Holographic Algorithms Beyond Matchgates

TL;DR

This paper extends holographic algorithms by replacing matchgates with affine and product-type functions, providing polynomial-time algorithms for deciding and finding holographic reductions for these broader classes, including symmetric functions.

Contribution

It introduces new tractable classes of holographic reductions using affine and product-type functions, with algorithms for decision and search problems, and applies algebraic techniques to analyze complexity.

Findings

Polynomial-time algorithms for affine and product-type holographic reductions

Decidability results for symmetric functions in holographic algorithms

Efficiently decidable dichotomy for Holant problems with symmetric constraints

Abstract

Holographic algorithms introduced by Valiant are composed of two ingredients: matchgates, which are gadgets realizing local constraint functions by weighted planar perfect matchings, and holographic reductions, which show equivalences among problems with different descriptions via certain basis transformations. In this paper, we replace matchgates in the paradigm above by the affine type and the product type constraint functions, which are known to be tractable in general (not necessarily planar) graphs. More specifically, we present polynomial-time algorithms to decide if a given counting problem has a holographic reduction to another problem defined by the affine or product-type functions. Our algorithms also find a holographic transformation when one exists. We further present polynomial-time algorithms of the same decision and search problems for symmetric functions, where the complexity is measured in terms of the (exponentially more) succinct representations. The algorithm for the symmetric case also shows that the recent dichotomy theorem for Holant problems with symmetric constraints is efficiently decidable. Our proof techniques are mainly algebraic, e.g., using stabilizers and orbits of group actions.