Quantum Optimization for the Steiner Traveling Salesman Problem with Time Windows and Pickup and Delivery

We present the Steiner Traveling Salesman Problem with Time Windows and Pickup and Delivery, an advanced and practical extension of classical routing models. This variant integrates the characteristics of the Steiner Traveling Salesman Problem with time-window constraints, pickup and delivery operations and vehicle capacity limitations. These features closely mirror the complexities of contemporary logistics challenges, including last-mile distribution, reverse logistics and on-demand service scenarios. To tackle the inherent computational difficulties of this NP-hard problem, we propose two specialized mathematical formulations: an arc-based model and a node-oriented model, each designed to capture distinct structural aspects of the problem. Both models are implemented on D-Wave's LeapCQMHybrid platform, which combines quantum and classical techniques for solving constrained optimization tasks. We further introduce a preprocessing reduction method that eliminates redundant arcs, significantly enhancing computational performance and scalability. Experimental results demonstrate that hybrid quantum approaches are capable of solving problem instances of realistic size, underscoring their potential as a transformative tool for next-generation routing optimization.