The theory of holographic algorithms, which are polynomial time algorithms for certain combinatorial counting problems, surprised the complexity community by showing certain problems, very similar to #P complete problems, were in fact in the class P. In particular, the theory produces algebraic tests for a problem to be in P. In this article we describe the geometric basis of these algorithms by (i) replacing the construction of graph fragments in the procedure by the direct construction of a skew symmetric matrix, and (ii) replacing the computation of weighted perfect matchings of an auxiliary graph by computing the Pfaffian of the directly constructed skew-symmetric matrix. This procedure indicates a more geometric approach to complexity classes. It also leads to more general constructions where one replaces the "Grassmann-Plücker identities" which test for admissibility by other algebraic tests. Natural problems treatable by these methods have been previously considered in a different context, and we present one such example.
All Science Journal Classification (ASJC) codes
- Algebra and Number Theory
- Numerical Analysis
- Geometry and Topology
- Discrete Mathematics and Combinatorics