Magnus Wahlström

Representative Sets and Irrelevant Vertices: New Tools for Kernelization

The existence of a polynomial kernel for Odd Cycle Transversal was a notorious open problem in parameterized complexity. Recently, this was settled by the present authors (Kratsch and Wahlstrom, SODA 2012), with a randomized polynomial kernel for the problem, using matroid theory to encode How questions over a set of terminals in size polynomial in the number of terminals (rather than the total graph size, which may be superpolynomially larger). In the current work we further establish the usefulness of matroid theory to kernelization by showing applications of a result on representative sets due to Lovasz (Combinatorial Surveys 1977) and Marx (TCS 2009). We show how representative sets can be used to give a polynomial kernel for the elusive Almost 2-sat problem (where the task is to remove at most k clauses to make a 2-CNF formula satisfiable), solving a major open problem in kernelization. We further apply the representative sets tool to the problem of finding irrelevant vertices in graph cut problems, that is, vertices which can be made undeletable without affecting the status of the problem. This gives the first significant progress towards a polynomial kernel for the Multiway Cut problem; in particular, we get a polynomial kernel for Multiway Cut instances with a bounded number of terminals. Both these kernelization results have significant spin-off effects, producing the first polynomial kernels for a range of related problems. More generally, the irrelevant vertex results have implications for covering min-cuts in graphs. In particular, given a directed graph and a set of terminals, we can find a set of size polynomial in the number of terminals (a cut-covering set) which contains a minimum vertex cut for every choice of sources and sinks from the terminal set. Similarly, given an undirected graph and a set of terminals, we can find a set of vertices, of size polynomial in the number of terminals, which contains a minimum multiway cut for every partition of the terminals into a bounded number of sets. Both results are polynomial time. We expect this to have further applications; in particular, we get direct, reduction rule-based kernelizations for all problems above, in contrast to the indirect compression-based kernel previously given for Odd Cycle Transversal. All our results are randomized, with failure probabilities which can be made exponentially small in the size of the input, due to needing a representation of a matroid to apply the representative sets tool.

Counting models for 2 SAT and 3 SAT formulae

We here present algorithms for counting models and max-weight models for 2SAT and 3SAT formulae. They use polynomial space and run in O(1.2561n) and O(1.6737n) time, respectively, where n is the number of variables. This is faster than the previously best algorithms for counting nonweighted models for 2SAT and 3SAT, which run in O(1.3247n) and O(1.6894n) time, respectively. In order to prove these time bounds, we develop new measures of formula complexity, allowing us to conveniently analyze the effects of certain factors with a large impact on the total running time. We also provide an algorithm for the restricted case of separable 2SAT formulae, with fast running times for well-studied input classes. For all three algorithms we present interesting applications, such as computing the permanent of sparse 0/1 matrices.

Compression via Matroids: A Randomized Polynomial Kernel for Odd Cycle Transversal

The Odd Cycle Transversal problem (OCT) asks whether a given undirected graph can be made bipartite by deleting at most k of its vertices. In a breakthrough result, Reed, Smith, and Vetta (Operations Research Letters, 2004) gave a O(4kkmn) time algorithm for it; this also implies that instances of the problem can be reduced to a so-called problem kernel of size O(4k). Since then, the existence of a polynomial kernel for OCT (i.e., a kernelization with size bounded polynomially in k) has turned into one of the main open questions in the study of kernelization, open even for the special case of planar input graphs. This work provides the first (randomized) polynomial kernelization for OCT. We introduce a novel kernelization approach based on matroid theory, where we encode all relevant information about a problem instance into a matroid with a representation of size polynomial in k. This represents the first application of matroid theory to kernelization.

On Problems as Hard as CNF-SAT

The field of exact exponential time algorithms for NP-hard problems has thrived over the last decade. While exhaustive search remains asymptotically the fastest known algorithm for some basic problems, difficult and non-trivial exponential time algorithms have been found for a myriad of problems, including Graph Coloring, Hamiltonian Path, Dominating Set and 3-CNF-Sat. In some instances, improving these algorithms further seems to be out of reach. The CNF-Sat problem is the canonical example of a problem for which the trivial exhaustive search algorithm runs in time O(2^n), where n is the number of variables in the input formula. While there exist non-trivial algorithms for CNF-Sat that run in time o(2^n), no algorithm was able to improve the growth rate 2 to a smaller constant, and hence it is natural to conjecture that 2 is the optimal growth rate. The strong exponential time hypothesis (SETH) by Impagliazzo and Paturi [JCSS 2001] goes a little bit further and asserts that, for every epsilon

Two Edge Modification Problems without Polynomial Kernels

Given a graph G and an integer k, the ? Edge Completion/Editing/Deletion problem asks whether it is possible to add, edit, or delete at most k edges in G such that one obtains a graph that fulfills the property ?. Edge modification problems have received considerable interest from a parameterized point of view. When parameterized by k, many of these problems turned out to be fixed-parameter tractable and some are known to admit polynomial kernelizations, i.e., efficient preprocessing with a size guarantee that is polynomial in k. This paper answers an open problem posed by Cai (IWPEC 2006), namely, whether the ? Edge Deletion problem, parameterized by the number of deletions, admits a polynomial kernelization when ? can be characterized by a finite set of forbidden induced subgraphs. We answer this question negatively based on recent work by Bodlaender et al. (ICALP 2008) which provided a framework for proving polynomial lower bounds for kernelizability. We present a graph H on seven vertices such that

Clique Cover and Graph Separation: New Incompressibility Results

\mathcal{H}

A tighter bound for counting max-weight solutions to 2SAT instances

-free Edge Deletion and H-free Edge Editing do not admit polynomial kernelizations, unless

Exact algorithms for finding minimum transversals in rank-3 hypergraphs

\mbox{NP}\subseteq \mbox{coNP}/\mbox{poly}

Preprocessing of min ones problems: a dichotomy

. The application of the framework is not immediate and requires a lower bound for a Not-1-in-3 SAT problem that may be of independent interest.

Half-integrality, LP-branching, and FPT Algorithms

The field of kernelization studies polynomial-time preprocessing routines for hard problems in the framework of parameterized complexity. In this article, we show that, unless the polynomial hierarchy collapses to its third level, the following parameterized problems do not admit a polynomial-time preprocessing algorithm that reduces the size of an instance to polynomial in the parameter: ---Edge Clique Cover, parameterized by the number of cliques, ---Directed Edge/Vertex Multiway Cut, parameterized by the size of the cutset, even in the case of two terminals, ---Edge/Vertex Multicut, parameterized by the size of the cutset, and ---k-Way Cut, parameterized by the size of the cutset.