Deterministic polynomial-time approximation algorithms for partition functions and graph polynomials

TL;DR

This paper introduces a new deterministic polynomial-time approximation method for complex evaluations of a broad class of graph polynomials on bounded degree graphs, linking zero-free regions to efficient algorithms.

Contribution

It defines a large class of graph polynomials and establishes a connection between their zero-free regions and the existence of polynomial-time approximation algorithms.

Findings

Provides deterministic algorithms for Tutte and independence polynomials

Establishes a link between zero-free regions and approximation efficiency

Extends Barvinok's approach to a broader class of graph polynomials

Abstract

In this paper we show a new way of constructing deterministic polynomial-time approximation algorithms for computing complex-valued evaluations of a large class of graph polynomials on bounded degree graphs. In particular, our approach works for the Tutte polynomial and independence polynomial, as well as partition functions of complex-valued spin and edge-coloring models. More specifically, we define a large class of graph polynomials $\mathcal C$ and show that if $p\in \cal C$ and there is a disk $D$ centered at zero in the complex plane such that $p(G)$ does not vanish on $D$ for all bounded degree graphs $G$, then for each $z$ in the interior of $D$ there exists a deterministic polynomial-time approximation algorithm for evaluating $p(G)$ at $z$. This gives an explicit connection between absence of zeros of graph polynomials and the existence of efficient approximation algorithms, allowing us to show new relationships between well-known conjectures. Our work builds on a recent line of work initiated by. Barvinok, which provides a new algorithmic approach besides the existing Markov chain Monte Carlo method and the correlation decay method for these types of problems.