Stochastic shortest path with unlimited hops

We present new results for the Stochastic Shortest Path problem when an unlimited number of hops may be used. Nodes and links in the network may be congested or uncongested, and their states change over time. The goal is to adaptively choose the next node to visit such that the expected cost to the destination is minimized. Since the state of a node may change, having the option to revisit a node can improve the expected cost. Therefore, the option to use an unbounded number of hops may be required to achieve the minimum expected cost. We show that when revisits are prohibited, the optimal routing problem is np-hard. We also prove properties about networks for which continual improvement may occur.We study the related routing problem which asks whether it is possible to determine the optimal next node based on the current node and state, when an unlimited number of hops is allowed. We show that as the number of hops increases, this problem may not converge to a solution.