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 $p=\Omega(1/\sqrt{\log n});$ this input graph $G$ has $O(n^{2}/\sqrt{\log n})$ 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.