DD2445/FDD3445 Complexity Theory Autumn 2015
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This webpage provides all documentation and information about the course, so there is no separate course memo ("kurs-PM") PDF file.
Computers are everywhere today—at work, in our cars, in our living rooms, and in our pockets—and have changed the world beyond our wildest imagination. Yet these marvellous devices are, at the core, amazingly simple and stupid: all they can do is to mechanically shuffle zeros and ones around. What is the true potential of such automated computational devices? And what are the limits of what can be done by mechanical calculations?
Complexity theory gives these deep and fascinating philosophical questions a crisp mathematical meaning. A computational problem is any task that is in principle amenable to being solved by a computer—i.e., it can be solved by mechanical application of mathematical steps. Complexity theory focuses on classifying computational problems according to their inherent difficulty, and on relating those classes of problems to each other. The goal is to understand the power of computers but also—and above all—the limitations of what problems can be solved by them, or more broadly by any type of automated computational process. A problem is regarded as inherently difficult if its solution requires unreasonably large resources regardless of which approach is used to solve it (i.e., no matter which algorithm is employed). Complexity theory formalizes this notion by introducing mathematical models of computation and quantifying the amount of resources needed to solve the problems, such as running time, memory usage, parallelism, communication, et cetera.
This course will give an introduction to computational complexity theory, survey some of the major research results, and present some of the open problems that are the focus of current research. While the intention is to give a fairly broad coverage, there will probably be a slight bias towards areas where the Theory Group at KTH has made significant contributions to the state of the art.
This course was given in periods 1-2 in the autumn of 2015. We had a total of 23 lectures, with 2 lectures per week on average. In accordance with the academic quarter tradition at KTH, 10 am in the schedule actually means 10:15 am et cetera. See the list of rooms at KTH to locate the different lecture rooms, which are mostly the seminar rooms on the 5th floor at Lindstedtsvägen 3/5. Chapter numbers in the course planning below refer to the Arora-Barak textbook.
The general idea behind the course was to first go over (most of) the first 9 chapters in the textbook, getting a fairly good general overview of computational complexity theory, and then spend some time on a selection of more "advanced" topics, where the textbook was followed less closely or not at all.
The main lecturer on the course was Jakob Nordström, who was responsible for all aspects of the course.
Ilario Bonacina was co-instructor and was among other things taking care of grading of the problem sets.
We used Piazza for teacher-student interaction on the course. Thus, questions should not be e-mailed to the instructor, but instead posted on Piazza (where you can send private notes to the instructor if needed, and also ask questions anonymously).
All course participants need to be signed up at Piazza to receive announcements related to the course (but note that this does not replace the official course registration at KTH).
Formal RequirementsThe course is open to anyone, but the main target audience are Master's students in computer science. The course is also suitable for PhD students in mathematics or computer science who have not previously taken a dedicated course on computational complexity theory. As to formal requirements, you need to have taken DD1352 Algorithms, Data Structures, and Complexity or DD2352 Algorithms and Complexity, or corresponding courses at other universities, and should feel comfortable with that material. There are no additional formal prerequisites on top of what is stated in the Study Handbook, but you will need mathematical maturity and a willingness to learn new stuff.
All lectures are given in English.
Although the formal prerequisites are very limited, it should be noted that this is a somewhat demanding course. (But hopefully even more fun!)
Before the Course StartsThere is no need to register beforehand in order to start attending the course—you can just show up at the first lecture (or send an e-mail to the instructor if you have questions).
RegistrationYou will need to formally register for the course in order for the instructor to be able to report your results. Students should register as soon as possible using the course web (earlier called my pages). Different categories of students take this course and might face different administrative problems. Please make sure that you get officially registered for the course as appropriate for your study program.
After having completed the course, the student should be able to:
Problem Sets and Peer EvaluationTo pass the course, students will be required to solve and hand in solutions to four problem sets. The scores on the problem sets are what will mainly determine the final grades on the course (except as explained below).
For each each problem set, each student will also have to evaluate the solutions of a fellow student (but not assign grades), who will be randomly chosen by the instructor. This part of the examination will only be pass/fail as discussed in the description of the grading process below.
The KTH CSC code of honour applies to all aspects of this course including the problem sets. There are also some additional rules specific to (the problem sets of) this course. These rules are available below on this webpage and are stated on each problem set.
The grades for Master's students taking DD2445 are determined according to the following principles:
Note that solutions to the problem sets should be handed in strictly by the deadlines. Being able to work towards a deadline and deliver the best possible result within a given time frame (rather than a 100% polished product that arrives too late) is an important skill, and is something that you will have the opportunity to practise during the course. Having said that, exceptional circumstances, such as severe illness, can be accepted as an excuse for late problem set solutions. It should be emphasized, however, that lack of time due to work outside the university or due to many parallel courses is not considered as a legitimate reason for handing in problem set solutions late.
If you are a PhD student, you can take this course as a research-level doctoral course with a course code FDD3445. The requirements are slightly tougher and the grading is only pass/fail, but the course counts fully towards the course credit requirements in the PhD program.
For PhD students taking the research-level course, in addition to fulfilling the above requirements the average grade on the problem sets should be at least C and the oral presentation of a research article is mandatory. PhD students also have the option to take the Master's level course.
As far as we are aware, there are no students from previous years having unfinished parts of this course, and we do not expect this to be a problem this year either. Students who are motivated and strong enough to take this course also tend to finish it. Since there is no exam on the course but only problem sets and peer evaluation, any students who do not complete the course requirements in time will have to be dealt with on a case-by-case basis. Any bonus points collected during the course (as explained below) will be voided once the course has ended.
