Sublinear Algorithms for MAXCUT and Correlation Clustering

We study sublinear algorithms for two fundamental graph problems, MAXCUT and correlation clustering. Our focus is on constructing core-sets as well as developing streaming algorithms for these problems. Constant space algorithms are known for dense graphs for these problems, while Ω(n) lower bounds exist (in the streaming setting) for sparse graphs. Our goal in this paper is to bridge the gap between these extremes. Our first result is to construct core-sets of size Õ(n) for both the problems, on graphs with average degree n (for any δ > 0). This turns out to be optimal, under the exponential time hypothesis (ETH). Our core-set analysis is based on studying random-induced sub-problems of optimization problems. To the best of our knowledge, all the known results in our parameter range rely crucially on near-regularity assumptions. We avoid these by using a biased sampling approach, which we analyze using recent results on concentration of quadratic functions. We then show that our construction yields a 2-pass streaming (1 + ε)-approximation for both problems; the algorithm uses Õ(n) space, for graphs of average degree n.