Holographic algorithm with matchgates is universal for planar #CSP over boolean domain

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).Proof. One direction is trivial, since (= 4 ) ∈ EQ 2 . For the other direction we prove by induction. For k = 1, we have (= 2 ) ∈ EQ. For k = 2, we have (= 4 ) given. Assume that we have (= 2(k−1) ). Then connecting (= 2(k−1) ) and (= 4 ) by one edge we get (= 2k ).By (2.

On blockwise symmetric matchgate signatures and higher domain #CSP

“…This fact, that under the Hadamard transformation all Equalities' become matchgate signatures is proved to be the universal reason that #CSP problems that are #P-hard can become P-time tractable on planar structures. This is proved in [2].…”

“…It is #P-hard over planar graphs. Moreover, category (2) consists precisely of those problems that are holographically reducible to the FKT algorithm, whereby all constraint functions in F and Equality functions of all arities are transformed to matchgate signatures. Theorem 1.1 shows that holographic algorithms with matchgates form a universal strategy, that applies a holographic transformation whereby we transform all Equality functions to matchgates, for all problems in this framework that are #P-hard in general but solvable in polynomial time on planar graphs.…”

“…By Theorem 2.1 and EQ ⊆ M , if F ⊆ M , then Pl-#CSP(F) can be computed in polynomial time by the holographic transformation using H 2 and the FKT algorithm. More precisely, we have the following complexity trichotomy theorem for #CSP on the Boolean domain [10,14,2]: every single problem in this class can be classified into one of three types. The first type can be solved in polynomial time over arbitrary structures.…”

“…The following classification theorem for #CSP on the Boolean domain is proved in gradually increasing generalities [10,14,2], reaching full generality in [2]:…”

Perfect Matchings, Rank of Connection Tensors and Graph Homomorphisms

“…Valiant introduced holographic algorithms to significantly extend the reach of this methodology [33,32,12]. It is proved that for all #CSP where variables are Boolean (but constraint functions can take complex values), the methodology of holographic algorithms is universal [8]. More precisely, we can prove that (A) the three-way classification above holds, and (B) the problems that belong to type (2) are precisely those that can be captured by this single algorithmic approach, namely a holographic reduction to Kasteleyn's algorithm.…”