Optimal LP Rounding and Linear-Time Approximation Algorithms for Clustering Edge-Colored Hypergraphs

TL;DR

This paper improves approximation algorithms for clustering edge-colored hypergraphs, providing tighter guarantees, matching integrality gaps, and establishing hardness results, with applications in machine learning and data mining.

Contribution

It introduces improved LP-based approximation guarantees, a linear-time combinatorial 2-approximation, and new connections to classical problems like vertex cover.

Findings

Enhanced approximation guarantees for hypergraph clustering

Matching integrality gap proofs for the LP relaxations

A linear-time combinatorial 2-approximation algorithm

Abstract

We study the approximability of an existing framework for clustering edge-colored hypergraphs, which is closely related to chromatic correlation clustering and is motivated by machine learning and data mining applications where the goal is to cluster a set of objects based on multiway interactions of different categories or types. We present improved approximation guarantees based on linear programming, and show they are tight by proving a matching integrality gap. Our results also include new approximation hardness results, a combinatorial 2-approximation whose runtime is linear in the hypergraph size, and several new connections to well-studied objectives such as vertex cover and hypergraph multiway cut.