Finding minimum cost to time ratio cycles with small integral transit times

Let D = (V, E) be a digraph with n vertices and m arcs. For each e E E there is an associated cost ce and a transit time te; Ce can be arbitrary, but we require t to be a non-negative integer. The cost to time ratio of a cycle C is X(C) = 3 ec ceCeec t. Let E' c E denote the set of arcs e with te > 0, let T = max{tv: (u, v) E} for each vertex u, and let T = uev T. We give a new algorithm for finding a cycle C with the minimum cost to time ratio X(C). The algorithm's (T(m + n log n)) running time is dominated by O(T) shortest paths calculations on a digraph with non-negative arc lengths. Further, we consider early termination of the algorithm and a faster O(Tm) algorithm in case E E' is acyclic, i.e., in case each cycle has a strictly positive transit time, which gives an O(n ) algorithm for a class of cyclic staffing problems considered by Bartholdi et al. The algorithm can be seen to be an extension of the O(nm) algorithm of Karp for the case in which t = 1 for all e E E, which is the problem of calculating a minimum mean cycle. Our algorithm can also be modified to solve the related parametric shortest paths problem in O(T(m + n log n)) time. © 1993 by John Wiley & Sons, Inc.