Sublinear Time and Space Algorithms for Correlation Clustering via Sparse-Dense Decompositions

TL;DR

This paper introduces sublinear algorithms for correlation clustering that operate efficiently in time and space, using novel sparse-dense graph decompositions, achieving constant approximation guarantees.

Contribution

It presents the first sublinear-time and semi-streaming algorithms for correlation clustering with approximation guarantees, leveraging a new connection to sparse-dense graph decompositions.

Findings

Sublinear-time algorithm achieves constant approximation in O(n log^2 n) time.

Semi-streaming algorithm achieves constant approximation in O(n log n) space.

First application of sparse-dense decompositions beyond graph coloring.

Abstract

We present a new approach for solving (minimum disagreement) correlation clustering that results in sublinear algorithms with highly efficient time and space complexity for this problem. In particular, we obtain the following algorithms for $n$-vertex $(+/-)$-labeled graphs $G$: -- A sublinear-time algorithm that with high probability returns a constant approximation clustering of $G$ in $O(n\log^2{n})$ time assuming access to the adjacency list of the $(+)$-labeled edges of $G$ (this is almost quadratically faster than even reading the input once). Previously, no sublinear-time algorithm was known for this problem with any multiplicative approximation guarantee. -- A semi-streaming algorithm that with high probability returns a constant approximation clustering of $G$ in $O(n\log{n})$ space and a single pass over the edges of the graph $G$ (this memory is almost quadratically smaller than input size). Previously, no single-pass algorithm with $o(n^2)$ space was known for this problem with any approximation guarantee. The main ingredient of our approach is a novel connection to sparse-dense graph decompositions that are used extensively in the graph coloring literature. To our knowledge, this connection is the first application of these decompositions beyond graph coloring, and in particular for the correlation clustering problem, and can be of independent interest.