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Exponential Time Approximation for Coloring 3-Colorable Graphs

Venkatesan Guruswami, Rhea Jain

TL;DR

This work considers exponential time approximations, and builds on tools developed in (Bansal et al., Algorithmic, 2019) to obtain an algorithm to color a 3-colorable graphs with O(r)$ colors in asymptotically improving upon the bound given by Atserias and Dalmau.

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 $2^{Θ(n^{1-2\varepsilon}\log(n))}$ 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.

Exponential Time Approximation for Coloring 3-Colorable Graphs

TL;DR

This work considers exponential time approximations, and builds on tools developed in (Bansal et al., Algorithmic, 2019) to obtain an algorithm to color a 3-colorable graphs with O(r)$ colors in asymptotically improving upon the bound given by Atserias and Dalmau.

Abstract

The problem of efficiently coloring -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 , we aim to develop an -approximation algorithm with the best possible runtime, providing a tradeoff between runtime and approximation ratio. In this vein, an algorithm to -color a 3-colorable graphs in time 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 -colorable graphs with colors in time, asymptotically improving upon the bound given by Atserias and Dalmau.
Paper Structure (7 sections, 5 theorems, 5 equations, 1 algorithm)

This paper contains 7 sections, 5 theorems, 5 equations, 1 algorithm.

Table of Contents

  1. Introduction
  2. Promise Constraint Satisfaction Problems and Width-based Algorithms
  3. Definitions and Notation
  4. Our Result and Techniques
  5. Improved Approximation for Coloring 3-Colorable Graphs
  6. Independent Set
  7. Graph Coloring

Key Result

Theorem 1.1

Given a 3-colorable graph $G = (V, E)$ and a parameter $r > 0$, there exists an algorithm that, with high probability, outputs a valid coloring of $G$ using at most $O(r)$ colors in time $O^*\left(\exp\left(\tilde{O}\left(\frac{n\log^{11/2}r}{r^3}\right)\right)\right)$.

Theorems & Definitions (13)

  • Theorem 1.1
  • Lemma 2.1
  • Lemma 2.2: bansal2019new
  • Lemma 2.3: karger1994approximate
  • Corollary 2.4
  • proof
  • proof : Proof of Lemma \ref{['lem:is_result']}
  • Claim 2.5
  • proof
  • Claim 2.6
  • ...and 3 more