Optimal Sample Complexity for Max-Cut

We determine the optimal sample size for constructing value coresets for Max-Cut via uniform vertex sampling. That is, sampling $O(\varepsilon^{-2})$ vertices uniformly at random in $n$-vertex graphs with edge weights in $[0,1]$ yields a reweighted subgraph whose Max-Cut value is an additive $\varepsilon n^2$-approximation to that of the original graph, with constant success probability. This improves the previously known $O(\varepsilon^{-4})$ upper bound and matches the known lower bound. The same guarantee holds also for importance sampling, where vertices are sampled proportionally to their weighted degrees, and we further prove a matching lower bound for this sampling method. Our analysis extends directly to metric Max-Cut, showing that importance sampling of $O(\varepsilon^{-2})$ vertices yields $(1\pm\varepsilon)$-approximation to the Max-Cut value, and to Max-Cut in graphs with edge weights in $[0,1]$ and average weighted degree $\bar d$, achieving the same $(1\pm\varepsilon)$-approximation by either uniform or importance sampling of $O(\varepsilon^{-2}n/\bar d)$ vertices. We additionally show how to use this coreset to recover a cut of expected value at least $OPT-\varepsilon n^2$, by making $O(n\varepsilon^{-2})$ additional edge-weight queries. Finally, our result extends to Max-$r$-Cut, giving an additive $\varepsilon n^2$-approximation using a sample of $O(\varepsilon^{-2}\log(2r))$ vertices.