KTH)
The averagecase complexity of a problem is the complexity of solving
instances of it picked from a specific distribution. In a seminal
work, Ajtai [STOC '96] showed a relation between the averagecase
complexity and the worstcase complexity of lattice problems. This
result is central to cryptography implemented using lattices.
Subsequently, Micciancio and Regev [FOCS '04] improved the parameters
involved in the proof while also simplifying the presentation.
In this talk, I will describe the result along the lines of the work
of Micciancio and Regev. Time permitting, I will describe the recent
improvements achieved by Micciancio and Peikert [CRYPTO '13].

Monday August 26 at 12:00 in 1537
Weak pigeonhole principles, circuits for approximate counting, and propositional proofs of bounded depth
(Albert Atserias, Universitat Politècnica de Catalunya)
The pigeonhole principle (PHP) states that m pigeons cannot sit
injectively into n holes if m is bigger than n. An often quoted result
of the research in propositional proof complexity is that the PHP with
m = n+1 does not have small proofs in proof systems that "lack the
ability to count". These include resolution [Haken] and, more
generally, proof systems that manipulate formulas with a bounded
number of alternations between disjunctions and conjunctions
(a.k.a. boundeddepth formulas) [Ajtai]. In contrast, for proof
systems that manipulate arbitrary propositional formulas, or even
boundeddepth formulas with "threshold gates", the PHP with m = n+1
admits small proofs [Buss]. For weaker PHPs where m is at least as
large as a constant factor larger than n, the situation is much less
understood. On one hand it looks clear that the ability to count
approximately should be enough to establish these weaker PHPs. On the
other, while boundeddepth circuits exist for approximate counting
[Stockmeyer, Ajtai], their mere existence is not enough to get
boundeddepth small proofs since one would also need elementary
(i.e. comparably complex) proofs of their correctness.
In this talk we will survey the status of this classical problem in
propositional proof complexity. Along the way we will discuss some new
recent related results showing that a close variant of the weak PHP
admits and requires quasipolynomialsize depth2 proofs. To our
knowledge, this is the first natural example that requires
superpolynomial but not exponential proofs in a natural proof
system. It also shows that the method of proof is in some sense "the
right method"; a remarkable and rather unexpected fact by itself.
Theory reading group meetings spring 2013

Monday May 13 at 12:00 in 1537
On fitting too many pigeons into too few holes
(Mladen Mikša, Theory Group, KTH)
If m pigeons want to nest in n separate pigeonholes, they will run
into serious problems if m > n. Although this might seem trivial, it
is one of the most extensively studied (and fascinating) combinatorial
principles in all of proof complexity.
In a breakthrough result, Haken (1985) showed that it is exponentially
hard for the resolution proof system to prove that n+1 pigeons don't
fit into n holes. However, what happens when the number of pigeons
increases? Since it becomes increasingly obvious that not every pigeon
can get a different hole, perhaps one can find shorter proofs as the
number of pigeons goes to infinity? This notorious open problem was
finally resolved by Raz (2001), who established that even for
infinitely many pigeons one still needs proofs of exponential
length. Raz's result was subsequently simplified and strengthened by
Razborov.
In the lunch seminar part of this talk, we will give an overview of
Razborov's proof. The presentation will be selfcontained and no
prerequisites in proof complexity are required. After the break, we
will go into technical details and prove the full result.

Monday April 22 at 12:00 in 4523
Approximability of some constraint satisfaction problems
(Sangxia Huang, Theory Group, KTH)
I will talk about approximability of Constraint Satisfaction Problems
(CSPs). In particular, we focus on CSPs of "sparse" Boolean
predicates. This is also related to other optimization problems, such
as finding maximum independent set. For CSP instances that are almost
satisfiable, Siu On Chan proved recently that there is a predicate on
k variables with k+1 accepting assignments that is NPhard to
approximate better than (k+1)/2^k. The case of approximation on
satisfiable instances is rather different. This is also an important
question, and I will describe many recent developments, including my
own work.
For the first part, I will give an overview of the state of the art
and some techniques — mainly reductions for showing hardness. In the
second part, I will go over some proofs in Chan's paper and prove part
of the main result.
The talk does not require prior experience in the PCP business.

