Simulated annealing for graph bisection

We resolve in the affirmative a question of R.B. Boppana and T. Bui: whether simulated annealing can with high probability and in polynomial time, find the optimal bisection of a random graph an G/sub npr/ when p-r=(/spl Theta/n/sup /spl Delta/-2/) for /spl Delta//spl les/2. (The random graph model G/sub 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/sup 2+/spl epsi//) steps with very high probability, provided /spl Delta/>11/6. (By using a slightly modified neighborhood structure, the number of steps can be reduced to O(n/sup 1+/spl epsi//).) We leave open the question of whether annealing is effective for /spl Delta/ in the range 3/2</spl les/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.<<ETX>>