Abstract.It is well known that many local graph problems, like Vertex Cover and Dominating Set, can be solved in 2 O(tw) n O(1) time for graphs with a given tree decomposition of width tw. However, for nonlocal problems, like the fundamental class of connectivity problems, for a long time it was unknown how to do this faster than tw O(tw) The rank-based approach introduces a new technique to speed up dynamic programming algorithms which is likely to have more applications. The determinant-based approach uses the Matrix Tree Theorem for deriving closed formulas for counting versions of connectivity problems; we show how to evaluate those formulas via dynamic programming.

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 Wahlström, SODA 2012), with a randomized polynomial kernel for the problem, using matroid theory to encode flow 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 Lovász (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 kernel of O(k s+1 ) vertices for Multiway Cut instances with at most s 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. For a directed graph G = (V, E) and sets S, T ⊆ V , let r be the size of a minimum (S, T )-vertex cut (which may intersect S and T ). We can find a set Z ⊆ V of size O(|S| · |T | · r) which contains a minimum (A, B)-vertex cut for every A ⊆ S, B ⊆ T . Similarly, for an undirected graph G = (V, E), a set of terminals X ⊆ V , and a constant s, we can find a set Z ⊆ V of size O(|X| s+1 ) which contains a minimum multiway cut for any partition of X into at most s pairwise disjoint subsets (see the paper for a detailed description). Both results are polynomial time. We expect this to have further applications; in particular, we get direct, reduction rulebased 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 n, due to needing a representation of a matroid to apply the representative sets tool.

We introduce the cross-composition framework for proving kernelization lower bounds. A classical problem L and/or-cross-composes into a parameterized problem Q if it is possible to efficiently construct an instance of Q with polynomially bounded parameter value that expresses the logical and or or of a sequence of instances of L. Building on work by Bodlaender et al. (ICALP 2008) and using a result by Fortnow and Santhanam (STOC 2008) with a refinement by Dell and van Melkebeek (STOC 2010), we show that if an NPhard problem or-cross-composes into a parameterized problem Q then Q does not admit a polynomial kernel unless NP ⊆ coNP/poly and the polynomial hierarchy collapses. Similarly, an and-cross-composition for Q rules out polynomial kernels for Q under Bodlaender et al.'s and-distillation conjecture.Our technique generalizes and strengthens the recent techniques of using composition algorithms and of transferring the lower bounds via polynomial parameter transformations. We show its applicability by proving kernelization lower bounds for a number of important graphs problems with structural (non-standard) parameterizations, e.g., Clique, Chromatic Number, Weighted Feedback Vertex Set, and Weighted Odd Cycle Transversal do not admit polynomial kernels with respect to the vertex cover number of the input graphs unless the polynomial hierarchy collapses, contrasting the fact that these problems are trivially fixed-parameter tractable for this parameter. We have similar lower bounds for Feedback Vertex Set and Odd Cycle Transversal under structural parameterizations.After learning of our results, several teams of authors have successfully applied the cross-composition framework to different parameterized problems. For completeness, our presentation of the framework includes several extensions based on this follow-up work. For example, we show how a relaxed version of or-cross-compositions may be used to give lower bounds on the degree of the polynomial in the kernel size.

The Odd Cycle Transversal problem (OCT) asks whether a given 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(4 k kmn) time algorithm for it, the first algorithm with polynomial runtime of uniform degree for every fixed k. It is known that this implies a polynomial-time compression algorithm that turns OCT instances into equivalent instances of size at most O(4 k ), a so-called kernelization. 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. Despite the impressive progress in the area, including the recent development of lower bound techniques (Bodlaender et al., ICALP 2008; Fortnow and Santhanam, STOC 2008) and meta-results on kernelizations for graph problems on planar and other sparse graph classes (Bodlaender et al., FOCS 2009; Fomin et al., SODA 2010), the existence of a polynomial kernel for OCT has remained open, even when the input is restricted to be planar.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. For OCT, the matroid is built to allow us to simulate the computation of the iterative compression step of the algorithm of Reed, Smith, and Vetta, applied (for only one round) to an approximate odd cycle transversal which it is aiming to shrink to size k. The process is randomized with one-sided error exponentially small in k, where the result can contain false positives but no false negatives, and the size guarantee is cubic in the size of the approximate solution. Combined with an O( √ log n)-approximation (Agarwal et al.,STOC 2005), we get a reduction of the instance to size O(k 4.5 ), implying a randomized polynomial kernelization. Interestingly, the known lower bound techniques can be seen to exclude randomized kernels that produce no false negatives, as in fact they exclude even co-nondeterministic kernels (Dell and van Melkebeek, STOC 2010). Therefore, our result also implies that deterministic kernels for OCT cannot be excluded by the known machinery.

