The Metropolis algorithm for graph bisection

Abstract

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 script G signnpr when p - r = Θ(nΔ-2) for Δ ≤ 2. (The random graph model script G signnpr 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(n2+ε) steps with very high probability, provided Δ > 11/6. (By using a slightly modified neighborhood structure, the number of steps can be reduced to O(n1+ε).) We leave open the question of whether annealing is effective for Δ in the range 2/3