Decentralised multi-robot task allocation algorithms.

Multi Robot Systems (MRS) are gaining increasing popularity both in the research community and in industry. A fundamental problem that underpins the effective coordinated operation of these systems is Task Allocation. To solve this problem, the MRS should be able to find an answer quickly, reliably, and effectively to the question: "given the robots available in the system and the tasks that ought to be carried out, what is the best allocation of these tasks among us?". In this thesis we focus on solving this problem in the decentralised setting, that is, when each agent only has access to its own utility function and does not have any knowledge of the functions corresponding to other agents, i.e.the utility function is local or private. Our algorithms are based on improved versions of the measured continuous greedy algorithm for general matroid-constrained submodular maximisation. The first improvement is a new and smoother increment rule that enables us to reduce the number of steps required to solve the relaxation. The second improvement is to adapt the Decreasing-Threshold procedure for monotone submodular functions to work with non-monotone submodular functions. Then, we present the first decentralised algorithm with constant-factor approximation guarantees for general submodular task allocation. Our algorithm provides an approximation factor of 1-1/e-4ϵ (≈63%) for monotone submodular utilities, and a factor of (1/e-3ϵ) (≈37%) for non-monotone submodular functions. To illustrate the possibilities enabled by non-monotone submodular task allocation, we present a submodular task allocation model for a multi-UAV surveillance mission. Our model features the allocation of heterogeneous surveillance tasks to a heterogeneous multi-UAV team under risk of enemy detection. We develop the model and present proofs to show that it is non-monotone submodular. Then, we run numerical experiments to study the effect of different parameters of our algorithm and compare its performance against the state-of-the- art. To conclude the thesis, we take a completely different approach, the key idea is to trade constant-factor approximation guarantees in exchange for flexibility. We present a preliminary framework based on combinatorial auctions that can transfer centralised solution method to the decentralised Task Allocation domain while requiring a polynomial number of communication rounds. In other words, our framework provides a way to transfer successful methods to solve NP-Hard problems such as Metaheuristics, Mixed-Integer Programming, Constraint Programming, etc. to the decentralised setting.