# Efficiently list-edge coloring multigraphs asymptotically optimally

## TL;DR

This paper develops polynomial-time algorithms for asymptotically optimal list-edge coloring of multigraphs, transforming non-constructive probabilistic proofs into constructive algorithms using advanced local search and correlation decay techniques.

## Contribution

It introduces a constructive approach to Kahn's asymptotic results on edge coloring multigraphs by leveraging local search analysis and correlation decay methods.

## Key findings

- Algorithms match Kahn's asymptotic bounds
- Use of local search analysis for constructive algorithms
- Correlation decay exploited for efficiency

## Abstract

We give polynomial time algorithms for the seminal results of Kahn, who showed that the Goldberg-Seymour and List-Coloring conjectures for (list-)edge coloring multigraphs hold asymptotically. Kahn's arguments are based on the probabilistic method and are non-constructive. Our key insight is to show that the main result of Achlioptas, Iliopoulos and Kolmogorov for analyzing local search algorithms can be used to make constructive applications of a powerful version of the so-called Lopsided Lovasz Local Lemma. In particular, we use it to design algorithms that exploit the fact that correlations in the probability spaces on matchings used by Kahn decay with distance.