Exact Algorithms for Distance to Unique Vertex Cover

TL;DR

This paper explores the computational complexity of ensuring a unique minimum vertex cover in graphs, extending previous work by relaxing certain constraints and providing algorithms for specific graph classes.

Contribution

It introduces the MU-VC problem, proves its $ ext{Sigma}^2_P$-completeness, and offers fixed-parameter algorithms for trees, bounded treewidth, and clique-width graphs.

Findings

MU-VC is $ ext{Sigma}^2_P$-complete even for planar graphs of degree 5.

Linear-time algorithm for MU-VC on trees.

FPT algorithms for graphs with bounded treewidth and clique-width.

Abstract

Horiyama et al. (AAAI 2024) studied the problem of generating graph instances that possess a unique minimum vertex cover under specific conditions. Their approach involved pre-assigning certain vertices to be part of the solution or excluding them from it. Notably, for the \textsc{Vertex Cover} problem, pre-assigning a vertex is equivalent to removing it from the graph. Horiyama et al.~focused on maintaining the size of the minimum vertex cover after these modifications. In this work, we extend their study by relaxing this constraint: our goal is to ensure a unique minimum vertex cover, even if the removal of a vertex may not incur a decrease on the size of said cover. Surprisingly, our relaxation introduces significant theoretical challenges. We observe that the problem is $\Sigma^2_P$-complete, and remains so even for planar graphs of maximum degree 5. Nevertheless, we provide a linear time algorithm for trees, which is then further leveraged to show that MU-VC is in \textsf{FPT} when parameterized by the combination of treewidth and maximum degree. Finally, we show that MU-VC is in \textsf{XP} when parameterized by clique-width while it is fixed-parameter tractable (FPT) if we add the size of the solution as part of the parameter.