Finding Dense Subgraphs in G(n,1/2)

Abstract

Finding the largest clique in random graphs is a well known hard problem. It is known that a random graph G(n, 1/2) almost surely has a clique of size about 2logn. A simple greedy algorithm finds a clique of size logn, and it is a long-standing open problem to find a clique of size (1 + ε)logn in randomized polynomial time. In this paper, we study the generalization of finding the largest subgraph of any given edge density. We show that a simple modification of the greedy algorithm finds a subset of 2logn vertices with induced edge density at least 0.951. We also show that almost surely there is no subset of 2.784logn vertices whose induced edge density is at least 0.951.