Representative Sets and Irrelevant Vertices: New Tools for Kernelization

Abstract

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.

On Group Feedback Vertex Set Parameterized by the Size of the Cutset

“…A similar reduction was also used recently by Kratsch and Wahlström in the kernelization algorithm for Group Feedback Vertex Set parameterized by k with constant |Σ| [20]. Currently the fastest FPT algorithm for Multiway Cut is due to Cygan et al [9], and it solves the problem in O * (2 k ) time and polynomial space.…”

“…As noted in [20], GFVS subsumes not only the classical FVS problem, but also OCT (with Σ = Z 2 ) and MWC (with Σ being an arbitrary group of size not smaller than the number of terminals). We note that if Σ is given in the blackbox, Group Feedback Vertex Set subsumes also Edge Subset Feedback Vertex Set, which is equivalent to Subset Feedback Vertex Set [10].…”

“…When parameterized by k, Guillemot showed a fixed-parameter algorithm for the easier edge-deletion variant of GFVS, running in time O * (2 O(k log k) ). Recently, Kratsch and Wahlström presented a randomized kernelization algorithm that reduces the size of a GFVS instance to O(k 2|Σ| ) [20].…”

“…In order to prove Lemma 4 we use the concept of untangling, previously used by Kratsch and Wahlström [20]. We transform an instance of C-GFVS to ensure that each arc (u, v) with both endpoints in V (G) \ Z is labeled 1 Σ by Λ.…”

“…In each step of the iterative compression, we solve a Compression Group Feedback Vertex Set problem, where we are given a solution Z of size a bit too large -k + 1 -and we are to find a new solution disjoint from it. We first prepare the Compression Group Feedback Vertex Set instance by untangling it in Section 3.2, in the same manner as it is done in the kernelization algorithm of [20]. The main step of the algorithm is done in Section 3.3, where we provide a set of reduction rules that enable us for each vertex v ∈ Z to limit the number of choices for a value of a consistent labeling on v to polynomial in k. Finally, we iterate over all O * (2 O(k log k) ) remaining labelings of Z and, for each labeling, reduce the instance to Multiway Cut (Section 3.4).…”