Wednesday April 17 at 12:00 in 4523
Rediscovering the joys of pebbling
(Jakob Nordström, Theory Group, KTH)
In the early 1970s, combinatorial pebble games played on directed
acyclic graphs were introduced as a way of studying programming
languages and compiler construction. These games later found a broad
range of applications in computational complexity theory and were
extensively studied up to ca 1985. Then they were mercifully
forgotten, more or less…
Until during the last decade, when pebbling quite surprisingly turned
out to be very useful in proof complexity. In this talk, we will
describe the connections between the two settings and how tight they
are, present an improved reduction, and discuss the gap that remains.
In order to use this reduction to obtain interesting results in proof
complexity, one needs pebbling results with quite specific (and rather
strong) properties. We will also discuss a new such result, that broke
the 25year hiatus in the pebbling literature by appearing in CCC '10.
This seminar is intended to be completely selfcontained. In
particular, no prerequisites in proof complexity or pebbling are
required. After the break, in the technical sequel we intend to have
some fun and actually prove some pebbling timespace tradeoff
results.

Tuesday April 9 at 12:00 in 1537
On the complexity of finding widthk resolution refutations
(Christoph Berkholz, RWTH Aachen University)
One approach to attack NPhard satisfiability problems such as 3SAT or
the Constraint Satisfaction Problem (CSP) is to design algorithms that
run in polynomial time but do not always succeed. In this talk I
gently introduce the approach of searching for widthk resolution
refutations for 3SAT (also known as kconsistency test in the
CSPcommunity). This technique can be implemented in time n^O(k),
hence is in polynomial time for every fixed k.
One drawback of this approach is that the degree of the polynomial
increases with the parameter k. My main result is a lower bound
showing that this cannot be avoided: Deciding whether there is a
widthk resolution refutation requires time n^{ck} for an absolute
constant c>0. Furthermore, the problem is EXPTIMEcomplete (if k is
part of the input).

Monday March 25 at 12:00 in 4523
Faster Hamiltonicity
(Per Austrin, Theory Group, KTH)
I will talk about exact algorithms for Hamiltonicity and Travelling
Salesperson. The classic Bellman/HeldKarp dynamic programming
algorithm for these problems has a running time of O(n^2*2^n). This
remained unbeaten for almost half a century until 2010 when Björklund
gave an algorithm with running time O(2^{0.73n}) for undirected
Hamiltonicity. Since then there has been a flurry of generalizations
and simplifications, but many open questions remain.
My aim for the first part is to describe the state of the art and some
general techniques that are used, and for the second part to go over
one or two algorithms in detail.
The talk should be accessible without any specific background
knowledge and in particular you don't need to know what the
Hamiltonicity and Travelling Salesperson problems are beforehand.

Monday March 18 at 12:00 in 4523
Subgraphs satisfying MSO properties on ztopologically orderable digraphs
(Mateus de Oliveira Oliveira, Theory Group, KTH)
We introduce the notion of ztopological orderings for digraphs. We
prove that given a digraph G admitting a ztopological ordering, we
may count the number of subgraphs of G that satisfy a monadic second
order formula φ and that are the union of k directed paths in time
f(φ, k, z)·n^O(k·z) . Our result implies the polynomial time
solvability of a vast number of natural counting problems on digraphs
admitting z topological orderings for constant z. For instance, we
are able to answer in polynomial time questions of the form "How many
planar subgraphs of G are the union of k directed paths?" Concerning
the relationship between ztopological orderability and other digraph
measures, we observe that any digraph of directed pathwidth d has a
ztopological ordering for z ≤ 2d + 1. Since graphs of directed
pathwidth can have both arbitrarily large undirected tree width and
arbitrarily large clique width, our result provides for the first time
a suitable way of partially transposing metatheorems developed in the
context of the monadic second order logic of graphs of bounded
undirected tree width and bounded clique width to the realm of digraph
width measures that are closed under taking subgraphs and whose
constant levels incorporate families of graphs of arbitrarily large
tree width and arbitrarily large clique width.

