Edge-Colored Clustering in Hypergraphs: Beyond Minimizing Unsatisfied Edges

2025-02-18

AI Summary

This paper investigates clustering in edge-colored hypergraphs—i.e., partitioning vertices (nodes) into color classes based on the types of multi-way interactions they participate in. Unlike conventional approaches that minimize unsatisfied hyperedges (those containing vertices of conflicting colors), we systematically generalize the paradigm: (1) we design the first constant-factor approximation algorithm for maximizing satisfied hyperedges in hypergraphs; (2) we improve the best-known approximation ratio for the corresponding graph problem; and (3) we introduce a novel objective function balancing group-level balance and individual fairness, establishing its NP-hardness, constant-factor approximability, and fixed-parameter tractability (FPT). Our methodology integrates hypergraph theory, combinatorial optimization, approximation algorithms, and parameterized complexity analysis.

Technology Category

• Constraint Satisfaction and Optimization: Satisfiability
• Search and Optimization: Combinatorial Optimization
• Machine Learning: Clustering

Application Category

• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs
• Economics, Online Markets and Human Computation: Fairness, privacy, and diversity in economic environments
• User Modeling, Personalization and Recommendation: Fairness-aware retrieval and ranking

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.

Problem

• Clustering edge-colored hypergraphs
• Maximizing satisfied hyperedges
• Incorporating balance and fairness

Innovation

• First hypergraph approximation algorithm
• Improved graph approximation factor
• Balanced and fair objective functions