The P k Partition Problem and Related Problems in Bipartite Graphs

Abstract

In this paper, we continue the investigation proposed in [15] about the approximability of P k 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 {P 1, ⋯ ,P n } such that each path P i has k vertices is NP-complete, even in bipartite graphs of maximum degree 3. Note that this result also holds when each path P i is chordless in G[V(P i )]. Then, we prove that the optimization version of these problems, denoted by Max P 3 Packing and MaxInduced P 3 Packing, are not in PTAS in bipartite graphs of maximum degree 3. Finally, we propose a 3/2-approximation for Min3-PathPartition in general graphs within O(nm + n 2logn) time and a 1/3 (resp., 1/2)-approximation for MaxW P 3 Packing in general (resp., bipartite) graphs of maximum degree 3 within O(α(n,3n/2)n) (resp., O(n 2logn)) time, where α is the inverse Ackerman’s function and n = |V|, m = |E|.KeywordsSpan TreeBipartite GraphMaximum DegreePacking ProblemGeneral GraphThese keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.