Algorithm for optimal chance constrained linear assignment

In this paper, we design provably-good algorithms for task allocation in multi-robot systems in the presence of payoff uncertainty. We consider a group of robots that has to perform a given set of tasks where each robot performs at most one task. The payoffs of the robots doing the tasks are assumed to be Gaussian random variables with known mean and variances. The total payoff of the robots is a sum of the individual payoffs of all the robots. The goal is to find an assignment with maximum payoff that can be achieved with a specified probability irrespective of the realization of the random variable. This problem can be formulated as a chance constrained combinatorial optimization problem. We develop a novel deterministic technique to solve this chance constrained optimization problem that ensures that the chance constraints are always satisfied. Adopting the notion of risk-aversion from the economics literature, we formulate a risk-averse task allocation problem, which is a deterministic integer optimization problem. We prove that by repeatedly solving the risk-averse task allocation problem using a one-dimensional search on the risk aversion parameter we find a solution for the chance constrained optimization formulation of the linear assignment problem with uncertain payoffs. We provide simulation results on randomly generated data to demonstrate our approach and also compare our method to existing approaches.