Why Colors Make Clustering Harder:Global Integrality Gaps, the Price of Fairness, and Color-Coupled Algorithms in Chromatic Correlation Clustering

2026-04-17

AI Summary

This work addresses the challenges in chromatic correlation clustering (CCC)—specifically, cross-color interference and irreducible color mismatch costs arising from edge-color semantics—by analyzing its linear programming relaxation and identifying additional error sources introduced by color constraints. The authors propose the Color-Coupled Correlation Clustering (C4) framework, which integrates a global integrality gap decomposition theorem and a step-function formulation to quantify color interference costs for the first time. Novel techniques, including the constraint $\sum_c x_{uv}^c \geq L-1$ and interval packing rounding, are introduced to tighten the relaxation. Theoretically, C4 achieves an approximation ratio of 2.0967 for the two-color case and recovers the optimal 2.06 ratio under refined analysis. Experiments confirm that the observed LP gaps align with step-function predictions and demonstrate strong performance even under fairness constraints.

Technology Category

• Constraint Satisfaction and Optimization: Distributed CSP/Optimization
• Search and Optimization: Combinatorial Optimization
• Knowledge Representation and Reasoning: Computational Complexity of Reasoning

Application Category

• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs
• Search and Retrieval-Augmented AI: Web learning to rank, online learning, and counterfactual learning for ranking
• Economics, Online Markets and Human Computation: Data quality aspects of human-annotated datasets

Abstract

Chromatic Correlation Clustering (CCC) extends Correlation Clustering by assigning semantic colors to edges and requiring each cluster to receive a single color label. Unlike standard CC, whose LP relaxation has integrality gap 2 on complete graphs and admits a 2.06-approximation, the analogous LP for CCC has a strict lower bound of 2.11, and the best known LP-rounding algorithm achieves 2.15. We explain this gap by isolating the source of difficulty: cross-edge chromatic interference. Neutral edges, whose color does not match the candidate cluster color, create an irreducible cost absent from standard CC and force any color-independent rounding scheme to pay an additional mismatch penalty. We make four contributions. First, we prove a Global Integrality Gap Decomposition Theorem showing that the gap of any color-independent CCC rounding algorithm equals the standard CC gap plus an irreducible chromatic penalty Delta(L) > 0. Second, we solve the associated min-max problem and derive the staircase formula Delta(L) = ((L-1)/L) Delta_infinity, where Delta_infinity is approximately 0.0734. In particular, the two-color gap is 2.0967, separating CCC from standard CC already at L = 2. Third, we introduce Color-Coupled Correlation Clustering (C4). Adding the valid global constraint sum_c x_uv^c >= L-1 and a correlated interval-packing rounding scheme makes neutral edges behave like classical negative edges, recovering the optimal 2.06 approximation and bypassing the 2.11 lower bound for the uncoupled LP. Fourth, experiments on extremal instances, real multi-relational networks, and fairness benchmarks validate the theory: empirical LP gaps follow the predicted staircase, and C4 matches the unconstrained approximation ratio under fairness constraints.

Problem

• Chromatic Correlation Clustering
• integrality gap
• color constraints
• approximation hardness
• fairness

Innovation

• Chromatic Correlation Clustering
• Integrality Gap Decomposition
• Color-Coupled Algorithms
• Cross-Edge Chromatic Interference
• Fairness in Clustering