Holographic Algorithms with Matchgates Capture Precisely Tractable Planar #CSP

TL;DR

This paper characterizes the exact class of counting problems solvable by holographic algorithms with matchgates, showing they precisely capture problems that are #P-hard in general but polynomial-time solvable on planar graphs.

Contribution

It establishes a complexity dichotomy theorem for counting CSP problems, classifying them into three explicit categories based on their computational complexity.

Findings

Problems in the class are either polynomial-time solvable, #P-hard on general graphs but tractable on planar graphs, or #P-hard even on planar graphs.

Holographic algorithms with matchgates exactly solve the class of problems that are #P-hard in general but tractable on planar graphs.

The classification criteria for these problem categories are explicitly defined.

Abstract

Valiant introduced matchgate computation and holographic algorithms. A number of seemingly exponential time problems can be solved by this novel algorithmic paradigm in polynomial time. We show that, in a very strong sense, matchgate computations and holographic algorithms based on them provide a universal methodology to a broad class of counting problems studied in statistical physics community for decades. They capture precisely those problems which are #P-hard on general graphs but computable in polynomial time on planar graphs. More precisely, we prove complexity dichotomy theorems in the framework of counting CSP problems. The local constraint functions take Boolean inputs, and can be arbitrary real-valued symmetric functions. We prove that, every problem in this class belongs to precisely three categories: (1) those which are tractable (i.e., polynomial time computable) on general graphs, or (2) those which are \#P-hard on general graphs but ractable on planar graphs, or (3) those which are #P-hard even on planar graphs. The classification criteria are explicit. Moreover, problems in category (2) are tractable on planar graphs precisely by holographic algorithms with matchgates.