The Parameterized Complexity of the Minimum Shared Edges Problem

Authors Till Fluschnik, Stefan Kratsch, Rolf Niedermeier, Manuel Sorge

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Till Fluschnik
Stefan Kratsch
Rolf Niedermeier
Manuel Sorge

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Till Fluschnik, Stefan Kratsch, Rolf Niedermeier, and Manuel Sorge. The Parameterized Complexity of the Minimum Shared Edges Problem. In 35th IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2015). Leibniz International Proceedings in Informatics (LIPIcs), Volume 45, pp. 448-462, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2015)


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 a subset of 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].
  • Parameterized complexity
  • kernelization
  • treewidth
  • treewidth reduction


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