On independent sets in random graphs

The independence number of a sparse random graph <i>G(n, m)</i> of average degree <i>d</i> = 2<i>m/n</i> is well-known to be α<i>(G(n, m))</i> ~ 2<i>n</i> ln<i>(d)/d</i> with high probability. Moreover, a trivial greedy algorithm w.h.p. finds an independent set of size (1 + <i>o</i>(1)) · <i>n</i> ln<i>(d)/d</i>, 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)<i>n</i> ln<i>(d)/d</i>, 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 <i>k</i> of the independent sets passes the point <i>k ~ n</i> ln<i>(d)/d.</i> Roughly speaking, we prove that independent sets of size <i>k</i> > (1 + ε)<i>n</i> ln<i>(d)/d</i> 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.