In the minimum-cost strong network orientation problem (MCSO), we are given an undirected graph G = (V, E) with nonnegative edge lengths l(e) and a transportation schedule T = {(s1,t1,w1), ..., (sk,tk,wk)}, where wi units of weight have to be transported from the source vertex si to the target vertex ti for i = 1, ..., k. Let Gσ be a strongly connected orientation of G and let Lσi be the length of the shortest (directed) path from si to ti in Gσ. The goal in the MCSO is to find a strongly connected orientation Gσ such that the overall cost of the orientation given by Σki=1 wiLσi (sum case) or maxi=1,...,kwiLσi (bottleneck case) is minimized. The strong network orientation problem is motivated by the practical problem of designing the optimal undirectional flow path of automated guided vehicles. In this paper, we investigate the MCSO from the algorithmic and complexity points of view and propose a classification scheme. In the first part of the paper, we identify several efficiently solvable cases of the MCSO with sum and bottleneck objective functions which arise if additional restrictions are imposed on the structure of the graph G, the edge lengths l(e), and/or the transportation schedule T. In the second part, we identify special cases of the MCSO which are NP-hard.
