A deterministic approximation algorithm for the Densest k-Subgraph Problem

摘要

In the Densest k-Subgraph Problem (DSP), we are given an undirected weighted graph G = (V, E) with n vertices (v{sub}1, ..., v{sub}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.