Improved Approximation for Time-Dependent Shortest Paths

Abstract

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 ε–approximation algorithm that runs $O( {\lambda \over \epsilon} \log ({T_{max} \over T_{min}}) \log({L \over \lambda \epsilon T_{min}}))$ shortest path computations, where λ 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 ε–approximation algorithms that improve upon Foschini et al.’s result. Our first algorithm runs $O({\lambda \over \epsilon}(\log ({T_{max}\over T_{min}})+ \log ({ L\over \lambda T_{min}})))$ shortest path computations at fixed departure times. In our second algorithm, we reduce the dependency on L, by using only $O(\lambda( {1 \over \epsilon} \log ({T_{max} \over T_{min}})+ \log ({ L \over \lambda \epsilon T_{min}})))$ total shortest path computations.