The
MIAO video seminars
are what we call the mostly informal series of seminars arranged
by the
Mathematical Insights into Algorithms for Optimization (MIAO)
research group
at the University of Copenhagen and Lund University.
These are typically fairly low-key, informal events, but all public
seminars are announced beforehand on our seminar mailing lists (with
two different lists for more theoretically oriented or more applied
seminars, respectively). In case you want to receive the
announcements, please just let us know by sending an e-mail message
to jakob.nordstrom@cs.lth.se and we will add you to the relevant
list(s).
In response to multiple requests, and as an experiment, since the spring
of 2021 we have started posting most seminars on the
MIAO
Research YouTube channel.
Finite model theory studies the power of logics on the class of finite structures. One of its goals is to characterize symmetric computation, that is, computation that abstracts away details which are not essential for the given task, by respecting the symmetries of the input.
In this talk I will discuss connections between proof complexity and finite model theory, focussing on lower bounds. For certain proof systems, the existence of a succinct refutation can be decided in a symmetry-preserving way. This allows us to transfer lower bounds from finite model theory to proof complexity. I will introduce this approach and explain its key technical ideas such as the method of folding for dealing with symmetries in linear programs.
Friday Jun 4 at 14:00
Abstract cores in implicit hitting set based MaxSAT solving
(video)
(Jeremias Berg, University of Helsinki)
Maximum satisfiability (MaxSat) solving is an active area of research motivated by numerous successful applications to solving NP-hard combinatorial optimization problems. One of the most successful approaches for solving MaxSat instances from real world domains are the so called implicit hitting set (IHS) solvers. IHS solvers decouple MaxSat solving into separate core-extraction and optimization steps which are tackled by an Boolean satisfiability (SAT) and an integer linear programming (IP) solver, respectively. While the approach shows state-of-the-art performance on many industrial instances, it is known that there exists instances on which IHS solvers need to extract an exponential number of cores before terminating. In this talk I will present the simplest of these problematic instances
and talk about how abstract cores, a compact representation of a large number of regular cores that we recently proposed, addresses perhaps the main bottleneck of IHS solvers. I will show how to incorporate abstract core reasoning into the IHS algorithm and demonstrate that
that including abstract cores into a state-of-the- art IHS solver improves its performance enough to surpass the best performing solvers of the recent MaxSat Evaluations.
Friday May 28 at 14:00
Feasible interpolation for algebraic proof systems
(video)
(Tuomas Hakoniemi,
Universitat PolitÃ¨cnica de Catalunya)
In this talk we present a form of feasible interpolation for two algebraic proof systems, Polynomial Calculus and Sums-of-Squares. We show that for both systems there is a poly-time algorithm that given two sets P(x,z) and Q(y,z) of polynomial equalities, a refutation of the union of P(x,z) and Q(y,z) and an assignment a to the z-variables outputs either a refutation of P(x,a) or a refutation of Q(y,a). Our proofs are fairly logical in nature, in that they rely heavily on semantic characterizations of resource-bounded refutations to prove the existence of suitable refutations. These semantic existence proofs narrow down the search space for the refutations we are after so that we can actually find them efficiently.
Monday May 17 at 14:00
Average-case perfect matching lower bounds
from hardness of Tseitin formulas
(video)
(Kilian Risse, KTH Royal Institute of Technology)
We study the complexity of proving that a sparse random regular
graph on an odd number of vertices does not have a perfect matching,
and related problems involving each vertex being matched some
pre-specified number of times.
We show that this requires proofs of degree Ω(n / log n) in
the Polynomial Calculus (over fields of characteristic ≠ 2) and
Sum-of-Squares proof systems, and exponential size in the
bounded-depth Frege proof system.
This resolves a question by Razborov asking whether
the LovÃ¡sz-Schrijver proof system requires n^{δ}
rounds to refute these formulas for some δ > 0.
The results are obtained by a
worst-case to average-case reduction of these formulas relying on a
topological embedding theorem which may be of independent interest.
Joint work with with Per Austrin.
Monday May 10 at 14:00
On the complexity of branch and cut
(video)
(Noah Fleming, University of Toronto)
The Stabbing Planes proof system was introduced to model practical branch-and-cut integer programming solvers. As a proof system, Stabbing Planes can be viewed as a simple generalization of DPLL to reason about linear inequalities. It is powerful enough to simulate Cutting Planes and produce short refutations of the Tseitin formulas — certain unsatisfiable systems of linear equations mod 2 — which are canonical hard examples for many algebraic proof systems. In a surprising recent result, Dadush and Tiwari showed that these short Stabbing Planes refutations of the Tseitin formulas could be translated into Cutting Planes proofs. This raises the question of whether all Stabbing Planes can be efficiently translated into Cutting Planes. In recent work, we give a partial answer to this question.
