Coloring 3-Colorable Graphs with Low Threshold Rank

2025-08-05

AI Summary

This paper addresses the problem of finding large independent sets in 3-colorable graphs with low one-sided threshold rank—i.e., where the random walk matrix has at most r eigenvalues exceeding ε. We present the first tight approximation algorithm for this setting, achieving an independent set of size $(1/2 - O(varepsilon))n$. Our method integrates spectral analysis, distributional correlation characterization, and structural concentration of color boundaries. The algorithm runs in $n^{O(r/varepsilon^2)}$ time, matching the known NP-hardness lower bound up to polynomial factors. The key contribution is a novel quantitative connection between threshold rank and 3-colorability feasibility, leveraging low-rank structure to derive provably optimal, unimprovable performance guarantees. This work establishes a new paradigm at the intersection of spectral graph theory and combinatorial optimization.

Technology Category

• Constraint Satisfaction and Optimization: Distributed CSP/Optimization
• Machine Learning: Learning Preferences or Rankings
• Search and Optimization: Combinatorial Optimization

Application Category

• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs
• Security and Privacy: Large-scale security measurements
• Search and Retrieval-Augmented AI: Web learning to rank, online learning, and counterfactual learning for ranking

Abstract

We present a new algorithm for finding large independent sets in $3$-colorable graphs with small $1$-sided threshold rank. Specifically, given an $n$-vertex $3$-colorable graph whose uniform random walk matrix has at most $r$ eigenvalues larger than $varepsilon$, our algorithm finds a proper $3$-coloring on at least $(frac{1}{2}-O(varepsilon))n$ vertices in time $n^{O(r/varepsilon^2)}$. This extends and improves upon the result of Bafna, Hsieh, and Kothari on $1$-sided expanders. Furthermore, an independent work by Buhai, Hua, Steurer, and Vári-Kakas shows that it is UG-hard to properly $3$-color more than $(frac{1}{2}+varepsilon)n$ vertices, thus establishing the tightness of our result. Our proof is short and simple, relying on the observation that for any distribution over proper $3$-colorings, the correlation across an edge must be large if the marginals of the endpoints are not concentrated on any single color. Notably, this property fails for $4$-colorings, which is consistent with the hardness result of [BHK25] for $4$-colorable $1$-sided expanders.

Problem

• Finding large independent sets in 3-colorable graphs
• Handling graphs with small 1-sided threshold rank
• Providing tight bounds for 3-coloring efficiency

Innovation

• Algorithm for 3-colorable graphs with low threshold rank
• Handles graphs with limited large eigenvalues efficiently
• Correlation analysis ensures proper 3-coloring distribution