Efficient Path Planning and Task Allocation Algorithm for Boolean Specifications

This paper presents a novel path-planning and task assignment algorithm for multi-robot systems that should fulfill a global Boolean specification. The proposed method is based on Integer Linear Programming (ILP) formulations, which are combined with structural insights from Petri nets to improve scalability and computational efficiency. By proving that the \emph{constraint matrix} is totally unimodular (TU) for certain classes of problems, the ILP formulation can be relaxed into a Linear Programming (LP) problem without losing the integrality of the solution. This relaxation eliminates complex combinatorial techniques, significantly reducing computational overhead and thus ensuring scalability for large-scale systems. Using the approach proposed in this paper, we can solve path-planning problems for teams made up to 500 robots. The method guarantees computational tractability, handles collision avoidance and reduces computational demands through iterative LP optimization techniques. Case studies demonstrate the efficiency of the algorithm in generating scalable, collision-free paths for large robot teams navigating in complex environments. While the conservative nature of collision avoidance introduces additional constraints, and thus, computational requirements, the solution remains practical and impactful for diverse applications. The algorithm is particularly applicable to real-world scenarios, including warehouse logistics where autonomous robots must efficiently coordinate tasks or search-and-rescue operations in various environments. This work contributes both theoretically and practically to scalable multi-robot path planning and task allocation, offering an efficient framework for coordinating autonomous agents in shared environments.