Stochastic Online Shortest Path Routing: The Value of Feedback

This paper studies online shortest path routing over multi-hop networks. Link costs or delays are time-varying and modeled by independent and identically distributed random processes, whose parameters are initially unknown. The parameters, and hence the optimal path, can only be estimated by routing packets through the network and observing the realized delays. Our aim is to find a routing policy that minimizes the regret (the cumulative difference of expected delay) between the path chosen by the policy and the unknown optimal path. We formulate the problem as a combinatorial bandit optimization problem and consider several scenarios that differ in where routing decisions are made and in the information available when making the decisions. For each scenario, we derive a tight asymptotic lower bound on the regret that has to be satisfied by any online routing policy. These bounds help us to understand the performance improvements we can expect when (i) taking routing decisions at each hop rather than at the source only, and (ii) observing per-link delays rather than end-to-end path delays. In particular, we show that (i) is of no use while (ii) can have a spectacular impact. Three algorithms, with a trade-off between computational complexity and performance, are proposed. The regret upper bounds of these algorithms improve over those of the existing algorithms, and they significantly outperform state-of-the-art algorithms in numerical experiments.Comment: 18 page

A Dynamic Boundary Guarding Problem with Translating Targets

We introduce a problem in which a service vehicle seeks to guard a deadline (boundary) from dynamically arriving mobile targets. The environment is a rectangle and the deadline is one of its edges. Targets arrive continuously over time on the edge opposite the deadline, and move towards the deadline at a fixed speed. The goal for the vehicle is to maximize the fraction of targets that are captured before reaching the deadline. We consider two cases; when the service vehicle is faster than the targets, and; when the service vehicle is slower than the targets. In the first case we develop a novel vehicle policy based on computing longest paths in a directed acyclic graph. We give a lower bound on the capture fraction of the policy and show that the policy is optimal when the distance between the target arrival edge and deadline becomes very large. We present numerical results which suggest near optimal performance away from this limiting regime. In the second case, when the targets are slower than the vehicle, we propose a policy based on servicing fractions of the translational minimum Hamiltonian path. In the limit of low target speed and high arrival rate, the capture fraction of this policy is within a small constant factor of the optimal.Comment: Extended version of paper for the joint 48th IEEE Conference on Decision and Control and 28th Chinese Control Conferenc

Game theoretic controller synthesis for multi-robot motion planning Part I : Trajectory based algorithms

We consider a class of multi-robot motion planning problems where each robot is associated with multiple objectives and decoupled task specifications. The problems are formulated as an open-loop non-cooperative differential game. A distributed anytime algorithm is proposed to compute a Nash equilibrium of the game. The following properties are proven: (i) the algorithm asymptotically converges to the set of Nash equilibrium; (ii) for scalar cost functionals, the price of stability equals one; (iii) for the worst case, the computational complexity and communication cost are linear in the robot number