Sublinear Algorithms for MAXCUT and Correlation Clustering

TL;DR

This paper develops near-optimal sublinear algorithms for MAXCUT and correlation clustering, constructing core-sets and streaming algorithms that work efficiently on graphs with varying densities, bridging the gap between dense and sparse cases.

Contribution

It introduces size-efficient core-sets and streaming algorithms for MAXCUT and correlation clustering on graphs with intermediate densities, avoiding regularity assumptions and matching ETH-based lower bounds.

Findings

Core-sets of size n^{1-\u03b4} are optimal under ETH.

A 2-pass streaming + approximation algorithm uses  O(n^{1-\u03b4}) space.

The approach works for graphs with average degree n^b4, bridging previous dense and sparse graph results.

Abstract

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 $\Omega(n)$ 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 $\tilde{O}(n^{1-\delta})$ for both the problems, on graphs with average degree $n^{\delta}$ (for any $\delta >0$). 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 $(1+\epsilon)$-approximation for both problems; the algorithm uses $\tilde{O}(n^{1-\delta})$ space, for graphs of average degree $n^\delta$.