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International Conference on Algorithmic Applications in Management

AAIM 2016: Algorithmic Aspects in Information and Management pp 162–172Cite as

Near-Optimal Dominating Sets via Random Sampling

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Part of the book series: Lecture Notes in Computer Science ((LNTCS,volume 9778))

Abstract

A minimum dominating set (MDS) of a simple undirected graph G is a dominating set with the smallest possible cardinality among all dominating sets of G and the MDS problem represents the problem of finding the MDS in a given input graph.

Motivated by the transportation, social and biological networks from a control theory perspective, the main result of this paper is the assertion that a random sampling is usable to find a near-optimal dominating set in an arbitrary connected graph. Our result might be of significance in particular contexts where exact algorithms cannot be run, e.g. in distributed computation environments. Moreover, the analysis of the relationship between the time complexity and the approximation ratio of the corresponding sequential algorithm exposes the counterintuitive behavior.

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Notes

  1. 1.

    For the time complexity, it is used a modified big-Oh notation \(O^*\) throughout this paper. For functions f and g, we write \(f(n) = O^*(g(n))\) if \(f(n) = O(g(n)poly(n))\), where poly(n) is a polynomial.

  2. 2.

    We use the Stirling’s formula in the form \(n! \approx \sqrt{2 \pi n}(n/e)^n\).

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Acknowledgement

The author gratefully acknowledge prof. J. Hromkovič and M. Demetrian for their valuable comments on the manuscript.

This research is supported by the MESRS of the Slovak Republic under the grants KEGA 047STU-4/2016 and VEGA 1/0026/16, respectively.

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Authors and Affiliations

  1. Institute of Information Engineering, Automation and Mathematics, Faculty of Chemical and Food Technology, Slovak University of Technology in Bratislava, Radlinského 9, 812 37, Bratislava, Slovak Republic

    Martin Nehéz

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  1. Martin Nehéz
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Correspondence to Martin Nehéz .

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Editors and Affiliations

  1. Università degli Studi di Bergamo , Bergamo, Italy

    Riccardo Dondi

  2. Université de Nantes , Nantes, France

    Guillaume Fertin

  3. Università degli Studi di Milano-Bicocca , Milano, Italy

    Giancarlo Mauri

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Nehéz, M. (2016). Near-Optimal Dominating Sets via Random Sampling. In: Dondi, R., Fertin, G., Mauri, G. (eds) Algorithmic Aspects in Information and Management. AAIM 2016. Lecture Notes in Computer Science(), vol 9778. Springer, Cham. https://doi.org/10.1007/978-3-319-41168-2_14

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  • DOI: https://doi.org/10.1007/978-3-319-41168-2_14

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  • Publisher Name: Springer, Cham

  • Print ISBN: 978-3-319-41167-5

  • Online ISBN: 978-3-319-41168-2

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