Improved SDP-Based Algorithm for Coloring 3-Colorable Graphs

2026-02-05

AI Summary

Efficient approximate coloring of 3-colorable graphs remains a classic challenge in graph theory and approximation algorithms. This work proposes a polynomial-time algorithm based on semidefinite programming (SDP), advancing SDP-based techniques to third-neighborhood analysis for the first time. By constructing a novel vector 5/2-coloring and extracting large independent sets from it, the method achieves significantly improved coloring efficiency. On an n-vertex 3-colorable graph, the algorithm uses only O(n^0.19539) colors, improving upon the previous best bound of Õ(n^0.19747) and breaking a two-decade-long technical barrier in this line of research.

Technology Category

• Machine Learning: Graph-based Machine Learning
• Constraint Satisfaction and Optimization: Distributed CSP/Optimization
• Search and Optimization: Combinatorial Optimization

Application Category

• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs
• Responsible Web: Human-perceived consequences of algorithmic deployment on the web
• Web Mining and Content Analysis: Robustness and generalizability of Web mining methods

Abstract

We present a polynomial-time algorithm that colors any 3-colorable $n$-vertex graph using $O(n^{0.19539})$ colors, improving upon the previous best bound of $\widetilde{O}(n^{0.19747})$ by Kawarabayashi, Thorup, and Yoneda [STOC 2024]. Our result constitutes the first progress in nearly two decades on SDP-based approaches to this problem. The earlier SDP-based algorithms of Arora, Chlamt\'a\v{c}, and Charikar [STOC 2006] and Chlamt\'a\v{c} [FOCS 2007] rely on extracting a large independent set from a suitably"random-looking"second-level neighborhood, under the assumption that the KMS algorithm [Karger, Motwani, and Sudan, JACM 1998] fails to find one globally. We extend their analysis to third-level neighborhoods. We then come up with a new vector $5/2$-coloring, which allows us to extract a large independent set from some third-level neighborhood. The new vector coloring construction may be of independent interest.

Problem

• graph coloring
• 3-colorable graphs
• semidefinite programming
• approximation algorithms
• independent set

Innovation

• semidefinite programming
• graph coloring
• independent set
• vector coloring
• 3-colorable graphs