Strongly polynomial-time and NC algorithms for detecting cycles in periodic graphs

Abstract

This paper is concerned with the problem of recognizing, in a graph with rational vector-weights associates with the edges, the existence of a cycle whose total weight is the zero vector. This problem is known to be equivalent to the problem of recognizing the existence of cycles in periodic (dynamic) graphs and to the validity of systems of recursive formulas. It was previously conjectured that combinatorial algorithms exist for the cases of two- and three-dimensional vector-weights. It is shown that strongly polynomial algorithms exist for any fixed dimension d. Moreover, these algorithms also establish membership in the class NC. On the other hand, it is shown that when the dimension of the weights is not fixed, the problem is equivalent to the general linear programming problem under strongly polynomial and logspace reductions