Combinatorial bounds via measure and conquer: Bounding minimal dominating sets and applications

Authors:
Fedor V. Fomin;Fabrizio Grandoni;Artem V. Pyatkin;Alexey A. Stepanov
Affiliations:
University of Bergen, Bergen, Norway;Università di Roma “Tor Vergata”, Roma, Italy;Sobolev Institute of Mathematics, Novosibirsk, Russia;University of Bergen, Bergen, Norway
Venue:
ACM Transactions on Algorithms (TALG)
Year:
2008

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Abstract

We provide an algorithm listing all minimal dominating sets of a graph on n vertices in time O(1.7159n). This result can be seen as an algorithmic proof of the fact that the number of minimal dominating sets in a graph on n vertices is at most 1.7159n, thus improving on the trivial O(2n/&sqrt;n) bound. Our result makes use of the measure-and-conquer technique which was recently developed in the area of exact algorithms. Based on this result, we derive an O(2.8718n) algorithm for the domatic number problem.