Exponential-Size Neighborhoods for the Pickup-and-Delivery Traveling Salesman Problem

Neighborhood search is a cornerstone of state-of-the-art traveling salesman and vehicle routing metaheuristics. Whereas neighborhood exploration procedures are well-developed for problems with individual services, their counterparts for one-to-one pickup-and-delivery problems are more scarcely studied. A direct extension of classic neighborhoods is often inefficient or complex because of the necessity of jointly considering service pairs. To circumvent these issues, we introduce major improvements to existing neighborhood searches for the pickup-and-delivery traveling salesman problem and new large neighborhoods. We show that the classic R elocate P air neighborhood can be fully explored in O ( n 2 ) instead of O ( n 3 ) time. We adapt the 4-O pt and B alas –S imonetti neighborhoods to consider precedence constraints. Moreover, we introduce an exponential-size neighborhood called 2 k -O pt , which includes all solutions generated by multiple nested 2-O pts and can be searched in O ( n 2 ) time using dynamic programming. We conduct extensive computational experiments, highlighting the significant contribution of these new neighborhoods and speedup strategies within two classical metaheuristics. Notably, our approach permits us to repeatedly solve small pickup-and-delivery problem instances to optimality or near-optimality within milliseconds, and therefore, it represents a valuable tool for time-critical applications, such as meal delivery or mobility on demand.