Faster Edge Coloring by Partition Sieving

TL;DR

This paper introduces a novel algorithm for Edge Coloring that operates faster than previous methods, using polynomial space and providing improved performance especially in graphs with bounded average degree.

Contribution

The paper presents the first polynomial-space algorithm for Edge Coloring that surpasses the traditional exponential-time bounds, with enhanced efficiency in graphs of bounded average degree.

Findings

Runs in $2^{m-3n/5}\text{poly}(n)$ time and polynomial space.

Achieves faster algorithms for List Edge Coloring with improved dependence on $k$.

Outperforms previous algorithms in both speed and space complexity.

Abstract

In the Edge Coloring problem, we are given an undirected graph $G$ with $n$ vertices and $m$ edges, and are tasked with finding the smallest positive integer $k$ so that the edges of $G$ can be assigned $k$ colors in such a way that no two edges incident to the same vertex are assigned the same color. Edge Coloring is a classic NP-hard problem, and so significant research has gone into designing fast exponential-time algorithms for solving Edge Coloring and its variants exactly. Prior work showed that Edge Coloring can be solved in $2^m\text{poly}(n)$ time and polynomial space, and in graphs with average degree $d$ in $2^{(1-\varepsilon_d)m}\text{poly}(n)$ time and exponential space, where $\varepsilon_d = (1/d)^{\Theta(d^3)}$. We present an algorithm that solves Edge Coloring in $2^{m-3n/5}\text{poly}(n)$ time and polynomial space. Our result is the first algorithm for this problem which simultaneously runs in faster than $2^m\text{poly}(m)$ time and uses only polynomial space. In graphs of average degree $d$, our algorithm runs in $2^{(1-6/(5d))m}\text{poly}(n)$ time, which has far better dependence in $d$ than previous results. We also generalize our algorithm to solve a problem known as List Edge Coloring, where each edge $e$ in the input graph comes with a list $L_e\subseteq\left\{1, \dots, k\right\}$ of colors, and we must determine whether we can assign each edge a color from its list so that no two edges incident to the same vertex receive the same color. We solve this problem in $2^{(1-6/(5k))m}\text{poly}(n)$ time and polynomial space. The previous best algorithm for List Edge Coloring took $2^m\text{poly}(n)$ time and space.