Simulated Annealing for Graph Bisection

W e resolve in the aff irmative a question of Boppana and B u i : whether simulated annealing can, w i th high probability and in polynomial t i m e , f ind the optimal bisection of a r a n d o m graph in Gnpr when p r = O(n*-’) f o r A 5 2 . ( T h e random graph model Gnpr specifies a “planted” bisection of density r , separating t w o n / 2 v e r t e x subsets of slightly higher density p . ) W e show that simulated “annealing” at a n appropriate fixed temperature (i.e., the Metropolis algorithm) f inds the unique smallest bisection in O(n2+‘) steps with very high probability, provided A > 1116. ( B y using a slightly modified neighborhood structure, the number of steps can be reduced t o O(n’+‘).) W e leave open the question of whether annealing i s effective f o r A in the range 312 < A 5 1116, whose lower l imit represents t he threshold at which the planted bisection becomes lost amongs t other r a n d o m smal l bisections. It also remains open whether hillclimbing (i.e., annealing at temperature 0) solves the same problem.