Computational complexity reductions using clifford algebras

Abstract

Given a computing architecture based on Clifford algebras, a natural context for determining an algorithm's time complexity is in terms of the number of geometric (Clifford) operations required. In this paper the existence of such a processor is assumed, and a number of graph-theoretical problems are considered. This paper is an extension of previous work, in which the authors defined the nilpotent adjacency matrix associated with a finite graph and showed that a number of graph problems of complexity class NP are polynomial in the number of Clifford operations required. Previous results are recalled and illustrated with Mathematica examples. New results are obtained, and old results are improved by the development of new techniques. In particular, a matrix-free approach is developed to count matchings, compute girth, and enumerate proper cycle covers of finite graphs. These new results and techniques are also illustrated with Mathematica examples. © 2010 Springer-Verlag London Limited.