2 research outputs found
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