An improved exact algorithm for the domatic number problem

Tobias Riege, Jörg Rothe, Holger Spakowski, Masaki Yamamoto

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 2.695n<math><msup is="true"><mn is="true">2.695</mn><mi is="true">n</mi></msup></math> (up to polynomial factors) and in polynomial space. This result improves the previous bound of 2.8805n<math><msup is="true"><mn is="true">2.8805</mn><mi is="true">n</mi></msup></math>, which is due to Björklund 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 Δ(G)<math><mi mathvariant="normal" is="true">Δ</mi><mo stretchy="false" is="true">(</mo><mi is="true">G</mi><mo stretchy="false" is="true">)</mo></math> by a randomized polynomial-space algorithm, whose running time is better than the previous bound due to Riege and Rothe whenever Δ(G)⩾5<math><mi mathvariant="normal" is="true">Δ</mi><mo stretchy="false" is="true">(</mo><mi is="true">G</mi><mo stretchy="false" is="true">)</mo><mo is="true">⩾</mo><mn is="true">5</mn></math>. Our new randomized algorithm employs Schöning's approach to constraint satisfaction problems.