On Conflict-Free Spanning Tree: Mapping tractable and hard instances through the lenses of graph classes

Abstract

A natural constraint in real-world applications is avoiding conflicting elements in problem solutions. Let $G=(V,E)$ be a graph where each edge $e\in E$ has a positive integer weight $\omega \left(e\right)$, and let $\hat{G}=(\hat{V},\hat{E})$ be a conflict graph such that $\hat{V}\subseteq E$ and each edge $\hat{e}=e_1{e}_2\in \hat{E}$ represents a conflict between two edges $e_1,e_2\in E$. In the Minimum Conflict-Free Spanning Tree (MCFST) problem, we are asked to find a spanning tree avoiding pairs of conflicting edges (if such a tree exists) with minimum cost. In contrast to the polynomial-time solvability of Minimum Spanning Tree, to determine whether an instance $(G,\hat{G})$ of MCFST admits a feasible solution is an $\mathrm{NP}$-complete problem. In this paper, we present a multivariate complexity analysis of MCFST by considering particular classes of graphs G and $\hat{G}$. We show that the problem of determining whether an instance $(G,\hat{G})$ of MCFST has a feasible solution is $\mathrm{NP}$-complete even if G is a bipartite planar subcubic graph, and $\hat{G}$ is a disjoint union of paths with three vertices. Contrastingly, we show that when G is complete and $\hat{G}$ is bipartite, then a solution for $(G,\hat{G})$ can be found in linear time, while the problem of finding an optimal solution is $\mathrm{NP}$-hard. Also, we present (in)approximability results for MCFST on complete graphs G, and a parameterized algorithm regarding the distance from the conflict graph to a hereditary graph class $F$ for which MCFST on $\hat{G}\in F$ is polynomial-time solvable.