An improved exact algorithm for the domatic number problem

  • Authors:

  • Tobias Riege;Jörg Rothe;Holger Spakowski;Masaki Yamamoto

  • Affiliations:

  • Institut für Informatik, Heinrich-Heine-Universität Düsseldorf, Düsseldorf, Germany;Institut für Informatik, Heinrich-Heine-Universität Düsseldorf, Düsseldorf, Germany;Institut für Informatik, Heinrich-Heine-Universität Düsseldorf, Düsseldorf, Germany;Department of Mathematical and Computing Science, Tokyo Institute of Technology, Tokyo, Japan

  • Venue:
  • Information Processing Letters
  • Year:
  • 2007

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Abstract

The 3-domatic number problem asks whether a given graph can be partitioned into three dominating sets. We prove that this problem can be solved by a deterministic algorithm in time 2.695^n (up to polynomial factors) and in polynomial space. This result improves the previous bound of 2.8805^n, which is due to Bjorklund and Husfeldt. To prove our result, we combine an algorithm by Fomin et al. with Yamamoto's algorithm for the satisfiability problem. In addition, we show that the 3-domatic number problem can be solved for graphs G with bounded maximum degree @D(G) by a randomized polynomial-space algorithm, whose running time is better than the previous bound due to Riege and Rothe whenever @D(G)=5. Our new randomized algorithm employs Schoning's approach to constraint satisfaction problems.