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.

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 < 1, there is a (large) integer k such that k-CNF-Sat cannot be computed in time 2 n . In this paper, we show that, for every < 1, the problems Hitting Set, Set Splitting, and NAE-Sat cannot be computed in time O(2 n ) unless SETH fails. Here n is the number of elements or variables in the input. For these problems, we actually get an equivalence to SETH in a certain sense. We conjecture that SETH implies a similar statement for Set Cover, and prove that, under this assumption, the fastest known algorithms for Steiner Tree, Connected Vertex Cover, Set Partitioning, and the pseudo-polynomial time algorithm for Subset Sum cannot be significantly improved. Finally, we justify our assumption about the hardness of Set Cover by showing that the parity of the number of solutions to Set Cover cannot be computed in time O(2 n ) for any < 1 unless SETH fails.

A recent trend in parameterized algorithms is the application of polytope tools (specifically, LPbranching) to FPT algorithms (e.g., Cygan et al., 2011;. Though the list of work in this direction is short, the results are already interesting, yielding significant speedups for a range of important problems. However, the existing approaches require the underlying polytope to have very restrictive properties, including half-integrality and Nemhauser-Trotter-style persistence properties. To date, these properties are essentially known to hold only for two classes of polytopes, covering the cases of Vertex Cover (Nemhauser and Trotter, 1975) and Node Multiway Cut (Garg et al., 1994).Taking a slightly different approach, we view half-integrality as a discrete relaxation of a problem, e.g., a relaxation of the search space from {0, 1}V to {0, 1 /2, 1} V such that the new problem admits a polynomial-time exact solution. Using tools from CSP (in particular Thapper andŽivný, 2012) to study the existence of such relaxations, we are able to provide a much broader class of half-integral polytopes with the required properties.Our results unify and significantly extend the previously known cases. In addition to the new insight into problems with half-integral relaxations, our results yield a range of new and improved FPT algorithms, including an O * (|Σ| 2k )-time algorithm for node-deletion Unique Label Cover with label set

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.

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 < 1, there is a (large) integer k such that k-CNF-Sat cannot be computed in time 2 n . In this paper, we show that, for every < 1, the problems Hitting Set, Set Splitting, and NAE-Sat cannot be computed in time O(2 n ) unless SETH fails. Here n is the number of elements or variables in the input. For these problems, we actually get an equivalence to SETH in a certain sense. We conjecture that SETH implies a similar statement for Set Cover, and prove that, under this assumption, the fastest known algorithms for Steiner Tree, Connected Vertex Cover, Set Partitioning, and the pseudo-polynomial time algorithm for Subset Sum cannot be significantly improved. Finally, we justify our assumption about the hardness of Set Cover by showing that the parity of the number of solutions to Set Cover cannot be computed in time O(2 n ) for any < 1 unless SETH fails.