Abstract.It is well known that many local graph problems, like Vertex Cover and Dominating Set, can be solved in 2 O(tw) n O(1) time for graphs with a given tree decomposition of width tw. However, for nonlocal problems, like the fundamental class of connectivity problems, for a long time it was unknown how to do this faster than tw O(tw) n O(1) until recently, when Cygan et al. (FOCS 2011) introduced the Cut&Count technique that gives randomized algorithms for a wide range of connectivity problems running in time c tw n O(1) for a small constant c.In this paper, we show that we can improve upon the Cut&Count technique in multiple ways, with two new techniques. The first technique (rank-based approach) gives deterministic algorithms with O(c tw n) running time for connectivity problems (like Hamiltonian Cycle and Stei-ner Tree) and for weighted variants of these; the second technique (determinant approach) gives deterministic algorithms running in time c tw n O(1) for counting versions, e.g., counting the number of Hamiltonian cycles for graphs of treewidth tw.The rank-based approach introduces a new technique to speed up dynamic programming algorithms which is likely to have more applications. The determinant-based approach uses the Matrix Tree Theorem for deriving closed formulas for counting versions of connectivity problems; we show how to evaluate those formulas via dynamic programming.

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For an even integer t ≥ 2, the Matching Connectivity matrix H t is a matrix that has rows and columns both labeled by all perfect matchings of the complete graph on t vertices; an entry H t [ M 1 , M 2 ] is 1 if M 1 and M 2 form a Hamiltonian cycle and 0 otherwise. Motivated by applications for the Hamiltonicity problem, we show that H t has rank exactly 2 t /2−1 over GF(2). The upper bound is established by an explicit factorization of H t as the product of two submatrices; the matchings labeling columns and rows, respectively, of the submatrices therefore form a basis X t of H t . The lower bound follows because the 2 t /2−1 × 2 t /2−1 submatrix with rows and columns labeled by X t can be seen to have full rank. We obtain several algorithmic results based on the rank of H t and the particular structure of the matchings in X t . First, we present a 1.888 n n O (1) time Monte Carlo algorithm that solves the Hamiltonicity problem in directed bipartite graphs. Second, we give a Monte Carlo algorithm that solves the problem in (2 + √ 2) pw n O (1) time when provided with a path decomposition of width pw for the input graph. Moreover, we show that this algorithm is best possible under the Strong Exponential Time Hypothesis, in the sense that an algorithm with running time (2 + √2 − ϵ) pw n O (1) , for any ϵ > 0, would imply the breakthrough result of a (2 − ϵ ′ ) n -time algorithm for CNF-Sat for some ϵ ′ > 0.

We study the complexity of the Graph Isomorphism problem on graph classes that are characterized by a finite number of forbidden induced subgraphs, focusing mostly on the case of two forbidden subgraphs. We show hardness results and develop techniques for the structural analysis of such graph classes, which applied to the case of two forbidden subgraphs give the following results: A dichotomy into isomorphism complete and polynomial-time solvable graph classes for all but finitely many cases, whenever neither of the forbidden graphs is a clique, a pan, or a complement of these graphs. Further reducing the remaining open cases we show that (with respect to graph isomorphism) forbidding a pan is equivalent to forbidding a clique of size three.