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

TL;DR

This paper establishes the tight threshold for perfect sampling of q-colorings in graphs using Coupling from the Past, proving that algorithms require at least 2.5 times the maximum degree in colors, and presents an optimal algorithm achieving this bound.

Contribution

The paper proves a lower bound of q ≥ 2.5Δ for bounding-chain-based CFTP algorithms and introduces an efficient algorithm that attains this asymptotic threshold.

Findings

Lower bound q ≥ 2.5Δ for all such algorithms

An efficient CFTP algorithm achieving q ≥ (2.5 + o(1))Δ

Optimal design of bounding chains for graph colorings

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 $\Delta$, the bounding chains of Huber (STOC 1998) provide a systematic framework for efficiently implementing CFTP algorithms within the classical regime $q \ge (1 + o(1))\Delta^2$. This was subsequently improved to $q > 3\Delta$ by Bhandari and Chakraborty (STOC 2020) and to $q \ge (8/3 + o(1))\Delta$ 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 $q \ge 2.5\Delta$, and we present an efficient CFTP algorithm that achieves this asymptotically optimal threshold $q \ge (2.5 + o(1))\Delta$ via an optimal design of bounding chains.