Structural Integrality in Task Assignment and Path Finding via Total Unimodularity of Petri Net Models

Task Assignment and Path Finding (TAPF) concerns computing collision-free motions for multiple robots while jointly selecting goal locations. In this paper, safety is enforced by requiring unit-capacity traversal between successive intermediate markings, yielding coordination strategies that are valid independently of any specific time interpretation. Existing optimization-based approaches typically rely on time-expanded network-flow models, which result in large mixed-integer programs and limited scalability. We instead develop a Petri net (PN)-based optimization framework that exploits structural properties of the motion model to improve computational efficiency without explicit time expansion. When robot motion is modeled by strongly connected state-machine PNs, we show that, once the congestion level (equivalently, the synchronization depth) is fixed to an integer value, the resulting motion-planning constraint matrix is totally unimodular. Consequently, the corresponding LP relaxation admits integral optimal solutions for the motion variables. When the estimated congestion exceeds one, we introduce a synchronization-on-demand mechanism based on intermediate markings; for a fixed number of synchronization stages, the associated constraint matrices remain totally unimodular, thereby preserving integrality of the motion variables. Finally, we extend TAPF to Boolean specifications over regions of interest and propose a two-stage LP/mixed-integer linear programming (MILP) scheme in which integrality is confined to task-selection variables. Simulations on large benchmarks demonstrate substantial scalability improvements over time-expanded optimization baselines.