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

TL;DR

This paper enhances neighborhood search techniques for the pickup-and-delivery traveling salesman problem, introducing efficient algorithms and a novel exponential-size neighborhood that significantly improves solution speed and quality for time-critical applications.

Contribution

It presents new algorithms for exploring neighborhoods efficiently and introduces the 2k-Opt neighborhood, enabling rapid near-optimal solutions for pickup-and-delivery problems.

Findings

Classical neighborhoods can be explored in quadratic time.

The 2k-Opt neighborhood includes all nested 2-Opt solutions.

Small instances can be solved to near-optimality within milliseconds.

Abstract

Neighborhood search is a cornerstone of state-of-the-art traveling salesman and vehicle routing metaheuristics. While neighborhood exploration procedures are well developed for problems with individual services, their counterparts for one-to-one pickup-and-delivery problems have been more scarcely studied. A direct extension of classic neighborhoods is often inefficient or complex due to 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 classical Relocate-Pair neighborhood can be fully explored in $O(n^2)$ instead of $O(n^3)$ 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 $O(n^2)$ time using dynamic programming. We conduct extensive computational experiments, highlighting the significant contribution of these new neighborhoods and speed-up strategies within two classical metaheuristics. Notably, our approach permits 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.