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Mixed Dominating Set: A Parameterized Perspective

  • Pallavi Jain
  • M. Jayakrishnan
  • Fahad Panolan
  • Abhishek SahuEmail author
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 10520)

Abstract

In the mixed dominating set (mds) problem, we are given an n-vertex graph G and a positive integer k, and the objective is to decide whether there exists a set \(S \subseteq V(G) \cup E(G)\) of cardinality at most k such that every element \(x \in (V(G) \cup E(G)) \setminus S\) is either adjacent to or incident with an element of S. We show that mds can be solved in time \({7.465^k n^{\mathcal {O}(1)}} \) on general graphs, and in time \(2^{\mathcal {O}(\sqrt{k})} n^{\mathcal {O}(1)}\) on planar graphs. We complement this result by showing that mds does not admit an algorithm with running time \(2^{o(k)} n^{\mathcal {O}(1)}\) unless the Exponential Time Hypothesis (ETH) fails, and that it does not admit a polynomial kernel unless coNP \( \subseteq \mathsf{NP / poly}\). In addition, we provide an algorithm which, given a graph G together with a tree decomposition of width \(\mathsf{tw}\), solves mds in time \(6^{\mathsf{tw}} n^{\mathcal {O}(1)}\). We finally show that unless the Set Cover Conjecture (SeCoCo) fails, mds does not admit an algorithm with running time \(\mathcal {O}((2-\epsilon )^{\mathsf{tw}(G)} n^{\mathcal {O}(1)})\) for any \(\epsilon >0\), where \(\mathsf{tw}(G)\) is the tree-width of G.

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Copyright information

© Springer International Publishing AG 2017

Authors and Affiliations

  • Pallavi Jain
    • 1
  • M. Jayakrishnan
    • 1
  • Fahad Panolan
    • 2
  • Abhishek Sahu
    • 1
    Email author
  1. 1.The Institute of Mathematical SciencesHBNIChennaiIndia
  2. 2.Department of InformaticsUniversity of BergenBergenNorway

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