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Improved SDP-Based Algorithm for Coloring 3-Colorable Graphs

Nikhil Bansal, Neng Huang, Euiwoong Lee

TL;DR

This work resolves coloring $3$-colorable graphs in polynomial time with $O(n^{0.19539})$ colors, advancing the SDP-based approach after a long period of stagnation. The authors extend the KMS framework from second- to third-level neighborhoods and introduce a novel vector $5/2$-coloring on $N^{(3)}(i)$ derived via the SoS/Lasserre hierarchy's local distributions, enabling extraction of large independent sets from deeper neighborhoods. The analysis combines refined pruning, cover composition, and a win-win argument to manage efficiency losses, culminating in a rigorous lower bound that yields the improved color bound. The result sharpens the state of the art in SDP-based graph coloring and may influence subsequent works on higher-level relaxations and their combinatorial consequences.

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, Chlamtáč, and Charikar [STOC 2006] and Chlamtáč [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.

Improved SDP-Based Algorithm for Coloring 3-Colorable Graphs

TL;DR

This work resolves coloring -colorable graphs in polynomial time with colors, advancing the SDP-based approach after a long period of stagnation. The authors extend the KMS framework from second- to third-level neighborhoods and introduce a novel vector -coloring on derived via the SoS/Lasserre hierarchy's local distributions, enabling extraction of large independent sets from deeper neighborhoods. The analysis combines refined pruning, cover composition, and a win-win argument to manage efficiency losses, culminating in a rigorous lower bound that yields the improved color bound. The result sharpens the state of the art in SDP-based graph coloring and may influence subsequent works on higher-level relaxations and their combinatorial consequences.

Abstract

We present a polynomial-time algorithm that colors any 3-colorable -vertex graph using colors, improving upon the previous best bound of 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, Chlamtáč, and Charikar [STOC 2006] and Chlamtáč [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 -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.
Paper Structure (21 sections, 34 theorems, 94 equations, 4 figures)

This paper contains 21 sections, 34 theorems, 94 equations, 4 figures.

Table of Contents

  1. Introduction
  2. Overview of Our Result
  3. Improvement over KMS from Second-Level Neighborhood
  4. Extending to Third-Level Neighborhood
  5. New Vector Coloring on the Third-Level Neighborhood
  6. Other Related Work
  7. Organization of the Paper
  8. Preliminaries
  9. SDP Relaxations and the KMS Algorithm
  10. Combining Combinatorial and SDP-Based Algorithms
  11. New Constructions of Vector Colorings
  12. Analyzing Second-Level Neighborhoods
  13. Covers and Packings
  14. Pruning Two-Step Walks
  15. Cover Composition and Finding a Large Vector 2-Colorable Set
  16. ...and 6 more sections

Key Result

Theorem 1.1

There is a polynomial time algorithm which, given a $3$-colorable graph $G$, produces a proper coloring with $O(n^{0.19539})$ colors.

Figures (4)

  • Figure 1: The ${\textsc{KMS}}$ Algorithm for 3-Colorable Graphs
  • Figure 2: Local distribution over an edge $\{\ell_1, \ell_2\}$ conditioned on $i$ being red
  • Figure 3: The ${\textsc{KMS}}$ Algorithm
  • Figure 4: The ${\textsc{KMS}'}$ Algorithm

Theorems & Definitions (64)

  • Theorem 1.1
  • Theorem 1.2: kawarabayashi_better_2024
  • Theorem 1.3: karger_approximate_1998
  • Theorem 1.4: kawarabayashi_better_2024
  • Definition 2.2
  • Theorem 2.3: karger_approximate_1998
  • Definition 2.4
  • proof
  • Theorem 2.6: cf. Proposition 7.1 in kawarabayashi_coloring_2017
  • Corollary 2.7
  • ...and 54 more