The complexity of minimum-length path decompositions

We consider a bi-criteria generalization of the pathwidth problem, where, for given integers k, l and a graph G, we ask whether there exists a path decomposition P of G such that the width of P is at most k and the number of bags in P, i.e., the length of P, is at most l. We provide a complete complexity classification of the problem in terms of k and l for general graphs. Contrary to the original pathwidth problem, which is fixed-parameter tractable with respect to k, we prove that the generalized problem is NP-complete for any fixed k ≥ 4, and is also NP-complete for any fixed l ≥ 2. On the other hand, we give a polynomial-time algorithm that, for any (possibly disconnected) graph G and integers k ≤ 3 and l > 0, constructs a path decomposition of width at most k and length at most l, if any exists. As a by-product, we obtain an almost complete classification of the problem in terms of k and l for connected graphs. Namely, the problem is NP-complete for any fixed k ≥ 5 and it is polynomial-time for any k ≤ 3. This leaves open the case k = 4 for connected graphs.