# Distributed Algorithm Design for Constrained Multi-robot Task Assignment

The task assignment problem is one of the fundamental combinatorial optimization problems. It has been extensively studied in operation research, management science, computer science and robotics. Task assignment problems arise in various applications of multi-robot systems (MRS), such as environmental monitoring, disaster response, extraterrestrial exploration, sensing data collection and collaborative autonomous manufacturing. In these MRS applications, there are realistic constraints on robots and tasks that must be taken into account both from the modeling perspective and the algorithmic perspective. From the modeling aspect, such constraints include (a) Task group constraints: where tasks form disjoint groups and each robot can be assigned to at most one task in each group. One example of the group constraints comes from tightly-coupled tasks, where multiple micro tasks form one tightly-coupled macro task and need multiple robots to perform each simultaneously. (b) Task deadline constraints: where tasks must be assigned to meet their deadlines. (c) Dynamically-arising tasks: where tasks arrive dynamically and the payoffs of future tasks are unknown. Such tasks arise in scenarios like searchrescue, where new victims are found dynamically. (d) Robot budget constraints: where the number of tasks each robot can perform is bounded according to the resource it possesses (e.g., energy). From the solution aspect, there is often a need for decentralized solution that are implemented on individual robots, especially when no powerful centralized controller exists or when the system needs to avoid single-point failure or be adaptive to environmental changes. Most existing algorithms either do not consider the above constraints in problem modeling, are centralized or do not provide formal performance guarantees. In this thesis, I propose methods to address these issues for two classes of problems, namely, the constrained linear assignment problem and constrained generalized assignment problem. Constrained linear assignment problem belongs to P, while constrained generalized assignment problem is NP-hard. I develop decomposition-based distributed auction algorithms with performance guarantees for both problem classes. The multi-robot assignment problem is decomposed into an optimization problem for each robot and each robot iteratively solving its own optimization problem leads to a provably good solution to the overall problem. For constrained linear assignment problem, my approaches provides an almost optimal solution. For constrained generalized assignment problem, I present a distributed algorithm that provides a solution within a constant factor of the optimal solution. I also study the online version of the task allocation problem with task group constraints. For the online problem, I prove that a repeated greedy version of my algorithm gives solution with constant factor competitive ratio. I include simulation results to evaluate the average-case performance of the proposed algorithms. I also include results on multi-robot cooperative package transport to illustrate the approach.