Monday February 18 at 12:00 in 4523
Exponential lower bounds for the PPSZ kSAT Algorithm
(Dominik Scheder, Aarhus University)
In 1998, Paturi, Pudlak, Saks, and Zane presented PPSZ, an elegant
randomized algorithm for kSAT. Fourteen years on, this algorithm is still
the fastest known worstcase algorithm. They proved that its expected
running time on kCNF formulas with n variables is at most 2^((1 
epsilon_k)n), where epsilon_k = Omega(1/k). So far, no exponential lower
bounds at all have been known.
We construct hard instances for PPSZ. That is, we construct satisfiable
kCNF formulas over n variables on which the expected running time is at
least 2^((1  epsilon_k)n), for epsilon_k in O(log^2 (k) / k).
This is joint work with Shiteng Chen, Navid Talebanfard, and Bangsheng Tang.

Monday February 11 at 12:00 in 4523
Modular Verification of Temporal Safety Properties of Procedural Programs
(Dilian Gurov, Theory Group, KTH)
Modularity as a software design principle aims at controlling the
complexity of developing and maintaining large software. When applied to
verification, modularity means that the individual modules are specified
and verified independently of each other, while global, systemwide
properties are verified relative to the specifications of the modules
rather than to their implementatons. Such a relativization is the key to
verifying sofware in the presence of variability, that is, of modules the
implementation of which is expected to either evolve or be dynamically
upgraded, or is not even available at verification time, or exists in
multiple versions as resulting from a software product line.
In this tutorial I will present modular verification in the context of
temporal safety properties and a program model of recursive programs that
abstracts away from data. The proposed method is algorithmic and is based
on the construction of maximal program models that replace the local
specifications when verifying global properties.

Monday February 4 at 12:00 in 4523
Boolean influence
(Erik Aas, Department of Mathematics, KTH)
The influence of a variable of a Boolean function is a measure of how
likely that variable is to control the output of the function. I'll
present some fundamental results concerning the influences of threshold
functions (a special kind of Boolean function). If time permits we will
prove the KahnKalaiLinial theorem, giving a lower bound for the largest
influence of a variable of a "balanced" Boolean function.

Tuesday January 22 at 12:00 in 1537
Verifying a faulttolerant distributed aggregation protocol with the Coq proof assistant
(Karl Palmskog, Theory Group, KTH)
Decentralized aggregation of data from network nodes is an important way
of determining systemwide properties in distributed systems, e.g. in
sensor networks. A scalable way of performing such aggregation is by
maintaining a network overlay in the form of a tree, with data flowing
from the leaves towards the root.
We describe a variant of the treebased Generic Aggregation Protocol which
is well suited to secure aggregation, and argue formally that the protocol
has the expected behaviour for networks of arbitrary size, even in the
presence of crash failures.
Practical verification techniques for communication protocols such as
model checking usually require models with bounded state space. To reason
about the protocol without such undue abstraction, we encode it as a
transition system in the inductive logical framework of the Coq proof
assistant and perform machineassisted proofs.
Using Coq for verification requires knowledge of many techniques and
theories unrelated to the problem domain, and we give an overview of the
libraries, tools, and tradeoffs involved.
Theory reading group meetings autumn 2012

Monday December 17 at 12:00 in 4523
Majority is stablest: Discrete and SoS
(Sangxia Huang, Theory Group, KTH)
Consider a vote by n people on a Yes/No question and a voting scheme that
decides the outcome of the vote based on the n inputs. The Majority is
Stablest Theorem says that among all voting schemes where the number of
possible ways to vote so that the result in Yes and so that the result in
No are the same, and that no voter has large influence on the result,
Majority is the stablest scheme (result least likely to be affected by
flipping a small number of inputs).
The Majority is Stablest Theorem has numerous applications in hardness of
approximation and social choice theory. The proof by [Mossel, O'Donnell,
Oleszkiewicz 10] uses deep results in Gaussian analysis and an Invariance
Principle that allows to deduce the discrete result from the Gaussian one.
Recently, Anindya De, Elchanan Mossel and Joe Neeman provided a short
general proof of the Majority is Stablest Theorem that does not rely on
Gaussian analysis and the Invariance Principle. The new proof can also be
transformed into a "Sum of Squares" proof, thus showing that the
KhotVishnoi instance of MaxCut does not provide an integrality gap
instance for MaxCut in the Lasserre hierarchy.
In this seminar, we introduce basic notions of Boolean function analysis.
We present the ideas of both the "Invariance Principle"based proof and
the new proof. In the reading group that follows the seminar, we discuss
details of the new proof as well as (time permits) the Sum of Squares
proof of the Majority is Stablest Theorem.

