Parameterized Complexity of (A, l)-Path Packing

Given a graph G = (V, E), A subset of V, and integers k and l, the (A, l)-PATH PACKING problem asks to find k vertex-disjoint paths of length exactly l that have endpoints in A and internal points in V\A. We study the parameterized complexity of this problem with parameters vertical bar A vertical bar, l, k, treewidth, pathwidth, and their combinations. We present sharp complexity contrasts with respect to these parameters. Among other results, we show that the problem is polynomial-time solvable when l <= 3, while it is NP-complete for constant l >= 4. We also show that the problem is W[1]-hard parameterized by pathwidth + vertical bar A vertical bar, while it is fixed-parameter tractable parameterized by treewidth + l. Additionally, we study a variant called SHORT A- PATH PACKING that asks to find k vertex-disjoint paths of length at most l. We show that all our positive results on the exact-length version can be translated to this version and show the hardness of the cases where vertical bar A vertical bar or l is a constant.