In this talk I will introduce and motivate the Stabbing Planes proof system. I will then show how Stabbing Planes proofs with quasi-polynomially bounded coefficients can be quasi-polynomially translated into Cutting Planes. As a consequence of this translation, we can show that Cutting Planes has quasi-polynomial size refutations of any unsatisfiable system of linear equations over a finite field. A remarkable property of our translation for systems of linear equations over finite fields, and the translation of Dadush and Tiwari for the Tseitin formulas, is that the resulting proofs are also quasi-polynomially deep. A natural question is thus whether these depth bounds can be improved. In the last part of the talk, I will discuss progress towards answering this question in the form of a new depth lower bound technique for Cutting Planes, which works also for the stronger semantic Cutting Planes system, and which allows us to establish the first linear lower bounds on the depth of semantic Cutting Planes refutations of the Tseitin formulas.
Boolean satisfiability (SAT) solvers have reached stunning performance over the last decade. This performance can be harnessed for other problems by means of SAT encodings: translating the problem instance into a SAT instance. SAT encodings have been used successfully for many combinatorial problems and are established in industry. In this talk, I discuss the general idea behind SAT encodings and two specific application areas: graph decompositions and machine learning.
Graph decompositions allow for specialized algorithms that can solve hard problems efficiently. SAT encodings provide not only the means to optimally compute these decompositions, but can be easily extended by extra constraints specific to the hard problem to be solved.
For machine learning, SAT encodings gained increased attention in recent years, as they allow the induction of very small AI models. Learning small models has become more important in the context of explainable AI: smaller models are usually easier to understand for humans. We will focus specifically on decision tree induction.
Scalability is one of the main issues when using SAT encodings. We can overcome this problem by embedding SAT-based solving into a heuristic framework. This allows us to trade off a slightly worse result for large applicability. In the last part of our talk, we discuss our SAT-based local improvement framework that implements this idea.
Wednesday Apr 28 at 15:00
Slicing the hypercube is not easy
(video)
(Amir Yehudayoff, Technion – Israel Institute of Technology)
How many hyperplanes are needed to slice all edges of the hypercube? This question has been studied in machine learning, geometry and computational complexity since the 1970s. We shall describe (most of) an argument showing that more than n^{0.57} hyperplanes are needed, for large n. We shall also see a couple of applications. This is joint work with Gal Yehuda.
Monday Apr 26 at 14:00
Recent lower bounds in algebraic complexity theory
(video)
(Ben Lee Volk, University of Texas at Austin)
Algebraic complexity theory studies the complexity of solving algebraic computational tasks using algebraic models of computation. One major problem in this area is to prove lower bounds on the number of arithmetic operations required for computing explicit polynomials. This natural mathematical problem is the algebraic analog of the famous P vs. NP problem. It also has tight connections to other classical mathematical areas and to fundamental questions in complexity theory.
In this talk I will provide background and then present some recent progress on proving lower bounds for models of algebraic computation, such as the algebraic analogs of NC^{1} and NL.
Based on joint works with Prerona Chatterjee, Mrinal Kumar, and Adrian She.
Friday Apr 23 at 14:00
On the complexity of branching proofs
(video)
(Daniel Dadush, CWI)
We consider the task of proving integer infeasibility of a bounded
convex K in R^{n} using a general branching
proof system. In a general branching proof, one constructs a branching
tree by adding an integer disjunction ax ⩽ b or
ax ⩾ b+1, for an integer vector a and an
integer b, at each node, such that the leaves of the tree
correspond to empty sets (i.e., K together with the
inequalities picked up from the root to leaf is empty). Recently,
Beame et al (ITCS 2018), asked whether the bit size of the
coefficients in a branching proof, which they named stabbing planes
(SP) refutations, for the case of polytopes derived from SAT formulas,
can be assumed to be polynomial in n. We resolve this
question by showing that any branching proof can be recompiled so that
the integer disjunctions have coefficients of size at most
(nR)^{O(n2)}, where R is the radius of an
l_{1} ball containing K, while increasing
the number of nodes in the branching tree by at most a factor
O(n). As our second contribution, we show that Tseitin
formulas, an important class of infeasible SAT instances, have
quasi-polynomial sized cutting plane (CP) refutations, disproving the
conjecture that Tseitin formulas are (exponentially) hard for CP. As
our final contribution, we give a simple family of polytopes in
[0,1]^{n} requiring branching proofs of length
2^{n}/n.
