On the Complexity of Cycle Enumeration for Simple Graphs

Abstract

We show how a number of combinatorial problems, such as determining the number of cycles in graphs, can be recast using a graded commutative algebra constructed within a real Grassmann exterior algebra. The computational complexity of this approach is then measured by considering operations at the basis blade level of the algebra. In particular, the worst-case time complexity of counting arbitrary length cycles in simple n-vertex graphs via nilpotent adjacency matrix methods is shown to be , where α≤3 is the exponent representing the complexity of matrix multiplication. The storage complexity of the nilpotent adjacency matrix approach is . A probabilistic model is used to describe a class of graphs in which the average-case time complexity of cycle enumeration is for fixed 0<q<1. For reference, experimental results detailing computation times (in seconds) are compared with algorithms based on the approaches of Bax and Tarjan.