Parameterized Complexity of $$(A,\ell )$$-Path Packing

Abstract

Given a graph G = (V, E), A ⊆ V , and integers k and , the (A,)-Path Packing problem asks to find k vertex-disjoint paths of length that have endpoints in A and internal points in V \ A. We study the parameterized complexity of this problem with parameters |A|, , 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 ≤ 3, while it is NP-complete for constant ≥ 4. We also show that the problem is W[1]-hard parameterized by pathwidth + |A|, while it is fixed-parameter tractable parameterized by treewidth + .