Reductions in Computational Complexity Using Clifford Algebras

Abstract

A number of combinatorial problems are treated using properties of abelian null-square-generated and idempotent-generated subalgebras of Clifford algebras. For example, the problem of deciding whether or not a graph contains a Hamiltonian cycle is known to be NP-complete. By considering entries of Λ k , where Λ is an appropriate nilpotent adjacency matrix, the kcycles in any finite graph are recovered. Within the algebra context (i.e., considering the number of multiplications performed within the algebra), these problems are reduced to matrix multiplication, which is in complexity class P. The Hamiltonian cycle problem is one of many problems moved from classes NP-complete and P-complete to class P in this context. Other problems considered include the set covering problem, counting the edge-disjoint cycle decompositions of a finite graph, computing the permanent of an arbitrary matrix, computing the girth and circumference of a graph, and finding the longest path in a graph 1 . Mathematics Subject Classification (2000). 68Q15, 05C50, 05C38, 60B99, 81P68.

CAUSTA: Clifford Algebra‐based Unified Spatio‐Temporal Analysis

“…Some early research on graph theory have proved that Clifford algebra can express the dynamic evolution of a directional and unidirectional finite graph, and can greatly reduce the computational complexity (e.g. Schott and Staples 2010). The geometric characteristics (direction, distance, topology and adjacent relationships) can be easily modeled by coding the nodes and routes with Clifford elements.…”

Zeons, lattices of partitions, and free probability

“…Their combinatorial properties have been applied to the study of graphs in a number of works by the current authors (cf. [11], [12], [9]), although the name "zeons" is attributed to Feinsilver [4]. Definition 2.1.…”

On computational complexity of Clifford algebra

“…Anyway we can conclude that any actual calculation performed by means of a Clifford algebra formulation (see e.g. [9] or [8]) is sandwiched between these lower and upper bounds, respectively:…”

“…⊗H 1 ⊗½ 2 ⊗H 1 )(γ⊗γ)(9) and it's easy to see, calling P 23 the symmetric permutation matrix corresponding to permutation {1, 3, 2, 4}, that P 23 (½ 2 ⊗ H 1 )P 23 = H 1 ⊗ ½ 2 so that we may write for the new transformation matrix H ⊗ H = ½ 2 ⊗ [P 23 (½ 2 ⊗ H 1 )P 23 ] ⊗ H 1 and, with easy passages,…”