Tuesday December 11 at 12:00 in 4523
Hypercontractivity, sumofsquares proofs, and their applications
(Rajsekar Manokaran, Theory Group, KTH)
In this talk, I will present the recent work of Barak, Brandao, Harrow,
Kelner, Steurer and Zhou on using the "Sum of Squares" semidefinite
programming hierarchy in solving the known hard instances of Unique Games
(UG) problem. The supposed hardness of this problem, known as the UG
Conjecture has been instrumental in proving a wide variety of tight
inapproximability results, hence warranting their study. [BBHKSZ '12]
show that the SoS SDP hierarchy can be used to refute the conjecture on
instances involving a noise operator on the boolean cube (which has been
the basis of hard instances to this problem till date).
I will first describe the "2 to 4 norm" problem, showing how it relates to
the UG conjecture, with a focus on the known family of hard instances.
Then, I will describe the SDP hierarchy followed by an overview of the
proof that the SDP does indeed refute the conjecture on this family of
instances. I will leave the details of the proofs to the session after
the seminar.

Monday November 26 at 12:00 in 4523
Monotone submodular maximization over a matroid using local search
(Yuval Filmus, University of Toronto)
Maximum coverage is the relative of set cover in which instead of
trying to cover the entire universe with as few sets as possible, we
are trying to cover as many elements as possible using a fixed number
of sets. The greedy algorithm gives a 11/e
approximation. Surprisingly, Feige proved that this ratio is optimal
unless P=NP.
Things get more complicated when we replace the cardinality constraint
on the sets with a matroid constraint (say, there are K types of sets,
and we should choose at most one of each type). The greedy algorithm
now gives only a 1/2 approximation. Calinescu et al. developed a
sophisticated algorithm that gives a 11/e approximation. Their
algorithm combines gradient ascent on a continuous relaxation with
optimal rounding, and also works in the more general setting of
monotone submodular maximization.
We show that nonoblivious local search also gives the optimal
approximation ratio. The auxiliary objective function, with respect to
which the local search proceeds, gives more weight to elements covered
multiple times in the current solution. We also extend our algorithm
to the setting of monotone submodular maximization.
Joint work with Justin Ward.

Wednesday November 14 at 12:00 in 4523
An information complexity approach to extended formulations
(Lukáš Poláček, Theory Group, KTH)
Lukáš will present the paper An Information Complexity Approach to
Extended Formulations by Mark Braverman and Ankur Moitra
(eccc.hpiweb.de/report/2012/131/).
The abstract of the paper is as
follows:
We prove an unconditional lower bound that any linear program that
achieves an O(n^{1?\eps}) approximation for clique has size
2^\Omega(n^\eps). There has been considerable recent interest in proving
unconditional lower bounds against any linear program. Fiorini et al
proved that there is no polynomial sized linear program for traveling
salesman. Braun et al proved that there is no polynomial sized
O(n^{1/2  \eps})approximate linear program for clique. Here we prove an
optimal and unconditional lower bound against linear programs for clique
that matches Håstad's celebrated hardness result. Interestingly, the
techniques used to prove such lower bounds have closely followed the
progression of techniques used in communication complexity. Here we
develop an information theoretic framework to approach these questions,
and we use it to prove our main result.
Also we resolve a related question: How many bits of communication are
needed to get \epsadvantage over random guessing for disjointness?
Kalyanasundaram and Schnitger proved that a protocol that gets constant
advantage requires \Omega(n) bits of communication. This result in
conjunction with amplification implies that any protocol that gets
\epsadvantage requires \Omega(\eps^2 n) bits of communication. Here we
improve this bound to \Omega(\eps n), which is optimal for any \eps > 0.

Wednesday November 6 at 13:15 in E53
Applying Boolean Groebner basis to cryptography and formal verification
(Alexander Dreyer, Fraunhofer Institute for Industrial Mathematics ITWM)
We will present and discuss some application examples for Boolean
Groebner basis from cryptography and formal verification.
Computing Groebner basis is of doubleexponential complexity in worst
case. But thanks to very sophisticated heuristics a plenty of practical
problem can be solved in reasonable time.
We will also talk about (still) open problems.

