TY - JOUR
T1 - Polynomial-time solvable #CSP problems via algebraic models and Pfaffian circuits
AU - Margulies, S.
AU - Morton, J.
N1 - Funding Information:
The authors would like to acknowledge the support of the Defense Advanced Research Projects Agency under Award No. N66001-10-1-4040 . We would also like to thank Tyson Williams and William Whistler for helpful comments. Finally, very special thanks to our anonymous reviewer.
Publisher Copyright:
© 2015 .
PY - 2016/5
Y1 - 2016/5
N2 - A Pfaffian circuit is a tensor contraction network where the edges are labeled with basis changes in such a way that a very specific set of combinatorial properties is satisfied. By modeling the permissible basis changes 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 basis change, and one that is possible under a heterogeneous basis change only. We enumerate a collection 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 basis changes. 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.
AB - A Pfaffian circuit is a tensor contraction network where the edges are labeled with basis changes in such a way that a very specific set of combinatorial properties is satisfied. By modeling the permissible basis changes 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 basis change, and one that is possible under a heterogeneous basis change only. We enumerate a collection 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 basis changes. 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.
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U2 - 10.1016/j.jsc.2015.06.008
DO - 10.1016/j.jsc.2015.06.008
M3 - Article
AN - SCOPUS:84948718401
VL - 74
SP - 152
EP - 180
JO - Journal of Symbolic Computation
JF - Journal of Symbolic Computation
SN - 0747-7171
ER -