Algorithm for shortest paths in bipartite digraphs with concave weight matrices and its applications

The traveling salesman problem on an n-point convex polygon and the minimum latency tour problem for n points on a straight line are two basic problems in graph theory and have been studied in the past. Previously, it was known that both problems can be solved in O(n²) time. However, whether they can be solved in o(n²) time was left open by Marcotte and Suri and Afrati et al., respectively. In this paper we show that both problems can be solved in O(n log n) time by reducing them to the following problem: Given an edge-weighted complete bipartite digraph G = (X, Y, E) with X = {x₀,..., x_n} and Y = {y₀,..., y_m}, we wish to find the shortest path from x₀ to x_n in G. This new problem requires Ω(nm) time to solve in general, but we show that it can be solved in O(n+m log n) time if the weight matrices A and B of G are both concave, where for 0≤i≤n and 0≤j≤m, A[i,j] and B[j,i] are the weights of the edges (x_i,y_j) and (y_j,x_i) in G, respectively. As demonstrated in this paper, the new problem is a powerful tool and we believe that it can be used to solve more problems.