On independent sets in random graphs

Abstract

The independence number of a sparse random graph G(n, m) of average degree d = 2m/n is well-known to be α(G(n, m)) ∼ 2n ln(d)/d with high probability. Moreover, a trivial greedy algorithm w.h.p. finds an independent set of size (1 + o(1)) · n ln(d)/d, i.e., half the maximum size. Yet in spite of 30 years of extensive research no efficient algorithm has emerged to produce an independent set with (1 + ε)n ln(d)/d, for any fixed ε > 0. 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 ∼ n ln(d)/d. Roughly speaking, we prove that independent sets of size k > (1 + ε)n ln(d)/d form an intricately ragged landscape, in which local search algorithms are bound to get stuck. We illustrate this phenomenon by providing an exponential lower bound for the Metropolis process, a Markov chain for sampling independents sets.