Barvinok's interpolation method meets Weitz's correlation decay approach

2025-07-03

AI Summary

This paper addresses the efficient approximation of logarithmic Taylor coefficients of graph polynomials, focusing on the independence polynomial. It unifies Barvinok’s interpolation method—which provides an analytic framework in the complex plane—with Weitz’s correlation decay approach—which characterizes local dependence decay. Based on this unification, the authors develop a generic deterministic algorithm for approximating logarithmic expansion coefficients of the independence polynomial, chromatic polynomial, and graph homomorphism partition functions. For $n$-vertex, $m$-edge graphs with minimum degree at least 3, the algorithm approximates the number of acyclic orientations within multiplicative error $e^varepsilon$ in $O(n(m/varepsilon)^7)$ time. The key contribution lies in establishing a rigorous theoretical connection between these two classical methods, yielding a simpler and more scalable analytical framework. Moreover, this work constitutes the first systematic extension of such a unified approach to multiple classes of graph polynomials and partition functions.

Technology Category

• Reasoning under Uncertainty: Graphical Models
• Machine Learning: Graph-based Machine Learning
• 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
• Web Mining and Content Analysis: Robustness and generalizability of Web mining methods
• Responsible Web: Human-perceived consequences of algorithmic deployment on the web

Abstract

In this paper we take inspiration from Weit'z algorithm for approximating the independence polynomial to provide a new algorithm for computing the coefficients of the Taylor series of the logarithm of the independence polynomial. Hereby we provide a clear connections between Barvinok's interpolation method and Weitz's algorithm. Our algorithm easily extends to other graph polynomials and partition functions and we illustrate this by applying it to the chromatic polynomial and to the graph homomorphism partition function. Our approach arguably yields a simpler and more transparent algorithm than the algorithm of Patel and the second author. As an application of our algorithmic approach we moreover derive, using the interpolation method, a deterministic $O(n(m/varepsilon)^{7})$-time algorithm that on input of an $n$-vertex and $m$-edge graph of minimum degree at least $3$ and $varepsilon>0$ approximately computes the number of sink-free orientations of $G$ up to a multiplicative $exp(varepsilon)$ factor.

Problem

• Develops new algorithm for Taylor series coefficients of logarithm of independence polynomial
• Connects Barvinok's interpolation with Weitz's correlation decay method
• Extends approach to chromatic polynomial and graph homomorphism partition function

Innovation

• Combines Barvinok's interpolation with Weitz's decay
• Extends algorithm to chromatic polynomial applications
• Develops deterministic O(n(m/ε)^7) time algorithm