A Collapse Theorem for Holographic Algorithms with Matchgates on Domain Size at Most 4

Holographic algorithms with matchgates are a novel approach to design polynomial time computation. It uses Kasteleyn’s algorithm for perfect matchings, and more importantly a holographic reduction. The two fundamental parameters of a holographic reduction are the domain size k of the underlying problem, and the basis size l. A holographic reduction transforms the computation to matchgates by a linear transformation that maps to (a tensor product space of) a linear space of dimension 2. We prove a sharp basis collapse theorem, that shows that for domain size 3 and 4, all non-trivial holographic reductions have basis size l collapse to 1 and 2 respectively. The main proof techniques are Matchgate Identities, and a Group Property of matchgate signatures. University of Wisconsin-Madison and Peking University. [email protected]. Supported by NSF CCF-0914969 and NSF CCF-1217549. ,Department of Computer Science and Engineering, Shanghai Jiao Tong University. [email protected]