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

TL;DR

We study Glauber dynamics for sampling $q$-edge colorings on trees with maximum degree $\Delta$. Using a novel inductive framework based on approximate tensorization of variance and a root-tensorization mechanism, the authors prove that for $q\ge\Delta+2$ the Glauber dynamics has relaxation time $T_{\text{rel}}=O(n)$, and they extend near-optimal results to the heat-bath neighboring-edge dynamics at $q=\Delta+1$. They further derive $O(n\log^2 n)$ mixing times on $\Delta$-regular complete trees and establish lower bounds indicating tightness in some regimes. The work introduces a canonical-path–coupling hybrid to bound variance via leaf-edge recolorings and provides a structured inductive method that may apply to related Gibbs distributions on sparse graphs.

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 $Δ$, when the number of colors $q$ satisfies $q\geqΔ+2$ 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 $Δ$ as Dyer, Goldberg, and Jerrum (2006) showed that the relaxation time is $Ω(n^3)$ when $q=Δ+1$. For the case $q=Δ+1$, we show that an alternative Markov chain which updates a pair of neighboring edges has relaxation time $O(n)$. Moreover, for the $Δ$-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 $\ell=O(Δ^2\log^2Δ)$, which we analyze using a canonical paths argument.

Figures (3)

• Figure 1: The figure illustrates the canonical path $\gamma^{\sigma,\tau}$. For simplicity, we only draw edges whose color will be used in the construction rather than the entire tree. The edge-coloring in is $\sigma$ and the coloring in is $\tau$. Let $a = $ and $b = $, we mark the vertices in $(a,b)$-alternating path $\mathcal{E}^*$ as $(v_0, v_1, v_2, v_3, v_4)$. Then, we fix the order $\mathcal{O}$ of colors as $\mathcal{O}=\{ \prec \prec \prec \prec \prec \}$; note, the colors $, $ are the last two colors in this order. The odd edges $e_i \in \mathcal{E}^*$ and their associated paths $\mathcal{E}_i$ are marked by dashed lines (and grey background). The coloring in is obtained after Stage-I, and hence after we recolored the edges in $\mathcal{E}$ in order from the leaves up. Then the coloring in is after Stage-II, and finally the coloring in is after Stage-III and is $\tau$.
• Figure 2: The figure details Stage-I of the canonical path. For convenience, we call the sub-figure located in the $i$-th row and $j$-th column as $\text{fig}(i, j)$. The canonical path is as follows. First, as in $\text{fig}(1, 1)$, we find the $(, )$-alternating path; this is marked with a shadowed background. Then, for the odd edges on the alternating path (i.e., the -edges), we want to change their color to some color other than and . To do that, we run Stage-I according to the priority of colors $ \prec \prec \prec \prec \prec $ (see also \ref{['fig:canonical-path']}) and get two lists of edges. We mark these lists $\mathcal{E}$ with dashed circles in $\text{fig}(1,2)$. Intuitively, to change the color of the odd edges on the alternating path, we need to change the color of edges in these lists. Then, as showed from $\text{fig}(1,3)$ to $\text{fig}(2,3)$, we change the color of the edges in the lists according to the order of their depth. For the edges on the same level, to break the tie, we first consider edges from left to right, but with an exception that we keep the edge in the alternating path to be the last one. We add number on these edges to denote this order. Lastly, when we change the color of some edge $i$, we always change its color to a different color which has the lowest priority among all the available colors.
• Figure 3: We consider two $3$-regular trees with , , , as avaliable colors. In both trees, we set $a = $ and $b = $. In the first example, we mark the odd numbered edge $e_1$ with corresponding $\mathcal{E}_1$ using shadowed background. This means, in order to change the color of $e_1$, by the construction of $\mathcal{E}_1$ we used in \ref{['sec:canonical-path']}, we want to change the color of the root edge. Since the root edge only have two avaliable color and , this is not possible. In the second example, we mark $e_1$ and $\mathcal{E}_1$ with shadowed background and we mark $e_3$ and $\mathcal{E}_3$ with dashed circle. It is then easy to see that these paths have neiboring edges. This means, after fliping the edges in $e_3 \cup \mathcal{E}_3$, the path $e_1 \cup \mathcal{E}_1$ will be different. This will fail the construction in \ref{['sec:canonical-path']}, which havily relies on these paths can be handled independently.

Theorems & Definitions (76)

• Theorem 1
• Theorem 2
• Theorem 3
• Theorem 4
• Theorem 5
• Definition 6
• Lemma 7: tetali2012phase
• Corollary 8
• Definition 9: $\mathbb{T}_k, \mathbb{T}^\star_k$
• Definition 10: $\mu_k, \mu_k^\star, \mu_k^{\star,1}$