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arXiv · 2406.15563

Exponential Time Approximation for Coloring 3-Colorable Graphs

Venkatesan Guruswami·Rhea Jain
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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.

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Venkatesan Guruswami, Rhea Jain. 2024-06-21. Exponential Time Approximation for Coloring 3-Colorable Graphs. https://arxiv.org/abs/2406.15563

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