We will mostly be following the booknotes by Johan Håstad used in previous offerings of the course that covers some of the basic material and that might be useful side reading.
While Arora-Barak is the recommended textbook, we comment briefly on some other alternatives below. Another recent textbook on computational complexity theory is
During the second half of the course, some lectures will be partly based on research articles. Below follows a list of links to these articles. The intention is that the lectures will cover the material in the papers that we need, so students are not required to read these papers—the references are provided for completeness and for students interested in further reading. However, for students interested in learning more, it should be noted that many of the proofs given in class are actually not those found in the papers, and more recent survey papers of an area are likely to be better reads than the original research articles. Please do not hesitate to contact the instructor if you are interested in specific references for some specific area.
Note that if you are not at KTH, or if you are connected to the KTH
network via wireless, then you might not be able to access the PDF
files with the articles linked to below.
One way around this problem is to search for the titles of the papers
in your favourite search engine—this should hopefully help you find
free versions of the same papers on the webpages of the
authors or similar.
Another, often better, solution to this problem
is to invoke the KTH library proxy server directly in the address field of
the browser. You do this by adding
Solutions to the
problems sets should be submitted as PDF files by e-mail to
When you are working on the problem sets, discussions of ideas in groups of two are allowed, but you should always write down your own solution individually and understand all aspects of it fully. You should also acknowledge any collaboration. For each problem set, state at the beginning if you have been collaborating and with whom.
You can (and should) ask the instructor if anything about the problem sets is unclear. Make sure to post private messages to the instructor on Piazza in that case, so that your questions do not accidentally give away unintended information about the problems to the other students. If there is some issue needing clarification regarding some problem, the instructor will make a public post on Piazza.
Some of the problems are "classic" and hence it might be easy to find solutions on the Internet, in textbooks or in research papers. It is not allowed to use such material in any way unless explicitly stated otherwise. You can, however, use in your solutions anything said during the lectures on in the lecture notes, unless you are specifically asked to show something that we claimed without proof in class. It is hard to pin down 100% formal rules on what all this means—when in doubt, ask the instructors.
The grading process will involve some peer evaluation (and hopefully tons of interaction among the students). All final grading will be done by the instructor, however. Here is how it is intended to work.
Step 1: Work on the Problem Set
Students solve the problem set, on their own or collaborating in pairs, and write down their own solutions and submit as a PDF file by e-mail before the deadline. During this phase no discussion of problem set solutions is allowed other than with the collaborating partner (but sending a private message on Piazza to the instructor asking for clarifications is of course OK).
Step 2: Discussion of Solutions
After the deadline, the instructor distributes the problem set solutions randomly to the students as PDF files by e-mail. All students will have a day or two to go over the received problem set solutions, compare with their own solutions, and try to figure out what might be good or bad approaches to solving the various problems on the problem set.
When a day or two has passed, the instructor gives the start signal for discussions of solutions to the problems on Piazza. During this phase all students on the course should work together to find solutions (possibly many different ones) to all problems on the problem set.
During the discussions a maximum of collaboration is allowed and encouraged. There is no incentive not to collaborate here since nothing that happens after this point can lower the grade of any student. Instead, there will be bonus points for writing down correct solutions on Piazza, as well as for helping to improve on not fully correct or complete solutions. Note that a student need not have solved the problem him- or herself in order to contribute a solution on Piazza. It is sufficient to have understood and be able to present a solution. (However, just quoting from the received set of solutions verbatim is not encouraged—the bonus points are meant to award understanding, not copying skills.)
The instructor will not take part too actively in the exchange of comments, but will try to nudge the discussions in the right direction if need be.
Step 3: Peer Evaluation
Simultaneously, and using material learned during the discussion phase, each student should evaluate all the solutions in the set of received problem set solutions and write down comments on a print-out. Even if the student solved the problem set in cooperation with another student, the peer evaluation should be performed individually, not cooperating in pairs.
Here is how the solutions should be evaluated:
The solutions together with the evaluation comments should be should be put in the main lecturer's mailbox on the 4th floor at Osquars backe 2 or handed to the lecturer before beginning of class on the day of the peer evaluation deadline.
Step 4: Final Grading
Finally, the instructors will grade all problem set solutions and assign scores, and will at the same time evaluate the evaluations. Each student will receive both the instructor grading comments and the peer evaluation copy.
The problem set will be regarded as a pass if it reaches the threshold for E as specified in the problem set.
The problem set evaluation will be regarded as a pass if at least 50% of the solutions marked as "correct" or "incorrect" are properly identified as such (with relevant explanations), and if no solution is marked "unknown" unless there is a convincing explanation as to why this solution was not possible to understand. In particular, the instructors may look at the discussions at Piazza to check if the student has made a good faith effort to get help to figure out what is going on in the solution.
Motivation for this Set-up
There are a number of reasons why this approach will be used. It is intended to:
This new approach will be thoroughly evaluated during (and certainly after) the course, and might be modified based on the conclusions from such evaluations. One of the reasons we are doing this is that a similar approach was used for the courses DD2446 Complexity Theory in 2013 and DD2442 Seminars on Theoretical Computer Science in 2014 and received overall very positive reviews. If you have any views or comments already now regarding this, please feel free to contact the instructor on Piazza or by e-mail.