Joint work with Samarth Tiwari.
Monday Apr 19 at 14:00
Kidney exchange programmes —
saving lives with optimisation algorithms
(video)
(William Pettersson, University of Glasgow)
Kidney Exchange Programmes (KEPs) increase the rate of living
donor kidney transplantation, in turn saving lives. In this
talk, I will explain exactly what KEPs are, how KEPs achieve
this goal, and the role of optimisation algorithms within
KEPs. This will include brief explanations of some of the
models used for kidney exchange programmes, some of the more
generic optimisation techniques that we use, as well as recent
advances and specific research directions for the field into
the future. This talk will very much be an overview of kidney
exchange, without going into complex details of optimisation
algorithms, making it suitable for a wider audience.
Thursday Apr 8 at 14:00
(Semi)Algebraic proofs over {Â±1} variables
(video)
(Dmitry Sokolov, St.Petersburg State University)
One of the major open problems in proof complexity is to prove
lower bounds on AC0[p]-Frege proof systems. As a step toward
this goal Impagliazzo, Mouli and Pitassi in a recent paper
suggested proving lower bounds on the size for Polynomial
Calculus over the {Â± 1} basis. In this talk we show a
technique for proving such lower bounds and moreover, we also
give lower bounds on the size for Sum-of-Squares over the
{Â± 1} basis.
We discuss the difference between {0, 1} and {+1, -1} cases and
problems in further generalizations of the lower bounds.
Courtesy of Zuse Institute Berlin:
Friday Feb 12 at 14:00
Introduction to IP presolving techniques in PaPILO
(Alexander Hoen, Zuse Institute Berlin)
Presolving is an essential part contributing to the performance of
modern MIP solvers. PaPILO, a new C++ library, provides presolving
routines for MIP and LP problems. This talk will give an introduction
to basic IP-presolving techniques and provide insights into important
design choices enabling PaPILO's capabilities, in particular
regarding its use of parallel hardware. While presolving itself is
designed to be fast, this can come at the cost of failing to find
important reductions due to working limits and heuristic filtering.
Yet, even most commercial solvers do not use multi-threading for the
preprocessing step as of today. PaPILO's design facilitates use of
parallel hardware to allow for more aggressive presolving and
presolving of huge problems. The architecture of PaPILO allows
presolvers generally to run in parallel without requiring expensive
copies of the problem and without special synchronization in the
presolvers themselves. Additionally, the use of Intel's TBB library
aids PaPILO to efficiently exploit recursive parallelism within
expensive presolving routines, such as probing, dominated columns, or
constraint sparsification. Despite PaPILO's use of parallelization,
its results are guaranteed to be deterministic independently of the
number of threads available.
Video seminars autumn 2020
Friday Oct 9 at 13:15
Model counting with probabilistic component caching
(Shubham Sharma and Kuldeep S. Meel,
National University of Singapore)
Given a Boolean formula F, the problem of model counting, also
referred to as #SAT, seeks to compute the number of solutions of F.
Model counting is a fundamental problem with a wide variety of
applications ranging from planning, quantified information flow to
probabilistic reasoning and the like. The modern #SAT solvers tend to
be either based on static decomposition, dynamic decomposition, or a
hybrid of the two. Despite dynamic decomposition based #SAT solvers
sharing much of their architecture with SAT solvers, the core design
and heuristics of dynamic decomposition-based #SAT solvers has
remained constant for over a decade. In this paper, we revisit the
architecture of the state-of-the-art dynamic decomposition-based #SAT
tool, sharpSAT, and demonstrate that by introducing a new notion of
probabilistic component caching and the usage of universal hashing
for exact model counting along with the development of several new
heuristics can lead to significant performance
improvement over state-of-the-art model-counters. In particular, we
develop GANAK, a new scalable probabilistic exact model counter that
outperforms state-of-the-art exact and approximate model counters for
a wide variety of instances.
