HOLOGRAPHIC ALGORITHM WITH MATCHGATES IS UNIVERSAL FOR PLANAR #CSP OVER BOOLEAN DOMAIN

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