Kernelization Complexity of Solution Discovery Problems

TL;DR

This paper investigates the kernelization complexity of solution discovery problems in graphs, focusing on token sliding variants for problems like Vertex Cover and Independent Set, with respect to parameters such as token count and graph structure.

Contribution

It provides the first complexity analysis of kernelization for solution discovery problems under token sliding, extending the framework to multiple classical graph problems.

Findings

Kernelization results for Vertex Cover, Independent Set, and others.

Complexity classifications depend on parameters like token number and pathwidth.

New insights into the parameterized complexity of solution discovery variants.

Abstract

In the solution discovery variant of a vertex (edge) subset problem $\Pi$ on graphs, we are given an initial configuration of tokens on the vertices (edges) of an input graph $G$ together with a budget $b$. The question is whether we can transform this configuration into a feasible solution of $\Pi$ on $G$ with at most $b$ modification steps. We consider the token sliding variant of the solution discovery framework, where each modification step consists of sliding a token to an adjacent vertex (edge). The framework of solution discovery was recently introduced by Fellows et al. [Fellows et al., ECAI 2023] and for many solution discovery problems the classical as well as the parameterized complexity has been established. In this work, we study the kernelization complexity of the solution discovery variants of Vertex Cover, Independent Set, Dominating Set, Shortest Path, Matching, and Vertex Cut with respect to the parameters number of tokens $k$, discovery budget $b$, as well as structural parameters such as pathwidth.