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

Abstract

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 Relocate Pair neighborhood can be fully explored in [Formula: see text] instead of [Formula: see text] time. We adapt the 4-Opt and Balas–Simonetti neighborhoods to consider precedence constraints. Moreover, we introduce an exponential-size neighborhood called 2k-Opt, which includes all solutions generated by multiple nested 2-Opts and can be searched in [Formula: see text] 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. Funding: This work was supported by Coordenação de Aperfeiçoamento de Pessoal de Nível Superior, Conselho Nacional de Desenvolvimento Científico e Tecnológico [Grants 308528/2018-2, 315361/2020-4, 422470/2021-0], and Fundação Carlos Chagas Filho de Amparo à Pesquisa do Estado do Rio de Janeiro [Grants E-26/202.790/2019, E-26/201.417/2022, E-26/010.002232/2019]. Supplemental Material: The electronic companion is available at https://doi.org/10.1287/trsc.2022.1176 .