The MAX-CUT of sparse random graphs

A k-cut of a graph G = (V, E) is a partition of its vertex set into k parts; the size of the k-cut is the number of edges with endpoints in distinct parts. MAX-k-CUT is the optimization problem of finding a k-cut of maximal size and the case where k = 2 (often called MAX-CUT) has attracted a lot of attention from the research community. MAX-CUT---more generally, MAX-k-CUT--- is NP-hard and it appears in many applications under various disguises. In this paper, we consider the MAX-CUT problem on random connected graphs C(n, m) and on Erdös-Rényi random graphs G(n, m). More specifically, we consider the distance from bipartiteness of a graph G = (V,E), the minimum number of edge deletions needed to turn it into a bipartite graph. If we denote this distance DistBip(G), the size of the MAX-CUT of a graph G = (V, E) is clearly given by |E| -- DistBip(G). Fix ε > 0. For random connected graphs, we prove that asymptotically almost surely (a.a.s) DistBip (C(n, m)) ~ m-n/4 whenever m = n + O(n^1−ε). For sparse random graphs G(n, m = n/2 + un^2/3/2) we show that DistBip (G(n, m)) is a.a.s about (2m-n)³/6n² + 1/12 log n − 1/4 log μ for any 1 << μ ≤ O(n^1/3−ε).