Edge-Colored Clustering in Hypergraphs: Beyond Minimizing Unsatisfied Edges

TL;DR

This paper introduces new algorithms and theoretical insights for clustering edge-colored hypergraphs, focusing on maximizing satisfied edges and incorporating fairness, extending beyond traditional minimization objectives.

Contribution

It presents the first approximation algorithm for hypergraph satisfaction maximization and extends clustering objectives to include balance and fairness considerations.

Findings

Developed the first approximation algorithm for hypergraph satisfaction maximization.

Improved the approximation factor for graph clustering.

Established hardness and fixed-parameter tractability results for new objectives.

Abstract

We consider a framework for clustering edge-colored hypergraphs, where the goal is to cluster (equivalently, to color) objects based on the primary type of multiway interactions they participate in. One well-studied objective is to color nodes to minimize the number of unsatisfied hyperedges -- those containing one or more nodes whose color does not match the hyperedge color. We motivate and present advances for several directions that extend beyond this minimization problem. We first provide new algorithms for maximizing satisfied edges, which is the same at optimality but is much more challenging to approximate, with all prior work restricted to graphs. We develop the first approximation algorithm for hypergraphs, and then refine it to improve the best-known approximation factor for graphs. We then introduce new objective functions that incorporate notions of balance and fairness, and provide new hardness results, approximations, and fixed-parameter tractability results.