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Dual Domination

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Combinatorial Algorithms (IWOCA 2019)

Part of the book series: Lecture Notes in Computer Science ((LNTCS,volume 11638))

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Abstract

Inspired by the feedback scenario, which characterizes online social networks, we introduce a novel domination problem, which we call Dual Domination (DD). We assume that the nodes in the input network are partitioned into two categories: Positive nodes (\(V^+\)) and negative nodes (\(V^-\)). We are looking for a set \(D\subseteq V^+\) that dominates the largest number of positive nodes while avoiding as many negative nodes as possible. In particular, we study the Maximum Bounded Dual Domination (MBDD) problem, where given a bound k, the problem is to find a set \(D\subseteq V^+\), which maximizes the number of nodes dominated in \(V^+,\) dominating at most k nodes in \(V^-.\) We show that the MBDD problem is hard to approximate to a factor better than \((1-1/e)\). We give a polynomial time approximation algorithm with approximation guaranteed \((1-e^{-1/\varDelta })\), where \(\varDelta \) represents the maximum number of neighbors in \(V^+\) of any node in \(V^-.\) Furthermore, we give an \(O(|V|k^2)\) time algorithm to solve the problem on trees.

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Notes

  1. 1.

    We can assume that no edge exists between two nodes in \(V^-\), since such edges are irrelevant for our problem.

  2. 2.

    (Notice that \(S^0 \subseteq \mathtt{OPT}\) and since \(\mathtt{OPT}\) is an optimal solution we have \(P_{\mathtt{OPT}} \subseteq \mathtt{OPT}\) and then \(P^0 \subseteq \mathtt{OPT}\).)

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Authors and Affiliations

  1. Università della Campania “Luigi Vanvitelli”, Caserta, Italy

    Gennaro Cordasco

  2. Department of Computer Science, Università di Salerno, Fisciano, Italy

    Luisa Gargano & Adele Anna Rescigno

Authors
  1. Gennaro Cordasco
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  2. Luisa Gargano
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  3. Adele Anna Rescigno
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Corresponding author

Correspondence to Gennaro Cordasco .

Editor information

Editors and Affiliations

  1. Arizona State University, Tempe, AZ, USA

    Charles J. Colbourn

  2. University of Pisa, Pisa, Italy

    Roberto Grossi

  3. University of Pisa, Pisa, Italy

    Nadia Pisanti

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Cordasco, G., Gargano, L., Rescigno, A.A. (2019). Dual Domination. In: Colbourn, C., Grossi, R., Pisanti, N. (eds) Combinatorial Algorithms. IWOCA 2019. Lecture Notes in Computer Science(), vol 11638. Springer, Cham. https://doi.org/10.1007/978-3-030-25005-8_14

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  • DOI: https://doi.org/10.1007/978-3-030-25005-8_14

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  • Publisher Name: Springer, Cham

  • Print ISBN: 978-3-030-25004-1

  • Online ISBN: 978-3-030-25005-8

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