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].Comment: 35 pages, 16 figure
