Go with the winners for graph bisection

We analyze “Go with the winners” for graph bisection. We introduce a weaker version of expansion called ‘<local expansion”. We show that “Go with the winners” works well in any search space whose sub-graphs with solutions at least as good as a certain threshold have local expansion, and where these sub-graphs do not shrink more than by a polynomial factor when the threshold is incremented. We give a general technique for showing that solution spaces for random instances of problems have local expansion. We apply this technique to the minimum bisection problem for random graphs. We conclude that “Go with the winners” approximates the best solution in random graphs of certain densities with planted bisections in polynomial time and finds the optimal solution in quasi-polynomial time. Although other methods also solve this problem for the same densities, the set of tools we develop may be useful in the analysis of similar problems. In particular, our results easily extend to hypergraph bisection, whereas it is not clear whether the other known techniques do.