CS266 -- Parameterized Algorithms and Complexity -- Spring 2013

Instructors: Ryan Williams and special guest lecturer Virginia Vassilevska Williams
Time: Tuesdays and Thursdays 2:15--3:30, McCullough Building, Room 122
Office Hours: By appointment/email

Description:

Many problems we want to solve are often NP-hard or worse, but somehow, they need to get solved anyway. What can we do? Over the years, multiple paradigms for coping with NP-hardness have been introduced: for instance, approximation algorithms, average-case analysis, and randomized algorithms were all borne out of a desire to solve hard problems by relaxing the problem or strengthening the model.

Within the last 20 years, a new paradigm has been introduced, where one measures the time complexity of an algorithm not just in terms of the input length but also a small side parameter. A priori, the parameter can be anything, but the interesting case is when complex instances of the problem still have relatively small parameter values. The overall goal is to identify interesting parameterizations of hard problems where we can design algorithms running in time polynomial in the input length but possibly exponential (or worse) in the small parameter. Such an algorithm is called "fixed-parameter tractable" and it is the gold standard for parameterized algorithms.

This is a subject which is still very new and fresh. Although many papers on the subject appear in the top theory conferences each year, there are few systematic courses within the United States. We will attempt to correct this. Parameterization has become an important topic that has bearing on other strands of theory as well (including circuit complexity). The course will be heavily research-oriented with lots of open problems, and it will introduce you to new ways of thinking about existing theory.

• Two (big) problem sets, about three weeks apart.
• In-class presentation on a topic of interest related to the course, or on your own research! These will be around 40 minutes.
• Scribe at least one lecture: you will need to take notes, write them up in LaTeX, and submit them to us within a week of the lecture

Prerequisites: This is technically a graduate course, but it is open to anyone. You should probably have a thorough and general understanding of algorithms and complexity theory -- otherwise, the course may be tough to follow in some places. We will try to recall the concepts needed along the way. There is no textbook for the course, but we will catalogue some reading material found on the web as we go.

Announcements:

• 4/23 The papers for your presentations are below

• 4/16 Here is the template to base your scribe notes on.

• 4/15 Scribe notes for lecture 1 are up. The other lectures will be posted soon

• 4/1 Look here for cool stuff to happen!

WARNING, INTERNET TRAVELERS: THE FOLLOWING SCRIBE NOTES HAVE ONLY BEEN LIGHTLY EDITED AND MAY CONTAIN ERRORS. YOU HAVE BEEN WARNED.

What we've done so far

• 4/2    Overview of course, some basic properties of FPT, kernels
[pdf] Scribe: Lucas
• 4/4    Vertex Cover: backtracking FPT algorithms, kernelization by graph theory and LP
[pdf] Scribe: Valeria
• 4/9    Hamiltonian Path and TSP: dynamic programming and inclusion-exclusion
[pdf] Scribe: Huacheng
• 4/11    $k$-Path: FPT algorithms, color-coding, random partitioning
[pdf] Scribe: Yongxing
• 4/16    $k$-Path: $O^{*}(2^k)$ time by vector coding and linear algebra mod 2
[pdf] Scribe: Robin
• 4/18    Algorithms for $k$-Clique, $k$-SUM, $k$-Dominating Set
[pdf] Scribe: Amir
• 4/23    SAT Algorithms: randomized reduction, branching algorithms, intro to local search
[pdf] Scribe: Chuanqi
• 4/25    Schoening's local search algorithm, intro to Parameterized Complexity
[pdf] Scribe: Hart
• 4/30    $W[1]$ and completeness: $k$-Clique, Weighted $c$-SAT
[pdf] Scribe: Lynelle
• 5/2    More on $W[1]$, $FPT=W[1]$ implies subexponential time 3SAT, $k$-Dominating Set is W[2]-complete
[pdf] Scribe: Kevin L.
• 5/7    $k$-Dominating Set in $n^{k-\varepsilon}$ implies faster CNF-SAT, $k$-SUM is $W[1]$-hard
[pdf] Scribe: Jack
• 5/9    The Exponential Time Hypothesis, Sparsification, and $k$-SUM
[pdf] Scribe: Josh
• 5/14    Solving $NP$-hard problems with bounded treewidth
[pdf] Scribe: Brian
• 5/16    Presentations: Hart, Chuanqi, Huacheng
[pdf] Scribe: Jeffrey
• 5/21    Presentations: Robin, Kevin L., Jeffrey
[none] No scribe
• 5/23    Presentations: Lucas, Yongxing, Josh (SAT Day)
[pdf] Scribe: Kevin M.

Upcoming topics (the rough plan)

• 5/28    Presentations: Jack, Lynelle, Amir
• 5/30    Presentations: Brian, Kevin M., Ryan and Virginia's last words
• Last week    No class, because everyone will be attending STOC and CCC which are both at Stanford!!

