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Journal Articles International Journal of Operational Research Year : 2008

A Deterministic Approximation Algorithm for the Densest k-Subgraph Problem

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.

Dates and versions

hal-00596169 , version 1 (26-05-2011)

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Alain Billionnet, Frédéric Roupin. A Deterministic Approximation Algorithm for the Densest k-Subgraph Problem. International Journal of Operational Research, 2008, 3 (3), pp.301-314. ⟨10.1504/IJOR.2008.017534⟩. ⟨hal-00596169⟩

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