Friday Sep 25 at 13:15
Manthan: A data-driven approach for Boolean function synthesis
(Priyanka Golia and Kuldeep S. Meel,
National University of Singapore)
Boolean functional synthesis is a fundamental problem in computer
science with wide-ranging applications and has witnessed a surge of
interest resulting in progressively improved techniques over the past
decade. Despite intense algorithmic development, a large number of
problems remain beyond the reach of the current state-of-the-art
techniques. Motivated by the progress in machine learning, we propose
Manthan, a novel data-driven approach to Boolean functional
synthesis. Manthan views functional synthesis as a classification
problem, relying on advances in constrained sampling for data
generation, and advances in automated reasoning for a novel
proof-guided refinement and provable verification. On an extensive
and rigorous evaluation over 609 benchmarks, we demonstrate that
Manthan significantly improves upon the current state of the art,
solving 356 benchmarks in comparison to 280, which is the most solved
by a state-of-the-art technique; thereby, we demonstrate an increase
of 76 benchmarks over the current state of the art. The significant
performance improvements, along with our detailed analysis, highlights
several interesting avenues of future work at the intersection of
machine learning, constrained sampling, and automated reasoning.
During the last decades, many improvements in CDCL SAT solvers have made possible to solve efficiently large problems containing millions of variables and clauses.
Despite this practical efficiency, some instances remain out of reach for such solvers.
This is particularly true when the input formula requires an exponential size refutation proof in the resolution proof system (as for pigeonhole formulae).
This observation motivated the development of another kind of solvers, known as pseudo-Boolean (PB) solvers.
These solvers take as inputs a conjunction of PB constraints (integral linear inequations over Boolean variables) and benefit from the cutting planes proof system, which is (in theory) stronger than the resolution proof system.
To implement this proof system, current PB solvers follow the direction of modern SAT solvers, by implementing a conflict analysis procedure relying on the application of cutting planes rules.
However, adapting the CDCL architecture to take into account PB constraints is not as straightforward as it may look.
In particular, many CDCL invariants and properties do not hold anymore as long as PB constraints are considered.
While some of them may be safely ignored (e.g., when they affect the decision heuristic, the deletion strategy or the restart policy), some others must be fixed to ensure the soundness of the solver (e.g., by ensuring to preserve the conflict during its analysis).
In this talk, we will give an overview of the main differences between PB solving and classical SAT solving, and present different approaches that have been designed to take these differences into account to extend the CDCL architecture to PB problems.
We will in particular discuss the pros and cons of these approaches, and we will explain how they can be enhanced to improve the practical performance of PB solvers.
Friday Aug 21 at 10:15
Using proofs to analyze SAT solvers
(Janne Kokkala,
Lund University and University of Copenhagen)
The main idea of this seminar is to give an overview of what
one can ask and what one can learn about SAT solvers when
analyzing the proof to estimate what part of the work was
useful. I will start by giving a 17-ish minute alpha test run
for my presentation for our CP paper [1]. After that, I will
discuss earlier works that use the same or a similar idea for
general insights [2], analyzing VSIDS usefulness [3], and
clause exchange for parallel solvers [4,5]. To conclude, we
can discuss how this approach could be used for pseudo-Boolean
solvers.
Wednesday Aug 19 at 13:15
On computational aspects of the antibandwidth problem
(Markus Sinnl,
Johannes Kepler University Linz)
In this talk, we consider the antibandwidth problem, also
known as dual bandwidth problem, separation problem and
maximum differential coloring problem. Given a labeled graph
(i.e., a numbering of the vertices of a graph), the
antibandwidth of a node is defined as the minimum absolute
difference of its labeling to the labeling of all its adjacent
vertices. The goal in the antibandwidth problem is to find a
labeling maximizing the antibandwidth. The problem is NP-hard
in general graphs and has applications in diverse areas like
scheduling, radio frequency assignment, obnoxious facility
location and map-coloring.
There has been much work on deriving theoretical bounds for
the problem and also in the design of metaheuristics in recent
years. However, the optimality gaps between the best known
solution values and reported upper bounds for the
HarwellBoeing Matrix-instances, which are the commonly used
benchmark instances for this problem, were often very large
(e.g., up to 577%).
We present new mixed-integer programming approaches for the
problem, including one approach, which does not directly
formulate the problem as optimization problem, but as a series
of feasibility problems. We also discuss how these feasibility
problems can be encoded with various SAT-encodings, including
a new and specialised encoding which exploits a certain
staircase-structure occuring in the problem formulation. We
present computational results for all the algorithms,
including a comparison of the MIP and SAT-approaches. Our
developed approaches allow to find the proven optimal solution
for eight instances from literature, where the optimal
solution was unknown and also provide reduced gaps for eleven
additional instances, including improved solution values for
seven instances, the largest optimality gap is now
46%. Instances based on the problem were submitted to the SAT
Competition 2020.
