Robust Reward Placement under Uncertainty

Reward placement is a common optimization problem in network diffusion processes, where a number of rewards are to be placed in a network so as to maximize the total reward obtained as agents move randomly in it. In many settings, the precise mobility network might be one of several possible, based on parameters outside our control, such as the weather conditions affecting peoples' transportation means. Solutions to the reward placement problem must thus be robust to this uncertainty, by achieving a high utility in all possible networks. To study such scenarios, we introduce the Robust Reward Placement problem (RRP). Agents move randomly on a Markovian Mobility Model that has a predetermined set of locations but its precise connectivity is unknown and chosen adversarialy from a known set Π of candidates. Network optimization is achieved by selecting a set of reward states, and the goal is to maximize the minimum, among all candidates, ratio of rewards obtained over the optimal solution for each candidate. We first prove that RRP is NP-hard and inapproximable in general. We then develop Ψ-Saturate, a pseudo-polynomial time algorithm that achieves an ϵ-additive approximation by exceeding the budget constraint by a factor that scales as O(ln|Π|/ϵ). In addition, we present several heuristics, most prominently one inspired from a dynamic programming algorithm for the max-min 0-1 Knapsack problem. We corroborate our theoretical findings with an experimental evaluation of the methods in both synthetic and real-world datasets.