The Pk partition problem and related problems in bipartite graphs

Abstract

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 the optimization version of these problems, denoted by MAXP 3PACKING and MAXINDUCEDP3PACKING, are not in PTAS in bipartite graphs of maximum degree 3. Finally, we propose a 3/2-approximation for MIN3PATHPARTITION in general graphs within O(nm + n2log n) time and a 1/3 (resp., l/2)-approximation for MAXWP3PACKING in general (resp., bipartite) graphs of maximum degree 3 within O(α(n, 3n/2)n) (resp., O(n2 log n)) time, where α is the inverse Ackerman's function and n = |V|, m=|E|. © Springer-Verlag Berlin Heidelberg 2007.