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
respect to pathwidth plus maximum degree plus solution size. These results
answer an open question raised in COCOON 2018. On the positive side, we show an
FPT algorithm parameterized by feedback vertex set plus maximum degree, and
also show an FPT algorithm parameterized by treewidth plus maximum degree plus
maximum length of a path in P. Both the positive results complement the
hardness of PATH SET PACKING with respect to any subset of the parameters used
in the FPT algorithms