Polynomial-time Solvable #CSP Problems via Algebraic Models and Pfaffian Circuits

A Pfaffian circuit is a tensor contraction network where the edges are labeled with changes of bases in such a way that a very specific set of combinatorial properties are satisfied. By modeling the permissible changes of bases as systems of polynomial equations, and then solving via computation, we are able to identify classes of 0/1 planar #CSP problems solvable in polynomial-time via the Pfaffian circuit evaluation theorem (a variant of L. Valiant's Holant Theorem). We present two different models of 0/1 variables, one that is possible under a homogeneous change of basis, and one that is possible under a heterogeneous change of basis only. We enumerate a series of 1,2,3, and 4-arity gates/cogates that represent constraints, and define a class of constraints that is possible under the assumption of a ``bridge" between two particular changes of bases. We discuss the issue of planarity of Pfaffian circuits, and demonstrate possible directions in algebraic computation for designing a Pfaffian tensor contraction network fragment that can simulate a swap gate/cogate. We conclude by developing the notion of a decomposable gate/cogate, and discuss the computational benefits of this definition.