Parameter Shortest Path Algorithms with an Application to Cyclic Staffing

Abstract

Let G = (V,E) be a digraph with n vertices including a special vertex s. Let E' C E be a designated subset of edges. For each e E E there is an associated real number fl(e). Furthermore, let 1 if e E E' f2(e): 0 if e E-E' The length of edge e is flpe)-Af2(e), where X is a parameter that takes on real values. Thus the length varies additively in X for each edge of E'. We shall present two algorithms for computing the shortest path from s to each vertex v E V parametrically in the parameter X, with respective running times O(n3 ) and O(nlE llogn). For dense digraphs the running time of the former algorithm is comparable to the fastest (non-parametric) shortest path algorithm known. This work generalizes the results of Karp [2] concerning the minimum cycle mean of a digraph, which reduces to the case that E' = E. Furthermore, the second parametric algorithm may be used in conjunction with a transformation given by Bartholdi, Orlin, and Ratliff [1] to give an O(n21logn) algorithm for the cyclic staffing problem.