[Paper Review] Representative sets and irrelevant vertices: New tools for kernelization

Stefan Kratsch, Magnus Wahlström|arXiv (Cornell University)|Nov 9, 2011

Advanced Graph Theory Research38 references4 citations

TL;DR

This paper introduces a novel application of representative sets from matroid theory to achieve polynomial kernelization for fundamental problems in parameterized complexity. By leveraging Lovász's and Marx's representative sets lemma, the authors develop a method to identify irrelevant vertices in graph cut problems, yielding the first polynomial kernel for Almost 2-SAT and a kernel of size O(ks+1) for Multiway Cut with s terminals, significantly advancing kernelization techniques for NP-hard problems.

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. 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. We further apply the representative sets tool to the problem of finding irrelevant vertices in graph cut problems, i.e., 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 \subseteq 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 \subseteq V of size O(|S|*|T|*r) which contains a minimum (A,B)-vertex cut for every A \subseteq S, B \subseteq T. Similarly, for an undirected graph G=(V,E), a set of terminals X \subseteq V, and a constant s, we can find a set Z\subseteq 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.

Motivation & Objective

• To resolve the long-standing open problem of polynomial kernelization for the Almost 2-SAT problem.
• To develop a new method for identifying irrelevant vertices in graph cut problems to enable efficient kernelization.
• To extend the applicability of matroid-theoretic tools—specifically representative sets—to kernelization in parameterized complexity.
• To provide the first polynomial kernel for Multiway Cut with a constant number of terminals.
• To construct small cut-covering sets that contain minimum (A,B)-vertex cuts and multiway cuts for all relevant partitions of terminals.

Proposed method

• Applying the representative sets lemma from matroid theory (Lovász and Marx) to identify vertices that can be made undeletable without affecting the solution.
• Constructing a gammoid structure across multiple layers to encode reachability and saturation conditions for potential cuts.
• Using the representative set of transversals to efficiently capture essential vertices that must lie in every minimum cut.
• Designing a randomized reduction rule that selects a small set Z of vertices containing all minimum cuts for any partition of terminals into at most s subsets.
• Formulating the kernelization process as an irrelevant vertex argument grounded in matroid representation and representative sets.
• Reducing the failure probability of the randomized construction to be exponentially small in the input size n.

Experimental results

Research questions

• RQ1 Can representative sets from matroid theory be used to construct polynomial kernels for NP-hard problems in parameterized complexity?
• RQ2 Is it possible to identify and remove irrelevant vertices in graph cut problems using representative sets to enable kernelization?
• RQ3 Can the technique be applied to achieve a polynomial kernel for the Almost 2-SAT problem?
• RQ4 Can a small set of vertices be computed that contains a minimum (A,B)-vertex cut for all subsets A,B of terminals in a graph?
• RQ5 Can a similar construction be used to cover all minimum multiway cuts for any partition of terminals into at most s parts?

Key findings

• The paper presents the first polynomial kernel for the Almost 2-SAT problem, resolving a major open problem in kernelization.
• A kernel of size O(ks+1) is achieved for Multiway Cut with s terminals, marking the first significant progress toward a polynomial kernel for this problem.
• A randomized polynomial-time algorithm is developed that computes a set Z of size O(|X|s+1) containing a minimum multiway cut for any partition of terminals into at most s disjoint subsets.
• For any directed graph G and sets S,T ⊆V, a set Z of size O(|S|·|T|·r) is computed in randomized polynomial time, containing a minimum (A,B)-vertex cut for all A⊆S, B⊆T.
• The method enables reduction rule-based kernelization for all covered problems, in contrast to the indirect compression-based approach used previously for Odd Cycle Transversal.
• The failure probability of the randomized construction is exponentially small in the input size n, ensuring high reliability.