Parameterized Complexity of Path Set Packing

In PATH SET PACKING, the input is an undirected graph G, a collection P of simple paths in G, and a positive integer k. The problem is to decide whether there exist k edge-disjoint paths in P. We study the parameterized complexity of PATH SET PACKING with respect to both natural and structural parameters. We show that the problem is W[1]-hard with respect to vertex cover plus the maximum length of a path in P, and W[1]-hard with respect to pathwidth plus maximum degree plus solution size. These results answer an open question raised in [17]. On the positive side, we present an FPT algorithm parameterized by feedback vertex set plus maximum degree, and also provide an FPT algorithm parameterized by treewidth plus maximum degree plus maximum length of a path in P.