Improved Algorithms for Overlapping and Robust Clustering of Edge-Colored Hypergraphs: An LP-Based Combinatorial Approach
Changyeol Lee, Yongho Shin, Hyung-Chan An
TL;DR
This work tackles edge-colored clustering on hypergraphs, focusing on overlapping variants Local ECC and Global ECC and outlier-robust Robust ECC. It introduces a primal-dual LP-based framework that combines LP accuracy with combinatorial efficiency, yielding a true $(b_ extsf{local}+1)$-approximation for Local ECC and true $(2(b_ extsf{robust}+1))$ and $(2(b_ extsf{global}+1))$-approximations for Robust ECC and Global ECC, respectively. The paper proves tight integrality-gap bounds and relevant inapproximability results, resolving open questions about bicriteria vs. true-approximation tradeoffs, and complements theory with extensive experiments showing fast, high-quality solutions compared to previous LP-rounding and greedy methods. Overall, the approach delivers scalable, high-quality clustering for higher-order, edge-colored data with configurable budgets, enabling practical deployment in domains with overlapping clusters and noise/outliers.
Abstract
Clustering is a fundamental task in both machine learning and data mining. Among various methods, edge-colored clustering (ECC) has emerged as a useful approach for handling categorical data. Given a hypergraph with (hyper)edges labeled by colors, ECC aims to assign vertex colors to minimize the number of edges where the vertex color differs from the edge's color. However, traditional ECC has inherent limitations, as it enforces a nonoverlapping and exhaustive clustering. To tackle these limitations, three versions of ECC have been studied: Local ECC and Global ECC, which allow overlapping clusters, and Robust ECC, which accounts for vertex outliers. For these problems, both linear programming (LP) rounding algorithms and greedy combinatorial algorithms have been proposed. While these LP-rounding algorithms provide high-quality solutions, they demand substantial computation time; the greedy algorithms, on the other hand, run very fast but often compromise solution quality. In this paper, we present an algorithmic framework that combines the strengths of LP with the computational efficiency of combinatorial algorithms. Both experimental and theoretical analyses show that our algorithms efficiently produce high-quality solutions for all three problems: Local, Global, and Robust ECC. We complement our algorithmic contributions with complexity-theoretic inapproximability results and integrality gap bounds, which suggest that significant theoretical improvements are unlikely. Our results also answer two open questions previously raised in the literature.