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A deterministic approximation algorithm for the Densest k-Subgraph Problem
by Frederic Roupin, Alain Billionnet
International Journal of Operational Research (IJOR), Vol. 3, No. 3, 2008


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.

Online publication date: Sat, 15-Mar-2008


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