On Conflict-Free Spanning Tree: Algorithms and Complexity

Abstract

AbstractA natural constraint in real-world applications is to avoid conflicting elements in the solution of problems. Given an undirected graph \(G=(V, E)\) where each edge \(e\in E\) has a positive integer weight \(\omega (e)\), and a conflict graph \(\hat{G}=(\hat{V}, \hat{E})\) 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 (if any) a spanning tree avoiding pairs of conflicting edges (conflict-free) with minimum cost, i.e., a minimum solution among spanning trees T such that E(T) is an independent set of \(\hat{G}\). A spanning tree T of G is a feasible solution for an instance \(I=(G,\hat{G})\) of MCFST if E(T) is an independent set of \(\hat{G}\). In contrast to the polynomial-time solvability of Minimum Spanning Tree, to determine whether an instance \(I=(G,\hat{G})\) of MCFST admits a feasible solution is \(\mathcal {NP}\)-complete. In this paper, we present a multivariate complexity analysis of MCFST by considering particular classes of graphs G and \(\hat{G}\). In particular, we show that the problem of determining whether an instance \(I=(G,\hat{G})\) of MCFST has a feasible solution is \(\mathcal {NP}\)-complete even if G is a bipartite planar subcubic graph, and \(\hat{G}\) is a disjoint union of paths of size three (\(P_3\)). Moreover, we show that whether G is a complete graph and \(\hat{G}\) is a disjoint union of stars, then a feasible solution for \(I=(G,\hat{G})\) can be found in polynomial time. In addition, we present (in)approximability results for MCFST on complete graphs G, and an FPT algorithm parameterized by the distance to \(\mathcal F\) of the conflict graph \(\hat{G}\), where \(\mathcal F\) is a hereditary graph class such that MCFST on conflict graphs \(\hat{G}\in \mathcal F\) can be solved in polynomial time.KeywordsConflict-freeSpanning treeApproximationFPT