A Duality Theory for Continuous-time Dynamic Shortest Paths with Negative Transit Times∗

This paper concerns the shortest path problem for a network in which arc costs can vary with time, each arc has a transit time, parking with a corresponding time-varying cost is allowed at the nodes, and time is modeled as a continuum. The resulting problem is called the continuoustime dynamic shortest path problem, which is well studied in the literature. However, the problem appears as a subproblem when one wishes to test, via an algorithm for dynamic shortest paths, the presence of negative cycles in the residual network in order to develop continuous-time analogues of several well-known optimality conditions for continuous-time dynamic network flow problems. But, in general, the residual network contains arcs with negative transit times and hence the results in the literature are useless for these purposes since all results are based on the assumption of positive transit times. In this paper, we relax this condition to allow negative transit times. We study a corresponding linear program in space of measures and prove the existence of an optimal extreme point solution. Moreover, we define a dual problem and establish a strong duality result that shows under certain assumptions the value of the linear program equals the value of the dual problem and both values are attained. We also present counterexamples to show that strong duality only holds under these assumptions.