Zeros of Holant problems: locations and algorithms

TL;DR

This paper develops efficient approximation algorithms for a broad class of Holant problems, leveraging polynomial zero analysis, and highlights a key computational complexity contrast with spin systems.

Contribution

It introduces a general approach using polynomial zeros to approximate Holant problems, extending the scope of efficient algorithms and contrasting with known phase transition phenomena.

Findings

Polynomial zero analysis enables approximation schemes for Holant problems.

Most Holant problems on cubic graphs are efficiently approximable, except for counting perfect matchings.

The approach reveals a sharp contrast with phase transitions in spin systems.

Abstract

We present fully polynomial-time (deterministic or randomised) approximation schemes for Holant problems, defined by a non-negative constraint function satisfying a generalised second order recurrence modulo a couple of exceptional cases. As a consequence, any non-negative Holant problem on cubic graphs has an efficient approximation algorithm unless the problem is equivalent to approximately counting perfect matchings, a central open problem in the area. This is in sharp contrast to the computational phase transition shown by 2-state spin systems on cubic graphs. Our main technique is the recently established connection between zeros of graph polynomials and approximate counting. We also use the "winding" technique to deduce the second result on cubic graphs.