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A deterministic approximation algorithm for the Densest k-Subgraph Problem

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  • Frederic Roupin
  • Alain Billionnet

Abstract

In the Densest k-Subgraph Problem (DSP), we are given an undirected weighted graph G = (V, E) with n vertices (v1,..., vn). We seek to find a subset of k vertices (k belonging to {1,..., n}) which maximises the number of edges which have their two endpoints in the subset. This problem is NP-hard even for bipartite graphs, and no polynomial-time algorithm with a constant performance guarantee is known for the general case. Several authors have proposed randomised approximation algorithms for particular cases (especially when k = n/c, c>1). But derandomisation techniques are not easy to apply here because of the cardinality constraint, and can have a high computational cost. In this paper, we present a deterministic max(d, 8/9c)-approximation algorithm for the DSP (where d is the density of G). The complexity of our algorithm is only that of linear programming. This result is obtained by using particular optimal solutions of a linear programme associated with the classical 0–1 quadratic formulation of DSP.

Suggested Citation

  • 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.
  • Handle: RePEc:ids:ijores:v:3:y:2008:i:3:p:301-314
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    Cited by:

    1. 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.

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