Improved SDP-Based Algorithm for Coloring 3-Colorable Graphs

TL;DR

This paper introduces an improved polynomial-time algorithm for coloring 3-colorable graphs with fewer colors, advancing SDP-based methods and extending previous analyses to third-level neighborhoods.

Contribution

It presents the first significant progress in nearly two decades on SDP-based coloring algorithms by extending analysis to third-level neighborhoods and developing a new vector coloring technique.

Findings

Achieved coloring with $O(n^{0.19539})$ colors, improving previous bounds.

Extended SDP analysis to third-level neighborhoods.

Developed a novel vector $5/2$-coloring method.

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.