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The k-hop connected dominating set problem: approximation and hardness

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Abstract

Let G be a connected graph and k be a positive integer. A vertex subset D of G is a k-hop connected dominating set if the subgraph of G induced by D is connected, and for every vertex v in G there is a vertex u in D such that the distance between v and u in G is at most k. We study the problem of finding a minimum k-hop connected dominating set of a graph (\({\textsc {Min}}k{\hbox {-}\textsc {CDS}}\)). We prove that \({\textsc {Min}}k{\hbox {-}\textsc {CDS}}\) is \(\mathscr {NP}\)-hard on planar bipartite graphs of maximum degree 4. We also prove that \({\textsc {Min}}k{\hbox {-}\textsc {CDS}}\) is \(\mathscr {APX}\)-complete on bipartite graphs of maximum degree 4. We present inapproximability thresholds for \({\textsc {Min}}k{\hbox {-}\textsc {CDS}}\) on bipartite and on (1, 2)-split graphs. Interestingly, one of these thresholds is a parameter of the input graph which is not a function of its number of vertices. We also discuss the complexity of computing this graph parameter. On the positive side, we show an approximation algorithm for \({\textsc {Min}}k{\hbox {-}\textsc {CDS}}\). Finally, when \(k=1\), we present two new approximation algorithms for the weighted version of the problem restricted to graphs with a polynomially bounded number of minimal separators.

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Acknowledgements

We thank the referees for the valuable comments and suggestions.

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Authors and Affiliations

  1. Instituto de Matemática e Estatística, Universidade de São Paulo, São Paulo, Brazil

    Rafael S. Coelho, Phablo F. S. Moura & Yoshiko Wakabayashi

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  1. Rafael S. Coelho
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  2. Phablo F. S. Moura
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  3. Yoshiko Wakabayashi
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Correspondence to Rafael S. Coelho.

Additional information

Research supported by CNPq (Proc. 456792/2014-7), FAPESP (Proc. 2013/03447-6) and MaCLinC project of NUMEC/USP. R.S. Coelho is supported by CAPES, P.F.S. Moura is supported by FAPESP (Proc. 2013/19179-0, 2015/11930-4) and Y. Wakabayashi is partially supported by CNPq Grant (Proc. 306464/2016-0).

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Coelho, R.S., Moura, P.F.S. & Wakabayashi, Y. The k-hop connected dominating set problem: approximation and hardness. J Comb Optim 34, 1060–1083 (2017). https://doi.org/10.1007/s10878-017-0128-y

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