On Independent Sets in Random Graphs

The independence number of a sparse random graph G(n,m) of average degree d = 2m/n is well-known to be (2-epsilon(d))nln(d)/d <=alpha(G(n,m))<=(2+epsilon(d))nln(d)/d with high probability, with epsilon(d) -> 0 in the limit of large d. Moreover, a trivial greedy algorithm w.h.p. finds an independent set of size nln(d)/d, i.e., about half the maximum size. Yet in spite of 30 years of extensive research no efficient algorithm has emerged to produce an independent set with size (1+epsilon)nln(d)/d for any fixed epsilon>0 (independent of both d and n). In this paper we prove that the combinatorial structure of the independent set problem in random graphs undergoes a phase transition as the size k of the independent sets passes the point k similar to nln(d)/d. Roughly speaking, we prove that independent sets of size k > (1+epsilon)nln(d)/d form an intricately rugged landscape, in which local search algorithms seem to get stuck. We illustrate this phenomenon by providing an exponential lower bound for the Metropolis process, a Markov chain for sampling independent sets. (c) 2014 Wiley Periodicals, Inc.