The complexity of the Pk partition problem and related problems in bipartite graphs

In this paper, we continue the investigation proposed in [15] about the approximability of Pk partition problems, but focusing here on their complexity. More precisely, we prove that the problem consisting of deciding if a graph of nk vertices has n vertex disjoint simple paths {P1, · · · , Pn} such that each path Pi has k vertices is NP-complete, even in bipartite graphs of maximum degree 3. Note that this result also holds when each path Pi is chordless in G[V (Pi)]. Then, we prove that MaxP3Packing and MaxInducedP3Packing in bipartite graphs of maximum degree 3 are not in PTAS. Finally, we propose a 3/2-approximation for Min3-PathPartition in general graphs within O(nm + n logn) time and a 1/3 (resp., 1/2)-approximation for MaxWP3Packing in general (resp., bipartite) graphs of maximum degree 3 within O(α(n, 3n/2)n) (resp., O(n logn)) time, where α is the inverse Ackerman’s function and n = |V |, m = |E|.