arXiv:2505.02485v1 Announce Type: cross Abstract: The Bus Driver Scheduling Problem (BDSP) is a combinatorial optimization problem with the goal to design shifts to cover prearranged bus tours. The objective takes into account the operational cost as well as the satisfaction of drivers. This problem is heavily constrained due to strict legal rules and collective agreements. The objective of this article is to provide state-of-the-art exact and hybrid solution methods that can provide high-quality solutions for instances of different sizes. This work presents a comprehensive study of both an exact method, Branch and Price (B&P), as well as a Large Neighborhood Search (LNS) framework which uses B&P or Column Generation (CG) for the repair phase to solve the BDSP. It further proposes and evaluates a novel deeper integration of B&P and LNS, storing the generated columns from the LNS subproblems and reusing them for other subproblems, or to find better global solutions. The article presents a detailed analysis of several components of the solution methods and their impact, including general improvements for the B&P subproblem, which is a high-dimensional Resource Constrained Shortest Path Problem (RCSPP), and the components of the LNS. The evaluation shows that our approach provides new state-of-the-art results for instances of all sizes, including exact solutions for small instances, and low gaps to a known lower bound for mid-sized instances. Conclusions: We observe that B&P provides the best results for small instances, while the tight integration of LNS and CG can provide high-quality solutions for larger instances, further improving over LNS which just uses CG as a black box. The proposed methods are general and can also be applied to other rule sets and related optimization problems