Papers for presentations

You may pick one or more papers to present, or you may present your own work, if it falls within the scope of the course! Email us ASAP with what you decide (first-come, first-serve); the deadline is May 2nd. We currently expect your presentations to be around 35 minutes long (note this estimate will only go down... it won't go up!).
The papers that have already been claimed will be crossed out.
1. Determinant Sums for Undirected Hamiltonicity, Andreas Bjorklund.
Summary: This paper presents an O*(1.66^n) time algorithm for Hamiltonian cycle in undirected graphs. This result was a breakthrough, giving the first improvement over the classic Held-Karp O*(2^n) time algorithm presented in class.
2. 3-SAT Faster and Simpler - Unique-SAT Bounds for PPSZ Hold in General, T. Hertli.
Summary: This presents the fastest algorithm for 3SAT to date, by analyzing an algorithm from the 90s in a new way.
3. Integer programming with a fixed number of variables, H.W. Lenstra.
Summary: This is a classic paper giving an FPT algorithm for integer programming where the parameter is the number of variables.
4. A Full Derandomization of Schoning's k-SAT Algorithm, R. Moser and D. Scheder.
Summary: This paper presents a deterministic k-SAT algorithm that has roughly the same running time as the randomized local search algorithm presented in class.
5. Iterative Compression and Exact Algorithms, F. V. Fomin, S. Gaspers, D. Kratsch, M. Liedloff, S. Saurabh.
Summary: This paper uses a technique called iterative compression to give new exact and parameterized algorithms for problems such as Max Independent Set and Min Hitting Set.
6. Infeasibility of Instance Compression and Succinct PCPs for NP, L. Fortnow and R. Santhanam.
Summary: This paper shows that several problems such as k-clique and k-dominating set do not have polynomial-size kernels under standard complexity assumptions (in particular, under the assumption that $coNP \not\subset NP/poly$).
7. New Limits to Classical and Quantum Instance Compression , A. Drucker.
Summary: This paper strengthens the results in the above Fortnow-Santhanam paper, showing that polynomial-size "instance compression" for satisfiability problems is even more unlikely.
8. Satisfiability Allows No Nontrivial Sparsification Unless The Polynomial-Time Hierarchy Collapses, H. Dell and D. van Melkebeek.
Summary: This paper considers the problem of finding small kernels for $d$-SAT on $n$ variables. A kernel of size $O(n^d log n)$ is easy to construct, and this paper shows that there is no $O(n^{d-\varepsilon})$-size kernel (under standard complexity assumptions).
9. On the Compressibility of NP Instances and Cryptographic Application, D. Harnik and M. Naor.
Summary: This paper shows that certain strong compressability results for SAT related to kernelization imply interesting things in cryptography.
10. On Problems Without Polynomial Kernels, H. Bodlaender, R. G. Downey, M. R. Fellows, D. Hermelin.
Summary: This paper shows that many FPT problems such as k-path don't admit polynomial kernels (under standard assumptions).
11. Parameterized complexity and approximation algorithms, D. Marx.
Summary: This paper presents relationships between parameterized complexity and approximation schemes.
12. Fourier Meets Mobius: Fast Subset Convolution, A. Bjorklund, T. Husfeldt, P. Kaski, M. Koivisto.
Summary: This paper presents an inclusion-exclusion based approach for obtaining $O^*(2^k)$ time algorithms for many problems where $O^*(3^k)$ is the easy algorithm. One example application is to $k$-Steiner tree.
13. Tight Lower Bounds for Certain Parameterized NP-Hard Problems, J. Chen. B. Chory, M. Fellows,X. Huang, D. Juedes, I. A. Kanj, G. Xia.
Summary: This paper shows that $n^{o(k)}$ time algorithms for many natural parameterized problems such as $k$-clique would imply unexpected results in parameterized complexity.
14. The parameterized complexity of counting problems, J. Flum and M. Grohe.
Summary: This paper defines the #W[t] hierarchy which is the analogue to the W hierarchy for counting problems.
15. Exponential Time Complexity of the Permanent and the Tutte Polynomial, H. Dell, T. Husfeldt, D. Marx, N. Taslaman, M. Wahlen.
Summary: This paper presents several conditional lower bounds based on ETH and its counting version, #ETH.
16. Fixed-Parameter Tractability and Parameterized Complexity, Applied to Problems From Computational Social Choice, C. Lindner and J. Rothe.
Summary: This paper shows some applications of parameterized complexity to computational problems in social choice, such as algorithmic questions in election manipulation.
17. A Measure & Conquer Approach for the Analysis of Exact Algorithms, F. Fomin, F. Grandoni, D. Kratsch.
Summary: The original paper introducing the Measure and Conquer Approach.
18. On Problems as Hard as CNF-SAT, M. Cygan, H. Dell, D. Lokshtanov, D. Marx, J. Nederlof, Y. Okamoto, R. Paturi, S. Saurabh, M. Wahlstrom.
Summary: This paper shows that designing $O^{*}((2-\varepsilon)^n)$ time algorithms for problems such as Hitting Set, Set Splitting, and NAE-Sat is equivalent to disproving the strong exponential time hypothesis (SETH).
19. Known Algorithms on Graphs of Bounded Treewidth are Probably Optimal, D. Lokshtanov, D. Marx, S. Saurabh.
Summary: This paper shows that the known algorithms for problems such as dominating set and independent set when parameterized by the graph treewidth are optimal unless the strong exponential time hypothesis fails.
20. Saving Space by Algebraization, D. Lokshtanov and J. Nederlof.
Summary: This paper gives interesting polynomial-space algorithms for problems such as Subset Sum, Knapsack, Traveling Salesman, Weighted Set Cover and Weighted Steiner Tree.