Generalized counting constraint satisfaction problems with determinantal circuits

Abstract

Available online xxxx Submitted by J.M. Landsberg MSC: 15A15 15A69 15A24 18D10 03D15Generalized counting constraint satisfaction problems include Holant problems with planarity restrictions; polynomialtime algorithms for such problems include matchgates and matchcircuits, which are based on Pfaffians. In particular, they use gates which are expressible in terms of a vector of sub-Pfaffians of a skew-symmetric matrix. We introduce a new type of circuit based instead on determinants, with seemingly different expressive power. In these determinantal circuits, a gate is represented by the vector of all minors of an arbitrary matrix. Determinantal circuits permit a different class of gates. Applications of these circuits include proofs of theorems from algebraic graph theory including the Chung-Langlands formula for the number of rooted spanning forests of a graph and computing Tutte polynomials of certain matroids. They also give a strategy for simulating quantum circuits with closed timelike curves. Monoidal category theory provides a useful language for discussing such counting problems, turning combinatorial restrictions into categorical properties. We introduce the counting problem in monoidal categories and count-preserving functors as a way to study FP subclasses of problems in settings which are generally #P-hard. Using this machinery we show that, surprisingly, determinantal circuits can be simulated by Pfaffian circuits at quadratic cost.

Belief propagation in monoidal categories

“…In this case it is sometimes called a contraction problem (Problem 1). For example, in semiringed categories, the word problem for morphisms in Mor(I, I) generalizes counting constraint satisfaction problems [25].…”

Polynomial-time solvable #CSP problems via algebraic models and Pfaffian circuits

“…Pfaffian circuits (Landsberg et al, 2013;Morton, 2011;Morton and Turner, 2015) are a simplified and extensible reformulation of Valiant's notion of a holographic algorithm, which builds on J.Y. Cai and V. Choudhary's work in expressing holographic algorithms in terms of tensor contraction networks (Cai and Choudhary, 2007).…”

“…Valiant's Holant Theorem (Valiant, 2002, 2008 is an equation where the left-hand side has both an exponential number of terms and an exponential number of term cancellations, whereas the right-hand side is computable in polynomial time. Extensions to holographic algorithms made possible by Pfaffian circuits include swapping out this equation for another combinatorial identity (Morton and Turner, 2015), viewing this equation as an equivalence of monoidal categories (Morton and Turner, 2015), or, as is done here, using heterogeneous changes of bases with the aid of computational commutative algebra.…”

Tensor Network Contractions for #SAT

“…There is body of recent work which has applied tensor network methods appearing in physics to solve combinatorial problems appearing in computer science [15][16][17][18][19][20][21]. Contracting a network to determine the norm of a Boolean tensor network state is equivalent to counting the number of satisfying solutions of a Boolean formula.…”