Holographic algorithms without matchgates

TL;DR

This paper simplifies holographic algorithms by removing matchgates, using matrix Pfaffians for counting solutions, and broadening their applicability with new algebraic tests, enhancing understanding and practical implementation.

Contribution

It introduces a matchgate-free approach to holographic algorithms, enabling new algebraic tests and expanding their applicability to more problems.

Findings

Simplified holographic algorithms using edge matrices and Pfaffians.

Broadened the class of problems solvable by these algorithms.

Enhanced understanding of the geometric and algebraic structure of complexity classes.

Abstract

The theory of holographic algorithms, which are polynomial time algorithms for certain combinatorial counting problems, yields insight into the hierarchy of complexity classes. In particular, the theory produces algebraic tests for a problem to be in the class P. In this article we streamline the implementation of holographic algorithms by eliminating one of the steps in the construction procedure, and generalize their applicability to new signatures. Instead of matchgates, which are weighted graph fragments that replace vertices of a natural bipartite graph G associated to a problem P, our approach uses only only a natural number-of-edges by number-of-edges matrix associated to G. An easy-to-compute multiple of its Pfaffian is the number of solutions to the counting problem. This simplification improves our understanding of the applicability of holographic algorithms, indicates a more geometric approach to complexity classes, and facilitates practical implementations. The generalized applicability arises because our approach allows for new algebraic tests that are different from the "Grassmann-Plucker identities" used up until now. Natural problems treatable by these new methods have been previously considered in a different context, and we present one such example.