Parameterized Complexity of Path Set Packing

In Path Set Packing, the input is an undirected graph G, a collection \(\mathcal{P}\) of simple paths in G, and a positive integer k. The problem is to decide whether there exist k edge-disjoint paths in \(\mathcal{P}\). 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 number, and W[1]-hard respect to pathwidth plus solution size when input graph is a grid. These results answer an open question raised in Xu and Zhang (in: Wang L, Zhu D (eds) Computing and combinatorics—24th international conference, COCOON 2018, Qing Dao, China, July 2–4, 2018, proceedings. Lecture notes in computer science, vol 10976, pp 305–315. Springer, 2018, https://doi.org/10.1007/978-3-319-94776-1_26). On the positive side, we present an FPT algorithm parameterized by feedback vertex number plus maximum degree, and present an FPT algorithm parameterized by treewidth plus maximum degree plus maximum length of a path in \(\mathcal{P}\). These positive results complement the hardness of Path Set Packing with respect to any subset of the parameters used in the FPT algorithms. We also give a 4-approximation algorithm for maximum path set packing problem which runs in FPT time when parameterized by feedback edge number.