Polynomial and analytic methods for classifying complexity of planar graph homomorphisms

TL;DR

This paper develops polynomial and analytic techniques to classify the computational complexity of planar graph homomorphisms, establishing a dichotomy for 4x4 matrices and demonstrating the universality of Valiant's holographic algorithms.

Contribution

It introduces new polynomial and analytic methods to handle infinite lattice conditions and characterizes the tractability of planar graph homomorphisms using tensor products of matchgates.

Findings

Proves a complexity dichotomy for 4x4 matrices in planar graph homomorphisms.

Shows Valiant's holographic algorithms are universal for planar tractability in this context.

Develops methods to handle infinitely many lattice conditions.

Abstract

We introduce some polynomial and analytic methods in the classification program for the complexity of planar graph homomorphisms. These methods allow us to handle infinitely many lattice conditions and isolate the new P-time tractable matrices represented by tensor products of matchgates. We use these methods to prove a complexity dichotomy for $4 \times 4$ matrices that says Valiant's holographic algorithm is universal for planar tractability in this setting.