Exponential Time Approximation for Coloring 3-Colorable Graphs

TL;DR

This paper presents a new exponential time approximation algorithm for coloring 3-colorable graphs, achieving a better tradeoff between runtime and number of colors compared to previous methods.

Contribution

It introduces an improved exponential time algorithm for coloring 3-colorable graphs with O(r) colors, surpassing prior bounds in runtime efficiency.

Findings

Achieves coloring with O(r) colors in subexponential time

Improves upon previous exponential time bounds

Provides a better tradeoff between runtime and approximation ratio

Abstract

The problem of efficiently coloring $3$-colorable graphs with few colors has received much attention on both the algorithmic and inapproximability fronts. We consider exponential time approximations, in which given a parameter $r$, we aim to develop an $r$-approximation algorithm with the best possible runtime, providing a tradeoff between runtime and approximation ratio. In this vein, an algorithm to $O(n^\varepsilon)$-color a 3-colorable graphs in time $2^{\Theta(n^{1-2\varepsilon}\log(n))}$ is given in (Atserias and Dalmau, SODA 2022.) We build on tools developed in (Bansal et al., Algorithmic, 2019) to obtain an algorithm to color $3$-colorable graphs with $O(r)$ colors in $\exp\left(\tilde{O}\left(\frac {n\log^{11/2}r} {r^3}\right)\right)$ time, asymptotically improving upon the bound given by Atserias and Dalmau.