Generalized counting constraint satisfaction problems with determinantal circuits

Jason Ryder Morton, Jacob Turner

Research output: Contribution to journalArticle

4 Citations (Scopus)

Abstract

Generalized counting constraint satisfaction problems include Holant problems with planarity restrictions; polynomial-time 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.

Original languageEnglish (US)
Pages (from-to)357-381
Number of pages25
JournalLinear Algebra and Its Applications
Volume466
DOIs
StatePublished - Feb 1 2015

Fingerprint

Counting Problems
Constraint satisfaction problems
Constraint Satisfaction Problem
Pfaffian
Networks (circuits)
Monoidal Category
Spanning Forest
Restriction
Skew symmetric matrix
Quantum Circuits
Tutte Polynomial
Category Theory
Polynomials
Planarity
Closed curve
Expressive Power
Matroid
Graph theory
Categorical
Functor

All Science Journal Classification (ASJC) codes

  • Algebra and Number Theory
  • Numerical Analysis
  • Geometry and Topology
  • Discrete Mathematics and Combinatorics

Cite this

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Generalized counting constraint satisfaction problems with determinantal circuits. / Morton, Jason Ryder; Turner, Jacob.

In: Linear Algebra and Its Applications, Vol. 466, 01.02.2015, p. 357-381.

Research output: Contribution to journalArticle

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