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Exact and superpolynomial approximation algorithms for the densest k-subgraph problem

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  • Bourgeois, Nicolas
  • Giannakos, Aristotelis
  • Lucarelli, Giorgio
  • Milis, Ioannis
  • Paschos, Vangelis Th.

Abstract

The densest k-subgraph problem is a generalization of the maximum clique problem, in which we are given a graph and a positive integer k, and we search among all the subsets of k vertices of the input graph for the subset which induces the maximum number of edges. densest k-subgraph is a well known optimization problem with various applications as, for example, in the design of public encryption schemes, the evaluation of certain financial derivatives, the identification of communities with similar characteristics, etc. In this paper, we first present algorithms for finding exact solutions for densest k-subgraph which improve upon the standard exponential time complexity of an exhaustive enumeration by creating a link between the computation of an optimum for this problem to the computation of other graph-parameters such as dominating set, vertex cover, longest path, etc. An FPT algorithm is also proposed which considers as a parameter the size of the minimum vertex cover. Finally, we present several approximation algorithms which run in moderately exponential or parameterized time, describing trade-offs between complexity and approximability. In contrast with most of the algorithms in the bibliography, our algorithms need only polynomial space.

Suggested Citation

  • Bourgeois, Nicolas & Giannakos, Aristotelis & Lucarelli, Giorgio & Milis, Ioannis & Paschos, Vangelis Th., 2017. "Exact and superpolynomial approximation algorithms for the densest k-subgraph problem," European Journal of Operational Research, Elsevier, vol. 262(3), pages 894-903.
    • Handle: RePEc:eee:ejores:v:262:y:2017:i:3:p:894-903
    DOI: 10.1016/j.ejor.2017.04.034
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    References listed on IDEAS

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    1. Ulrich Pferschy & Joachim Schauer, 2016. "Approximation of the Quadratic Knapsack Problem," INFORMS Journal on Computing, INFORMS, vol. 28(2), pages 308-318, May.
    2. Gerold Jäger & Anand Srivastav, 2005. "Improved Approximation Algorithms for Maximum Graph Partitioning Problems," Journal of Combinatorial Optimization, Springer, vol. 10(2), pages 133-167, September.
    3. N. Bourgeois & A. Giannakos & G. Lucarelli & I. Milis & V. T. Paschos & O. Pottié, 2012. "The max quasi-independent set problem," Journal of Combinatorial Optimization, Springer, vol. 23(1), pages 94-117, January.
    4. Frederic Roupin & Alain Billionnet, 2008. "A deterministic approximation algorithm for the Densest k-Subgraph Problem," International Journal of Operational Research, Inderscience Enterprises Ltd, vol. 3(3), pages 301-314.
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    Cited by:

    1. Alex Gliesch & Marcus Ritt, 2022. "A new heuristic for finding verifiable k-vertex-critical subgraphs," Journal of Heuristics, Springer, vol. 28(1), pages 61-91, February.
    2. Tesshu Hanaka, 2023. "Computing densest k-subgraph with structural parameters," Journal of Combinatorial Optimization, Springer, vol. 45(1), pages 1-17, January.

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