The peculiar phase structure of random graph bisection

The mincut graph bisection problem involves partitioning the $n$ vertices of a graph into disjoint subsets, each containing exactly $n/2$ vertices, while minimizing the number of “cut” edges with an endpoint in each subset. When considered over sparse random graphs, the phase structure of the graph bisection problem displays not only certain familiar properties but also some surprises. It is known that when the mean degree is below the critical value of $2log2$⁠, the cutsize is zero with high probability. We study how the minimum cutsize increases with mean degree above this critical threshold, finding a new analytical upper bound that improves considerably upon previous bounds. Combined with recent results on expander graphs, our bound suggests the unusual scenario that random graph bisection is replica symmetric up to and beyond the critical threshold, with a replica symmetry breaking transition possibly taking place above the threshold. An intriguing algorithmic consequence is that although the problem is NP-hard, we can conceivably find near-optimal cutsizes (whose ratio to the optimal value approaches 1 asymptotically) in polynomial time for typical instances near the phase transition.