The Pk Partition Problem and Related Problems in Bipartite Graphs

  • Authors:

  • Jérôme Monnot;Sophie Toulouse

  • Affiliations:

  • CNRS LAMSADE - UMR 7024, Université Paris-Dauphine, Place du Maréchal De Lattre de Tassigny, 75775 Paris Cedex 16, France;LIPN - UMR CNRS 7030, Institut Galilée, Université Paris 13, 99 av. Jean-Baptiste Clément, 93430 Villetaneuse, France

  • Venue:
  • SOFSEM '07 Proceedings of the 33rd conference on Current Trends in Theory and Practice of Computer Science
  • Year:
  • 2007

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Abstract

In this paper, we continue the investigation proposed in [15] about the approximability of Pkpartition problems, but focusing here on their complexity. More precisely, we prove that the problem consisting of deciding if a graph of nkvertices has nvertex disjoint simple paths {P1, 茂戮驴 ,Pn} such that each path Pihas kvertices is NP-complete, even in bipartite graphs of maximum degree 3. Note that this result also holds when each path Piis chordless in G[V(Pi)]. Then, we prove that the optimization version of these problems, denoted by MaxP3Packingand MaxInducedP3Packing, are not in PTASin bipartite graphs of maximum degree 3. Finally, we propose a 3/2-approximation for Min3-PathPartitionin general graphs within O(nm+ n2logn) time and a 1/3 (resp., 1/2)-approximation for MaxWP3Packingin general (resp., bipartite) graphs of maximum degree 3 within O(茂戮驴(n,3n/2)n) (resp., O(n2logn)) time, where 茂戮驴is the inverse Ackerman's function and n= |V|, m= |E|.