Cut Size Statistics of Graph Bisection Heuristics

We investigate the statistical properties of cut sizes generated by heuristic algorithms which solve the graph bisection problem approximately. On an ensemble of sparse random graphs, we find empirically that the distribution of the cut sizes found by “local” algorithms becomes peaked as the number of vertices in the graphs becomes large. Evidence is given that this distribution tends toward a Gaussian whose mean and variance scales linearly with the number of vertices of the graphs. Given the distribution of cut sizes associated with each heuristic, we provide a ranking procedure that takes into account both the quality of the solutions and the speed of the algorithms. This procedure is demonstrated for a selection of local graph bisection heuristics.