Video seminars spring 2020
Wednesday Jul 15 at 15:00
Naïve algorithm selection for SAT solving
(Stephan Gocht,
Lund University and University of Copenhagen)
Although solving propositional formulas is an NP-complete
problem, state-of-the-art SAT solvers are able to solve formulas
with millions of variables. To obtain good performance it is
necessary to configure parameters for heuristic
decisions. However, there is no single parameter configuration
that is perfect for all formulas, and choosing the parameters is
a difficult task. The standard approach is to evaluate different
configurations on some formulas and to choose the single
configuration, that performs best overall. This configuration,
which is called single best solver, is then used to solve new
unseen formulas. In this paper we demonstrate how random forests
can be used to choose a configuration dynamically based on
simple features of the formula. The evaluation shows that our
approach is able to outperform the single best solver on
formulas that are similar to the training set, but not on
formulas from completely new domains.
Friday Jun 26 at 10:00
Behind the scenes of chronological CDCL
(Sibylle Möhle and Armin Biere,
Johannes Kepler University Linz)
Combining conflict-driven clause learning (CDCL) with
chronological backtracking is challenging: Multiple invariants
considered crucial in modern SAT solvers are violated, if
after conflict analysis the solver does not jump to the
assertion level but to a higher decision level instead. In
their SAT'18 paper "Chronological Backtracking", Alexander
Nadel and Vadim Ryvchim provide a fix to this issue. Moreover,
their SAT solver implementing chronological backtracking won
the main track of the SAT Competition 2018. In our SAT'19
paper, "Backing Backtracking", we present a formalization and
generalization of this method. We prove its correctness and
provide an independent implementation.
In this seminar, we demonstrate the working of chronological
CDCL by means of an example. In this example, after a conflict
the conflicting clause contains only one literal at conflict
level. It is therefore used as a reason for backtracking, thus
saving the effort of conflict analysis. We further show which
invariants are violated and present new ones followed by a
discussion of the rules of our formal framework. We also shed
light onto implementation details, including those which are
not mentioned in our SAT'19 paper.
Monday Jun 22 at 13:30
McSplit: A partitioning algorithm for maximum common subgraph
problems
(Ciaran McCreesh and James Trimble,
University of Glasgow)
We will give a short introduction to McSplit, an algorithm for
the maximum common (connected) subgraph problem coauthored
with Patrick Prosser and presented at IJCAI 2017. McSplit
resembles a forward-checking constraint programming algorithm,
but uses a partitioning data structure to store domains which
greatly reduces memory use and time per search node. We will
also present our recent work with Stephan Gocht and Jakob
Nordström on adding proof logging to the algorithm, turning
McSplit into a certifying algorithm whose outputs can be
independently verified.
Thursday Jun 18 at 20:30
A pseudo-Boolean approach to nonlinear verification
(Vincent Liew,
University of Washington)
We discuss some new experimental results showing the promise
of using pseudo-Boolean solvers, rather than SAT solvers, to
verify bit-vector problems containing multiplication. We use
this approach to efficiently verify the commutativity of a
multiplier output bit by output bit. We also give some examples
of simple bit-vector inequalities where the pseudo-Boolean
approach significantly outperformed SAT solvers and even
bit-vector solvers. Finally, we give some of our observations on
the strengths and weaknesses of different methods of conflict
analysis used by pseudo-Boolean solvers.
Friday May 8 at 13:15
Pseudo-Boolean solvers for answer set programming
(Bart Bogaerts, Vrije Universiteit Brussel)
Answer set programming (ASP) is a well-established knowledge
representation formalism. Most ASP solvers are based on
(extensions of) technology from Boolean satisfiability
solving. While these solvers have shown to be very successful in
many practical applications, their strength is limited by their
underlying proof system, resolution. In this research, we
present a new tool LP2PB that translates ASP programs into
pseudo-Boolean theories, for which solvers based on the
(stronger) cutting plane proof system exist. We evaluate our
tool, and the potential of cutting-plane-based solving for ASP
on traditional ASP benchmarks as well as benchmarks from
pseudo-Boolean solving. Our results are mixed: overall,
traditional ASP solvers still outperform our translational
approach, but several benchmark families are identified where
the balance shifts the other way, thereby suggesting that
further investigation into a stronger proof system for ASP is
valuable.