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Flip Dynamics for Sampling Colorings: Improving $(11/6-ε)$ Using a Simple Metric

Charlie Carlson, Eric Vigoda

TL;DR

An optimal mixing time bound of $O(n\log{n})$ for the flip dynamics when $k \geq 1.809 \Delta$ is proved for the Glauber dynamics.

Abstract

We present improved bounds for randomly sampling $k$-colorings of graphs with maximum degree $Δ$; our results hold without any further assumptions on the graph. The Glauber dynamics is a simple single-site update Markov chain. Jerrum (1995) proved an optimal $O(n\log{n})$ mixing time bound for Glauber dynamics whenever $k>2Δ$ where $Δ$ is the maximum degree of the input graph. This bound was improved by Vigoda (1999) to $k > (11/6)Δ$ using a "flip" dynamics which recolors (small) maximal 2-colored components in each step. Vigoda's result was the best known for general graphs for 20 years until Chen et al. (2019) established optimal mixing of the flip dynamics for $k > (11/6 - ε) Δ$ where $ε\approx 10^{-5}$. We present the first substantial improvement over these results. We prove an optimal mixing time bound of $O(n\log{n})$ for the flip dynamics when $k \geq 1.809 Δ$. This yields, through recent spectral independence results, an optimal $O(n\log{n})$ mixing time for the Glauber dynamics for the same range of $k/Δ$ when $Δ=O(1)$. Our proof utilizes path coupling with a simple weighted Hamming distance for "unblocked" neighbors.

Flip Dynamics for Sampling Colorings: Improving $(11/6-ε)$ Using a Simple Metric

TL;DR

An optimal mixing time bound of for the flip dynamics when is proved for the Glauber dynamics.

Abstract

We present improved bounds for randomly sampling -colorings of graphs with maximum degree ; our results hold without any further assumptions on the graph. The Glauber dynamics is a simple single-site update Markov chain. Jerrum (1995) proved an optimal mixing time bound for Glauber dynamics whenever where is the maximum degree of the input graph. This bound was improved by Vigoda (1999) to using a "flip" dynamics which recolors (small) maximal 2-colored components in each step. Vigoda's result was the best known for general graphs for 20 years until Chen et al. (2019) established optimal mixing of the flip dynamics for where . We present the first substantial improvement over these results. We prove an optimal mixing time bound of for the flip dynamics when . This yields, through recent spectral independence results, an optimal mixing time for the Glauber dynamics for the same range of when . Our proof utilizes path coupling with a simple weighted Hamming distance for "unblocked" neighbors.
Paper Structure (25 sections, 17 theorems, 101 equations, 1 figure)

This paper contains 25 sections, 17 theorems, 101 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Proof Overview
  3. Preliminaries
  4. List Colorings
  5. Clusters, Flip Dynamics, and Mixing Time
  6. Path Coupling
  7. New Metric Definition
  8. Cluster and Flip Dynamics
  9. Our New Metric
  10. Coupling Analysis
  11. Relating $\widetilde{H}$ to $H$
  12. Analysis by Color
  13. Vertices Changing Between Blocked and Unblocked
  14. Bounding $\widetilde{W}^c_t$ and $\widetilde{Z}^c_t$
  15. Proof of \ref{['thrm:ColoringBound']}
  16. ...and 10 more sections

Key Result

Theorem 1.1

For all $\Delta\geq 125$, for all $k \geq 1.809 \Delta$, there exists a setting of the parameters for the flip dynamics with $P_j=0$ for all $j\geq 7$, so that for any graph $G$ on $n$ vertices with maximum degree $\Delta$, the flip dynamics has mixing time $O(n\log(n))$.

Figures (1)

  • Figure 1: A small graph showing how vertices can be unblocked, singly blocked, and multiblocked. The vertex $u_1$ is singly blocked with respect to $v^*$ since it has a neighbor $w_1$ that is colored $B$. The vertex $u_2$ is multiblocked with respect to $v^*$ since it has a $B$ and $R$ neighbor, $w_1$ and $w_2$ respectively. Likewise, $u_3$ is multiblocked since it has two $R$ neighbors, $w_3$ and $w_4$. Finally, $u_4$ is unblocked since it has no neighbors that are $R$ or $B$.

Theorems & Definitions (36)

  • Theorem 1.1
  • Corollary 1.2
  • Theorem 2.1
  • Lemma 3.1
  • proof
  • Lemma 4.1
  • proof : Proof of \ref{['lem:total-per-color']}
  • Definition 4.2
  • Lemma 4.3
  • Lemma 4.4
  • ...and 26 more