Improved Approximation for Time-Dependent Shortest Paths

We study the approximation of minimum travel time paths in time dependent networks. The travel time on each link of the network is a piecewise linear function of the departure time from the start node of the link. The objective is to find the minimum travel time to a destination node d, for all possible departure times at source node s. Dehne et al. proposed an exact output-sensitive algorithm for this problem [6, 7] that improves, in most cases, upon the existing algorithms. They also provide an approximation algorithm. In [10, 11], Foschini et al. show that this problem has super-polynomial complexity and present an epsilon-approximation(1) algorithm that runs O(lambda/epsilon log(T-max/T-min) log(L/lambda epsilon T-min)) shortest path computations, where O is the total number of linear pieces in travel time functions on links, L is the horizontal span of the travel time function and T-min and T-max are the minimum and maximum travel time values, respectively. In this paper, we present two T-approximation algorithms that improve upon Foschini et al.'s result. Our first algorithm runs O(lambda/epsilon (log(T-max/T-min) + log(L/lambda T-min))) shortest path computations at fixed departure times. In our second algorithm, we reduce the dependency on L, by using only O(lambda(1/epsilon log(T-max/T-min) + log(L/lambda epsilon T-min))) total shortest path computations.