Optimal Mixing for Randomly Sampling Edge Colorings on Trees Down to the Max Degree

TL;DR

This paper proves that certain Markov chains efficiently generate proper edge colorings of trees with a number of colors close to the maximum degree, achieving optimal or near-optimal convergence times.

Contribution

It establishes optimal relaxation times for Glauber dynamics in edge coloring of trees when q ≥ Δ+2, and introduces an alternative chain for q=Δ+1, with new analytical techniques.

Findings

Glauber dynamics has O(n) relaxation time for q ≥ Δ+2.

Alternative chain achieves O(n) relaxation time for q=Δ+1.

For Δ-regular trees, mixing time is O(n log^2 n).

Abstract

We address the convergence rate of Markov chains for randomly generating an edge coloring of a given tree. Our focus is on the Glauber dynamics which updates the color at a randomly chosen edge in each step. For a tree $T$ with $n$ vertices and maximum degree $\Delta$, when the number of colors $q$ satisfies $q\geq\Delta+2$ then we prove that the Glauber dynamics has an optimal relaxation time of $O(n)$, where the relaxation time is the inverse of the spectral gap. This is optimal in the range of $q$ in terms of $\Delta$ as Dyer, Goldberg, and Jerrum (2006) showed that the relaxation time is $\Omega(n^3)$ when $q=\Delta+1$. For the case $q=\Delta+1$, we show that an alternative Markov chain which updates a pair of neighboring edges has relaxation time $O(n)$. Moreover, for the $\Delta$-regular complete tree we prove $O(n\log^2{n})$ mixing time bounds for the respective Markov chain. Our proofs establish approximate tensorization of variance via a novel inductive approach, where the base case is a tree of height $\ell=O(\Delta^2\log^2{\Delta})$, which we analyze using a canonical paths argument.