The Metropolis algorithm for graph bisection

We resolve in the affirmative a question of Boppana and Bui: whether simulated annealing can, with high probability and in polynomial time, find the optimal bisection of a random graph in G(npr) when p - r = Theta(n(Delta-2)) for Delta less than or equal to 2. (The random graph model G(npr) specifies a "planted" bisection of density r, separating two n/2-vertex subsets of slightly higher density p.) We show that simulated "annealing" at an appropriate fixed temperature (i.e., the Metropolis algorithm) finds the unique smallest bisection in O(n(2+epsilon)) steps with very high probability, provided Delta > 11/6. (By using a slightly modified neighborhood structure, the number of steps can be reduced to O(n(1+epsilon)).) We leave open the question of whether annealing is effective for Delta in the range 3/2 < Delta less than or equal to 11/6, whose lower limit represents the threshold at which the planted bisection becomes lost amongst other random small bisections. It also remains open whether hillclimbing (i.e., annealing at temperature 0) solves the same problem; towards the latter result, Juels has recently extended our analysis and shown that random hillclimbing finds the minimum bisection with constant probability, when p - r = Omega(1) (corresponding to Delta = 2). (C) 1998 Elsevier Science B.V. All rights reserved.