Title: A deterministic approximation algorithm for the Densest k-Subgraph Problem

Authors: Frederic Roupin, Alain Billionnet

Addresses: CEDRIC, CNAM-IIE, 18 allee Jean Rostand 91025 Evry cedex, France. ' CEDRIC, CNAM-IIE, 18 allee Jean Rostand 91025 Evry cedex, France

Abstract: In the Densest k-Subgraph Problem (DSP), we are given an undirected weighted graph G = (V, E) with n vertices (v₁,..., v_n). 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.

Keywords: approximation algorithms; complexity; linear programming; quadratic programming; densest k-subgraph problem.

DOI: 10.1504/IJOR.2008.017534

International Journal of Operational Research, 2008 Vol.3 No.3, pp.301 - 314

Published online: 15 Mar 2008 *

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