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

2025-11-07

AI Summary

This work establishes the asymptotically tight threshold for coupling-from-the-past (CFTP) perfect sampling of graph $q$-colorings via boundary-condition chains. For graphs with maximum degree $Delta$, we seek the minimal $q$ ensuring efficient uniform sampling. Our method systematically analyzes contraction properties of boundary chains: we first prove that *any* boundary-chain algorithm satisfying standard contraction conditions must have $q geq 2.5Delta$; then, we construct an efficient CFTP algorithm achieving the asymptotically optimal threshold $q geq (2.5 + o(1))Delta$. This improves upon the previous best bound $q geq frac{8}{3}Delta$, approaches the theoretical lower bound, and precisely characterizes the fundamental performance limit of boundary-chain methods for perfect sampling in graph coloring.

Technology Category

• Constraint Satisfaction and Optimization: Distributed CSP/Optimization
• Reasoning under Uncertainty: Graphical Models
• Machine Learning: Probabilistic Circuits and Graphical Models

Application Category

• Graph Algorithms and Modeling for the Web: Querying, indexing, and retrieval in Web-related graphs
• Economics, Online Markets and Human Computation: Data quality aspects of human-annotated datasets
• Security and Privacy: Large-scale security measurements

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 $q>3Delta$ 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 $q ge 2.5Delta$, 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.

Problem

• Establish tight threshold for CFTP sampling of graph colorings
• Prove lower bound requiring q ≥ 2.5Δ for contraction property
• Design optimal bounding chains achieving q ≥ (2.5+o(1))Δ threshold

Innovation

• Establishes tight threshold for CFTP algorithms
• Proves lower bound requiring q ≥ 2.5Δ
• Achieves optimal threshold via bounding chain design