Complexity and approximate algorithm of shortest paths in dynamic networks

The shortest path problem of dynamic directed networks is significant in the disciplines of transportation and communication systems. In the classical models, the weight of each arc is invariant and usually given beforehand, but it may be varying in the practical problems. For instance, the running time of a car across a city block would be different according to the temporal traffic flow density. The shortest path problem in this context can be reduced to the Dynamic Single Source Shortest Path (DSSSP) problem. This paper first discusses the computational complexity and proves that the DSSSP problem is NP-hard. Then to aim to propose a new approximate algorithm for the DSSSP problem, the authors introduce the concept of the stability of the shortest path tree, and moreover, give the sufficient and necessary condition. The idea is as follows: first, a series of sectional linear functions are selected to approach the original nonlinear arc weight function. Then each corresponding linear time interval is partitioned into several stable subintervals, in which the dynamic shortest path tree maintains invariability. Finally, the holistic shortest path can be found by connecting the solutions in each stable subinterval. The effectiveness of the new algorithm is estimated by simulating experiments.