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Branch and Recharge: Exact Algorithms for Generalized Domination

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Abstract

In this paper we present branching algorithms for infinite classes of problems.

The novelty in the design and analysis of our branching algorithms lies in the fact that the weights are redistributed over the graph by the algorithms. Our particular setting to make this idea work is a combination of a branching approach with a recharging mechanism. We call it Branch & Recharge. To demonstrate this approach we consider a generalized domination problem.

Let σ and ϱ be two nonempty sets of nonnegative integers. A vertex subset S⊆V of an undirected graph G=(V(G),E(G)) is called a (σ,ϱ)-dominating set of G if |N(v)∩S|∈σ for all v∈S and |N(v)∩S|∈ϱ for all v∈V∖S. This notion generalizes many domination-type graph invariants.

We present Branch & Recharge algorithms enumerating all (σ,ϱ)-dominating sets of an input graph G in time O *(c n) for some c<2, if σ is successor-free, i.e., it does not contain two consecutive integers, and either both σ and ϱ are finite, or one of them is finite and σ∩ϱ=∅. Our algorithm implies a non trivial upper bound of O *(c n) on the number of (σ,ϱ)-dominating sets in an n-vertex graph under the above conditions on σ and ϱ.

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Authors and Affiliations

  1. Department of Informatics, University of Bergen, 5020, Bergen, Norway

    Fedor V. Fomin

  2. School of Engineering and Computing Sciences, Durham University, South Road, DH1 3LE, Durham, UK

    Petr A. Golovach

  3. Department of Applied Mathematics, and Institute for Theoretical Computer Science, Charles University, Malostranské nám. 25, 118 00, Praha 1, Czech Republic

    Jan Kratochvíl

  4. Laboratoire d’Informatique Théorique et Appliquée, Université Paul Verlaine—Metz, 57045, Metz Cedex 01, France

    Dieter Kratsch

  5. Laboratoire d’Informatique Fondamentale d’Orléans, Université d’Orléans, 45067, Orléans Cedex 2, France

    Mathieu Liedloff

Authors
  1. Fedor V. Fomin
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  2. Petr A. Golovach
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  3. Jan Kratochvíl
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  4. Dieter Kratsch
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  5. Mathieu Liedloff
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Corresponding author

Correspondence to Dieter Kratsch.

Additional information

A preliminary version of the paper appeared in the proceedings of WADS 2007 [7].

Research of J. Kratochvíl was supported by Czech research projects MSM0021620838 and 1M0545.

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Fomin, F.V., Golovach, P.A., Kratochvíl, J. et al. Branch and Recharge: Exact Algorithms for Generalized Domination. Algorithmica 61, 252–273 (2011). https://doi.org/10.1007/s00453-010-9418-9

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  • DOI: https://doi.org/10.1007/s00453-010-9418-9

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