Algorithms for Minimum-Cost Paths in Time-Dependent Networks with Waiting Policies

We study the problem of computing minimum-cost paths through a time-varying network, in which the travel time and travel cost of each arc are known functions of one's departure time along the arc. For some problem instances, the ability to wait at nodes may allow for less costly paths through the network. When waiting is allowed, it is constrained by a (potentially time-varying) waiting policy that describes the length of time one may wait and the cost of waiting at every node. In discrete time, time-dependent shortest path problems with waiting constraints can be optimally solved by straightforward dynamic programming algorithms; however, for some waiting policies these algorithms can be computationally impractical. In this article, we survey several broad classes of waiting policies and show how techniques for speeding up dynamic programming can be effectively applied to obtain practical algorithms for these different problem variants

Learning to Prune: Speeding up Repeated Computations

It is common to encounter situations where one must solve a sequence of similar computational problems. Running a standard algorithm with worst-case runtime guarantees on each instance will fail to take advantage of valuable structure shared across the problem instances. For example, when a commuter drives from work to home, there are typically only a handful of routes that will ever be the shortest path. A naive algorithm that does not exploit this common structure may spend most of its time checking roads that will never be in the shortest path. More generally, we can often ignore large swaths of the search space that will likely never contain an optimal solution. We present an algorithm that learns to maximally prune the search space on repeated computations, thereby reducing runtime while provably outputting the correct solution each period with high probability. Our algorithm employs a simple explore-exploit technique resembling those used in online algorithms, though our setting is quite different. We prove that, with respect to our model of pruning search spaces, our approach is optimal up to constant factors. Finally, we illustrate the applicability of our model and algorithm to three classic problems: shortest-path routing, string search, and linear programming. We present experiments confirming that our simple algorithm is effective at significantly reducing the runtime of solving repeated computations

On green routing and scheduling problem

The vehicle routing and scheduling problem has been studied with much interest within the last four decades. In this paper, some of the existing literature dealing with routing and scheduling problems with environmental issues is reviewed, and a description is provided of the problems that have been investigated and how they are treated using combinatorial optimization tools