Abstract

Given an n-vertex graph G and a parameter k, we are to find a k-vertex subgraph with the maximum number of edges. This problem is NP -hard. We show that the problem remains NP -hard even when the maximum degree in G is three. When G contains a k-clique, we give an algorithm that for any ffl ? 0 finds a k-vertex subgraph with at least (1 \Gamma ffl) \Gamma k 2 \Delta edges, in time n O((1+log n k )=ffl) . We study the applicability of semidefinite programming for approximating the dense k-subgraph problem. Our main result in this respect is negative, showing that for k ' n 1=3 , semidefinite programs fail to distinguish between graphs that contain k- cliques and graphs in which the densest k-vertex subgraph has average degree below log n. 1 Introduction We study the Densest k-subgraph (DkS) maximization problem, of computing the densest k-vertex subgraph of a given simple graph. That is, given an input graph G and a parameter k, we are interested in finding a set of...

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