Holographic Algorithm with Matchgates Is Universal for Planar $\#$CSP Over Boolean Domain

TL;DR

This paper provides a complete complexity classification for Boolean counting constraint satisfaction problems, showing that holographic algorithms with matchgates are universal for efficiently solvable planar instances.

Contribution

It establishes a comprehensive classification theorem for $\

Findings

Classifies all Boolean $\

paper_type":"theoretical"}}

Abstract

We prove a complexity classification theorem that classifies all counting constraint satisfaction problems ($\#$CSP) over Boolean variables into exactly three categories: (1) Polynomial-time tractable; (2) $\#$P-hard for general instances, but solvable in polynomial-time over planar graphs; and (3) $\#$P-hard over planar graphs. The classification applies to all sets of local, not necessarily symmetric, constraint functions on Boolean variables that take complex values. It is shown that Valiant's holographic algorithm with matchgates is a universal strategy for all problems in category (2).