Hill-climbing Nds Random Planted Bisections

We analyze the behavior of hill-climbing algorithms for the minimum bisection problem on instances drawn from the \planted bisection" ran-is one of the few problem distributions for which various popular heuris-tic methods, such as simulated annealing, have been proven to succeed. However, it has been open whether these sophisticated methods were necessary , or whether simpler heuristics would also work. Juels 15] made the rst progress towards an answer by showing that simple hill-climbing does suuce for very wide separations between p and q. Here we give a more complete answer. A simple, polynomial-time, hill-climbing algorithm for this problem is given and shown to succeed in nd-ing the planted bisection with high probability if p ? q = ? n ?1=2 ln 4 n. For dense graphs, this matches the condition for optimality of the planted bisection to within a poly-logarithmic factor. Furthermore, we show that a generic randomized hill-climbing algorithm succeeds in nding the planted bisection in polynomial time if p ? q = ? n ?1=4 ln 4 n. This algorithm is a degenerate case of both Metropolis and go-with-the-winners. The range analyzed here properly includes those in 12, 9, 15], extending and unifying those results. Thus, to get a provable distinction between simulated annealing and hill-climbing for natural problems will require considerable progress both on new positive results for SA and new negative results for hill-climbing methods.