The Parameterized Complexity of the Minimum Shared Edges Problem

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 ⊆ 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]. 1998 ACM Subject Classification F.2.2 Nonnumerical Algorithms and Problems, G.2.2 Graph Theory Keywords and phrases Parameterized complexity, kernelization, treewidth, treewidth reduction Digital Object Identifier 10.4230/LIPIcs.FS...

• Type: Journal
• Year: 2016
• Where: CORR
• Authors: Till Fluschnik, Stefan Kratsch, Rolf Niedermeier, Manuel Sorge
• Keywords: CORR 2016, Education