The Parameterized Complexity of the Minimum Shared Edges Problem

TL;DR

This paper investigates the computational complexity of the Minimum Shared Edges problem, revealing its hardness under certain parameters and its fixed-parameter tractability with respect to the number of paths, providing new insights into its algorithmic properties.

Contribution

It establishes the W[1]-hardness of MSE with combined parameters and proves fixed-parameter tractability with respect to the number of paths p.

Findings

MSE is W[1]-hard when parameterized by treewidth and shared edges k.

MSE is fixed-parameter tractable with respect to the number of paths p.

MSE does not admit a polynomial kernel unless NP ⊆ coNP/poly.

Abstract

We study the NP-complete Minimum Shared Edges (MSE) problem. Given an undirected graph, a source and a sink vertex, and two integers p and k, the question is whether there are p paths in the graph connecting the source with the sink and sharing at most k edges. Herein, an edge is shared if it appears in at least two paths. We show that MSE is W[1]-hard when parameterized by the treewidth of the input graph and the number k of shared edges combined. We show that MSE is fixed-parameter tractable with respect to p, but does not admit a polynomial-size kernel (unless NP is contained in coNP/poly). In the proof of the fixed-parameter tractability of MSE parameterized by p, we employ the treewidth reduction technique due to Marx, O'Sullivan, and Razgon [ACM TALG 2013].