Randomized Shortest-Path Problems: Two Seemingly Unrelated Models

Suppose you have to route agents through a network from a source node to a destination node in some optimal way; for instance by minimizing the total travel cost. Nothing new up to now – you could use a standard shortest-path algorithm. Suppose, however, that you want to avoid pure deterministic routing policies in order, for instance, to allow some exploration of the network, to avoid congestion, or to avoid complete predictability of your routing strategy. In other words, you want to introduce some randomness/unpredictability in the routing policy, i.e., the routing policy is randomized. This problem, which will be called the randomized shortest path problem, is introduced in this paper. The global degree of randomness of the routing policy will be quantified by the expected Shannon entropy spread into the network, and is provided a priori by the designer. Then, necessary conditions allowing to compute the optimal randomized policy – minimizing the expected routing cost – are derived. Iterating these necessary conditions, reminiscent of Bellman’s value iteration equations, allows to compute the optimal policy, that is, the set of conditional probabilities of following a link in each node. Interestingly and surprisingly enough, the proposed model, while formulated in a totally different framework, reduces to Akamatsu’s model [3], introduced in transportation science, for a special choice of the entropy constraint. We therefore revisit Akamatsu’s model by recasting it into a statistical physics formalism allowing to easily derive all the quantities of interest in an elegant, unified, way. For instance, the optimal policy can be obtained by solving a linear system of equations, which was not noticed by Akamatsu. Finally, simulation results show that the system behaves as expected.