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

Given a graph $$G = (V,E)$$ , $$A \subseteq V$$ , and integers k and $$\ell $$ , the $$(A,\ell )$$ -Path Packing problem asks to find k vertex-disjoint paths of length exactly $$\ell $$ that have endpoints in A and internal points in $$V{\setminus }A$$ . We study the parameterized complexity of this problem with parameters |A|, $$\ell $$ , 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 $$\ell \le 3$$ , while it is NP-complete for constant $$\ell \ge 4$$ . We also show that the problem is W[1]-hard parameterized by pathwidth $${}+|A|$$ , while it is fixed-parameter tractable parameterized by treewidth $${}+\ell $$ . Additionally, we study a variant called Short A -Path Packing that asks to find k vertex-disjoint paths of length at most $$\ell $$ . 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 |A| or $$\ell $$ is a constant.