Simulated Annealing for Graph Bisection

We resolve in the a rmative a question of Boppana and Bui: whether simulated annealing can, with high probability and in polynomial time, nd the optimal bisection of a random graph in Gnpr when p?r = (n ?2) for 2. (The random graph model Gnpr speci es a \planted" bisection of density r, separating two n=2vertex subsets of slightly higher density p.) We show that simulated \annealing" at an appropriate xed temperature (i.e., the Metropolis algorithm) nds the unique smallest bisection in O(n2+" ) steps with very high probability, provided > 11=6. (By using a slightly modi ed neighborhood structure, the number of steps can be reduced to O(n1+" ).) We leave open the question of whether annealing is e ective for in the range 3=2 < 11=6, whose lower limit represents the threshold at which the planted bisection becomes lost amongst other random small bisections. It remains open whether hillclimbing (i.e., annealing at temperature 0) solves the same problem.

• Type: Conference
• Year: 1993
• Where: FOCS
• Authors: Mark Jerrum, Gregory B. Sorkin
• Keywords: Bisection, FOCS 1993, Random Graph, Theoretical Computer Science, Unique Smallest Bisection