Representative Sets and Irrelevant Vertices

Abstract

We continue the development of matroid-based techniques for kernelization, initiated by the present authors [47]. We significantly extend the usefulness of matroid theory in kernelization by showing applications of a result on representative sets due to Lovász [51] and Marx [53]. As a first result, we show how representative sets can be used to derive 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. This result also yields a new O(√log OPT)-approximation for the problem, improving on the O(√log n)-approximation of Agarwal et al. [3] and an implicit O(log OPT)-approximation due to Even et al. [24]. We further apply the representative sets tool to the problem of finding irrelevant vertices in graph cut problems, that is, vertices that can be made undeletable without affecting the answer to 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) that 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 ) that contains a minimum multiway cut for every partition of X into at most s pairwise disjoint subsets. 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 [47]. All our results are randomized, with failure probabilities that can be made exponentially small in n , due to needing a representation of a matroid to apply the representative sets tool.

Subexponential Parameterized Algorithms for Cut and Cycle Hitting Problems on H-Minor-Free Graphs

“…EB and OCT: It follows from Theorems 6.9, and 6.7, which relies on Lemma 6.5 of [43]. Also, see [46,Lemma 3].…”

“…Proof sketch. Proof of this lemma is implicit in literature [43,61]. Known kernels for all these problems actually first compute a candidate set and then apply reduction rules (in fact, an appropriate torso operation) to reduce the graph size.…”

“…Known kernels for all these problems actually first compute a candidate set and then apply reduction rules (in fact, an appropriate torso operation) to reduce the graph size. The candidate sets for EB, OCT, VMWC, VMC, and GFVS follow from the work [43], while the candidate sets for EMWC, EMC, and GFES follow from the work [61]. See Appendix B for a more detailed discussion about the references for each of these problems.…”

“…First, we apply our framework on GFVS. We note that OCT is a specialization of GFVS when Σ = {0, 1} [43]. Indeed, our algorithm for GFVS is an extension of the algorithm for OCT.…”

Vertex Deletion Parameterized by Elimination Distance and Even Less

“…The notion of A-exhaustive families is similar in spirit to that of q-representative families for matroids, which have been used in recent algorithms working on graph decompositions [39,67] (cf. [26, §12.3]) to trim the set of partial solutions stored by a dynamic program while preserving the existence of an optimal solution.…”