Finding Colorings in One-Sided Expanders

2025-08-04

AI Summary

This work addresses graph coloring and independent set problems on one-sided expanders. For 3-colorable regular one-sided expanders, we design polynomial-time algorithms that either construct an independent set of relative size at least $1/2 - o(1)$ or produce a valid 3-coloring for all but an $o(1)$ fraction of vertices. Methodologically, we introduce a novel $k imes k$ coloring constraint matrix model, revealing an intrinsic connection between row repetition in this matrix and the NP-hardness of $k$-coloring; we further apply spectral graph theory to characterize the asymptotic behavior of the second eigenvalue of the normalized adjacency matrix and uncover new distributional properties of negative eigenvalues. Key contributions include improving the approximation ratio for vertex cover from $2-varepsilon$ to $5/3$, substantially strengthening lower bounds for independent sets and approximation guarantees for coloring, and establishing a new theoretical framework and efficient algorithms for combinatorial optimization on expander graphs.

Technology Category

• Constraint Satisfaction and Optimization: Other Foundations of Constraint Satisfaction
• Machine Learning: Graph-based Machine Learning
• 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
• Web Mining and Content Analysis: Models for Web evolution

Abstract

We establish new algorithmic guarantees with matching hardness results for coloring and independent set problems in one-sided expanders and related classes of graphs. For example, given a $3$-colorable regular one-sided expander, we compute in polynomial time either an independent set of relative size at least $1/2-o(1)$ or a proper $3$-coloring for all but an $o(1)$ fraction of the vertices, where $o(1)$ stands for a function that tends to $0$ with the second largest eigenvalue of the normalized adjacency matrix. This result improves on recent seminal work of Bafna, Hsieh, and Kothari (STOC 2025) developing an algorithm that efficiently finds independent sets of relative size at least $0.01$ in such graphs. We also obtain an efficient $1.6667$-factor approximation algorithm for VERTEX COVER in sufficiently strong regular one-sided expanders, improving over a previous $(2-ε)$-factor approximation in such graphs for an unspecified constant $ε>0$. We propose a new stratification of $k$-COLORING in terms of $k$-by-$k$ matrices akin to predicate sets for constraint satisfaction problems. We prove that whenever this matrix has repeated rows, the corresponding coloring problem is NP-hard for one-sided expanders under the Unique Games Conjecture. On the other hand, if this matrix has no repeated rows, our algorithms can solve the corresponding coloring problem on one-sided expanders in polynomial time. As starting point for our algorithmic results, we show a property of graph spectra that, to the best of our knowledge, has not been observed before: The number of negative eigenvalues smaller than $-τ$ is at most $O(1/τ^{2})$ times the number of eigenvalues larger than $τ^{2}/2$. While this result allows us to bound the number of eigenvalues bounded away from $0$ in one-sided spectral expanders, this property alone is insufficient for our algorithmic results.

Problem

• Develop algorithms for coloring and independent set in one-sided expanders
• Improve approximation for VERTEX COVER in regular one-sided expanders
• Analyze NP-hardness of k-COLORING via matrix stratification in expanders

Innovation

• Polynomial-time algorithm for expander graph coloring
• Improved vertex cover approximation in expanders
• New matrix-based stratification for coloring problems