Refined Notions of Parameterized Enumeration Kernels with Applications to Matching Cut Enumeration

TL;DR

This paper introduces refined notions of enumeration kernels for parameterized problems, demonstrating their effectiveness in data reduction and solution enumeration for the NP-hard Matching Cut problem with structural parameters.

Contribution

It proposes new versions of enumeration kernels that enable polynomial-time or polynomial-delay enumeration from kernel solutions, enhancing data reduction techniques.

Findings

New enumeration kernel concepts for parameterized problems.

Application to Matching Cut problem with structural parameters.

Efficient solution enumeration from kernels.

Abstract

An enumeration kernel as defined by Creignou et al. [Theory Comput. Syst. 2017] for a parameterized enumeration problem consists of an algorithm that transforms each instance into one whose size is bounded by the parameter plus a solution-lifting algorithm that efficiently enumerates all solutions from the set of the solutions of the kernel. We propose to consider two new versions of enumeration kernels by asking that the solutions of the original instance can be enumerated in polynomial time or with polynomial delay from the kernel solutions. Using the NP-hard Matching Cut problem parameterized by structural parameters such as the vertex cover number or the cyclomatic number of the input graph, we show that the new enumeration kernels present a useful notion of data reduction for enumeration problems which allows to compactly represent the set of feasible solutions.