Going for Speed: Sublinear Algorithms for Dense r-CSPs

TL;DR

This paper introduces new sublinear, parallel algorithms for dense r-CSPs that significantly improve runtime efficiency and approximation quality, especially for Max-Cut and k-Correlation Clustering problems.

Contribution

The paper presents the first sublinear, parallel algorithms for dense r-CSPs with optimal n dependence and near-optimal epsilon dependence, improving previous bounds.

Findings

Achieves O(n/ε^2) + 2^{O(1/ε^2)} runtime for Boolean r-CSPs.

Provides exponential improvement for Max-Cut approximation dependence on 1/ε.

Improves runtime for k-Correlation Clustering to O(k^4 n / ε^2) + k^{O(1/ε^2)}.

Abstract

We give new sublinear and parallel algorithms for the extensively studied problem of approximating n-variable r-CSPs (constraint satisfaction problems with constraints of arity r up to an additive error. The running time of our algorithms is O(n/\epsilon^2) + 2^O(1/\epsilon^2) for Boolean r-CSPs and O(k^4 n / \epsilon^2) + 2^O(log k / \epsilon^2) for r-CSPs with constraints on variables over an alphabet of size k. For any constant k this gives optimal dependence on n in the running time unconditionally, while the exponent in the dependence on 1/\epsilon is polynomially close to the lower bound under the exponential-time hypothesis, which is 2^\Omega(\epsilon^(-1/2)). For Max-Cut this gives an exponential improvement in dependence on 1/\epsilon compared to the sublinear algorithms of Goldreich, Goldwasser and Ron (JACM'98) and a linear speedup in n compared to the algorithms of Mathieu and Schudy (SODA'08). For the maximization version of k-Correlation Clustering problem our running time is O(k^4 n / \epsilon^2) + k^O(1/\epsilon^2), improving the previously best n k^{O(1/\epsilon^3 log k/\epsilon) by Guruswami and Giotis (SODA'06).