MAX-CUT on Samplings of Dense Graphs

The maximum cut problem finds a partition of a graph that maximizes the number of crossing edges. When the graph is dense or is sampled based on certain planted assumptions, there exist polynomial-time approximation schemes that given a fixed epsilon > 0, find a solution whose value is at least 1 - epsilon of the optimal value. This paper presents another random model relating to both successful cases. Consider an n-vertex graph G whose edges are sampled from an unknown dense graph H independently with probability p = Omega(1/root log n); this input graph G has O(n(2)/root log n) edges and is no longer dense. We show how to modify a PTAS by de la Vega for dense graphs to find an (1 - epsilon)-approximate solution for G. Although our algorithm works for a very narrow range of sampling probability p, the sampling model itself generalizes the planted models fairly well.