Solution Discovery for Vertex Cover, Independent Set, Dominating Set, and Feedback Vertex Set

TL;DR

This paper investigates the computational complexity of solution discovery problems for key graph problems, providing algorithms and hardness results across various graph classes and parameters.

Contribution

It introduces XP algorithms parameterized by clique-width, proves NP-completeness on certain graph classes, and offers polynomial and FPT algorithms for specific cases.

Findings

XP algorithms for all four problems based on clique-width

NP-completeness on chordal graphs and diameter-2 graphs

Polynomial-time solutions on split graphs

Abstract

In the solution discovery problem for a search problem on graphs, we are given an initial placement of $k$ tokens on the vertices of a graph and asked whether this placement can be transformed into a feasible solution by applying a small number of modifications. In this paper, we study the computational complexity of solution discovery for several fundamental vertex-subset problems on graphs, namely Vertex Cover Discovery, Independent Set Discovery, Dominating Set Discovery, and Feedback Vertex Set Discovery. We first present XP algorithms for all four problems parameterized by clique-width. We then prove that Vertex Cover Discovery, Independent Set Discovery, and Feedback Vertex Set Discovery are NP-complete for chordal graphs and graphs of diameter 2, which have unbounded clique-width. In contrast to these hardness results, we show that all three problems can be solved in polynomial time on split graphs. Furthermore, we design an FPT algorithm for Feedback Vertex Set Discovery parameterized by the number of tokens.