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Edge-Colored Clustering in Hypergraphs: Beyond Minimizing Unsatisfied Edges
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Studies edge-colored clustering in hypergraphs beyond min-unsatisfied objectives: gives the first hypergraph MaxECC approximation, a 154/405 graph MaxECC improvement, and balance/fairness objectives (p-mean, color-fair min-max, protected-color) with hardness, approximations, and FPT results.
Problem
Edge-colored clustering (ECC) on hypergraphs: color nodes to maximize satisfied hyperedges or minimize unsatisfied ones, while incorporating balance and fairness across edge colors.
Approach
A global random color ordering combined with independent color thresholds from LP rounding, plus an analysis based on mutually independent events, gives the first hypergraph MaxECC approximation and an improved graph factor; new fairness objectives (p-norm of per-color unsatisfied counts, min-max per color, protected-color bicriteria) are added with LP-based rounding.
Key results
- Hypergraph MaxECC approximation of (2/e)^r/(r+1), first for hypergraphs.
- Graph MaxECC approximation improved to 154/405 ≈ 0.38 from prior 0.3622.
- Color-fair MinECC admits a 2-approximation via LP rounding and a k-approximation combinatorially; NP-hard even for k=2 colors.
Measured effects reported in the paper
| What was changed | Where | Measured | Effect |
|---|---|---|---|
| Use a global random color ordering to assign wanted colors instead of assigning colors independently per node | Graph MaxECC · An edge whose endpoints want its color and at most one other color each | conditional edge-satisfaction probability: 1/4 -> 1/3 | Helped |
| Refine global-priority rounding by separating colors into strong and weak using the 2/3 threshold | Graph MaxECC · Graph MaxECC | approximation factor: 0.3622 -> 154/405 | Helped |
| Extend global-priority threshold rounding to hyperedges of rank r | Hypergraph MaxECC · Hypergraph MaxECC (rank r) | approximation factor: (2/e)^r(r+1)^{-1} | Helped |
| Use convex-relaxation rounding for the generalized ℓp unsatisfied-edge objective | pmeanECC · pmeanECC (p ≥ 1) | approximation factor: 2 | Helped |
| Use Lovász-extension rounding for the generalized ℓp unsatisfied-edge objective | pmeanECC · pmeanECC (0 < p < 1) | approximation factor: 2^{1/p} | Helped |
| Round the protected-color LP using thresholds determined by ρ | Protected-color ECC · Protected-color ECC | bicriteria approximation: (1/ρ, 1/(1 - ρ)) | Helped |
| Use a maximal matching in the conflict graph for a combinatorial solution | Color-fair MinECC · Color-fair MinECC | approximation factor: k | Helped |
Surprising findings
- Color-fair MinECC is NP-hard even with k=2 colors, although standard MinECC is polynomial for k=2.
- The natural convex relaxation for p-mean objectives has integrality gap converging to 2 as p grows.
Limitations
- Hypergraph MaxECC approximation factor decays exponentially with rank r.
- Several fairness algorithms rely on convex optimization subroutines.
Experimental setup
Datasets: benchmark ECC datasets
Benchmarks: benchmark ECC datasets
Metrics: number of unsatisfied edges max unsatisfied edges per color
Baselines: Algorithm of prior graph MaxECC 0.3622-approximation standard MinECC LP 2-approximation
Builds on: LP relaxation for MaxECC chromatic correlation clustering submodular vertex cover
Closest papers by meaning
- Optimal LP Rounding and Linear-Time Approximation Algorithms for Clustering Edge-Colored Hypergraphs (ICML 2023 · similarity 0.95)
- Improved Algorithms for Overlapping and Robust Clustering of Edge-Colored Hypergraphs: An LP-Based Combinatorial Approach (NeurIPS 2025 · similarity 0.93)
- Combinatorial Approximations for Cluster Deletion: Simpler, Faster, and Better (ICML 2024 · similarity 0.88)
- Optimal Graph Clustering without Edge Density Signals (NeurIPS 2025 · similarity 0.87)
- Improved Approximation Algorithms for Chromatic and Pseudometric-Weighted Correlation Clustering (NeurIPS 2025 · similarity 0.86)
- Hierarchical Overlapping Clustering on Graphs: Cost Function, Algorithm and Scalability (ICML 2025 · similarity 0.86)
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