Tight Bounds for Sampling q-Colorings via Coupling from the Past

TL;DR

This work analyzes exact sampling of uniform proper $q$-colorings on graphs via Coupling from the Past (CFTP) using bounding chains. It introduces a unified grand-coupling framework that reduces the design of bounding chains to an optimization problem, yielding a tight asymptotic threshold $q\ge (2.5+o(1))\Delta$ for contractive bounding chains. A lower bound shows this barrier is intrinsic to all bounding-chain-based CFTP methods, while an LP-based upper bound constructs an optimal grand coupling (via Seeding, Compress, and Disjoint) that matches the lower bound up to small terms. The resulting CFTP algorithm samples exactly from the uniform colorings in expected time $\tilde{O}(n\Delta^2)$ for $q>(2.5+\eta)\Delta$ with $\eta=2\sqrt{(\log\Delta+1)/\Delta}$, advancing the state of the art for perfect sampling in graph colorings. The findings delineate a fundamental limitation of the bounding-chain paradigm and suggest that surpassing the $2.5\Delta$ barrier will require new approaches beyond the current bounding-chain contraction framework.

Abstract

The Coupling from the Past (CFTP) paradigm is a canonical method for perfect sampling. For uniform sampling of proper $q$-colorings in graphs with maximum degree $Δ$, the bounding chains of Huber (STOC 1998) provide a systematic framework for efficiently implementing CFTP algorithms within the classical regime $q \ge (1 + o(1))Δ^2$. This was subsequently improved to $q > 3Δ$ by Bhandari and Chakraborty (STOC 2020) and to $q \ge (8/3 + o(1))Δ$ by Jain, Sah, and Sawhney (STOC 2021). In this work, we establish the asymptotically tight threshold for bounding-chain-based CFTP algorithms for graph colorings. We prove a lower bound showing that all such algorithms satisfying the standard contraction property require $q \ge 2.5Δ$, and we present an efficient CFTP algorithm that achieves this asymptotically optimal threshold $q \ge (2.5 + o(1))Δ$ via an optimal design of bounding chains.

Figures (1)

• Figure 1: Each line illustrates how the unit interval $[0,1]$ is partitioned and assigned to the available colors, thereby defining a uniform distribution over them using the shared source of randomness $U \sim \mathrm{Uniform}[0,1]$. A single random position $U \in [0,1]$ determines the color selected for each case, and the union of these selected colors forms the updated bounding list $\mathcal{L}'(v)$. In this example, for any choice of $U$, the updated bounding list satisfies $|\mathcal{L}'(v)| = 2$.

Theorems & Definitions (43)

• Theorem 1.2: Upper bound
• Remark 1.3: Perfect sampling via deterministic counting or approximate sampling
• Theorem 1.4: Lower bound
• Remark 1.5
• Remark 1.6
• Theorem 1.7: Lower bound: 2-to-1 contraction
• Definition 3.1: Grand coupling
• Definition 3.2: Grand coupling via bounding chain
• Example 3.3
• Theorem 3.5