Deterministic counting from coupling independence

2024-10-30

arXiv.org

AI Summary

This work presents the first deterministic fully polynomial-time approximation scheme (FPTAS) for spin systems satisfying coupling independence and bounded-degree (maximum degree Δ) constraints, breaking reliance on randomized algorithms. Methodologically, it introduces a novel deterministic counting framework built upon recursive decomposition, tree-like structure approximation, and rigorous coupling independence analysis, with tight control over error propagation. Key contributions include the first three deterministic FPTASes for the q-coloring problem: (1) for graphs with Δ ≥ 3 and q ≥ (11/6 − ε₀)Δ; (2) for Δ ≥ 125 and q ≥ 1.809Δ; and (3) for graphs of large girth and q ≥ Δ + 3. All degree–coloring thresholds match the best known bounds achieved by randomized algorithms, marking a fundamental advance in deterministic approximate counting for graph coloring.

Technology Category

• Constraint Satisfaction and Optimization: Other Foundations of Constraint Satisfaction
• Planning, Routing, and Scheduling: Deterministic Planning
• Knowledge Representation and Reasoning: Computational Complexity of Reasoning

Application Category

• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs
• Search and Retrieval-Augmented AI: Web learning to rank, online learning, and counterfactual learning for ranking
• Responsible Web: Human-perceived consequences of algorithmic deployment on the web

Abstract

We show that spin systems with bounded degrees and coupling independence admit fully polynomial time approximation schemes (FPTAS). We design a new recursive deterministic counting algorithm to achieve this. As applications, we give the first FPTASes for $q$-colourings on graphs of bounded maximum degree $Deltage 3$, when $qge (11/6-varepsilon_0)Delta$ for some small $varepsilon_0approx 10^{-5}$, or when $Deltage 125$ and $qge 1.809Delta$, and on graphs with sufficiently large (but constant) girth, when $qgeqDelta+3$. These bounds match the current best randomised approximate counting algorithms by Chen, Delcourt, Moitra, Perarnau, and Postle (2019), Carlson and Vigoda (2024), and Chen, Liu, Mani, and Moitra (2023), respectively.

Problem

• Develop FPTAS for spin systems with bounded degrees
• Design recursive deterministic counting algorithm for spin systems
• Match best randomized bounds for q-colorings on graphs

Innovation

• Recursive deterministic counting algorithm
• FPTAS for spin systems
• Bounded degree coupling independence