Tuesday November 6 at 12:00 in E53
Basics on Boolean Groebner basis and algebraic SAT solving
(Alexander Dreyer, Fraunhofer Institute for Industrial Mathematics ITWM)
We will give a short introduction into the topic of Groebner basis, in
particular we will scratch the surface of Boolean polynomials, i.e.
those polynomials with coefficients in {0,1} and a fixed degree bound
(per variable) of one.
This is accompanied with a brief presentation of the data structure
proposed for Boolean polynomials in our software framework PolyBoRi for
computing wit POlynomials in BOolean RIngs. These polynomials are the
algebraic representation of Boolean function. This enables us to use
techniques (like the Groebner basis formalism) from computational
algebra for solving Boolean problems. Their special properties allow for
representing them effectively as binary decision diagrams, but this
will need for new heuristics and algorithms.
Finally, the talk will connect Groebner basis with the classical SAT
solver approach for solving satisfiability problem of propositional logic.

Tuesday October 16 at 12:00 in 4523
Optimality of sizedegree tradeoffs for polynomial calculus
(Massimo Lauria, Theory Group, KTH)
Polynomial calculus is a way of refuting unsatisfiable CNF formulas by
translating them to polynomials and proving that these polynomials have no
common root. To show lower bounds on the size of such proofs, one usually
proves strong lower bounds on degree, and then uses a general sizedegree
tradeoff theorem saying that very high degree implies very high size.
There is an annoying gap in this theorem, however, in that fairly high
degree, up to sqrt(n), cannot be proven to say anything about size. A
natural question is whether this is inherent or whether the theorem could
be strengthened so that somewhat strong degree lower bounds would yield
somewhat strong size lower bounds.
We rule out this possibility by proving that the sizedegree tradeoff in
its current form is essentially optimal. We do so by studying formulas
encoding properties of linear orderings, which are known to have small
proofs, and showing that these formulas require sqrt(n) degree.
Joint work with Nicola Galesi.

Tuesday October 2 at 12:00 in 4523
On the virtue of succinct proofs: Amplifying communication complexity
hardness to timespace tradeoffs in proof complexity
(Jakob Nordström, Theory Group, KTH)
Motivated by the satisfiability problem and SAT solving, the last decade
has seen active research into timespace tradeoffs in proof systems for
propositional logic. So far most of the research has focused on the
resolution proof system operating with CNF formulas, whereas stronger
systems using algebraic and geometric methods of reasoning have remained
beyond reach.
In this work, we report the first tradeoff results for algebraic and
geometric proof systems. Our results are obtained by combining
combinatorial pebble games, extensively studied in the 1970s and 1980s,
with recently developed tools in communication complexity based on
information theory.
No prior knowledge of proof complexity or communication complexity will be
assumed.
This talk presents joint work with Trinh Huynh published in STOC '12.

Tuesday September 11 at 12:00 in 4523
Amazingly dense RuzsaSzemeredi graphs, plus applications
(Joshua Brody, Aarhus University)
An induced matching of a graph G=(V,E) is a matching M such that no two
edges of M are joined by an edge in G. A RuzsaSzemeredi graph is a graph
G, together with an edge coloring, such the set of edges of any particular
color forms an induced matching on G. Researchers have found surprising
applications of RuzsaSzemeredi graphs in a wide range of areas,
including communication over shared channels, linearity testing, and
communication complexity. For these applications, we generally want graphs
that (1) are as dense as possible, but (2) use as few colors as possible.
In this talk, I'll present a recent construction of Alon, Moitra, and
Sudakov from STOC 2012 which seems too good to be true. They construct a
graph which is nearly complete, but can be decomposed into a slightly
superlinear number of induced matchings. This is essentially the best one
can hope for. Time permitting, I'll also present an application or two.
This talk assumes only basic knowledge of graph theory. (i.e. I'll start
by describing what induced matchings are, and go from there)
This talk mostly covers work from the STOC 2012 paper Nearly Complete
Graphs Decomposable into Large Induced Matchings and their Applications by
Noga Alon, Ankur Moitra, and Benny Sudakov.
