Stability Representations of Many-to-One Matching Problems: An Integer Optimization Approach

We consider integer optimization models for finding stable solutions to many-to-one, utility-weighted matching problems with incomplete preference lists and ties. Whereas traditional algorithmic approaches for the stable many-to-one matching problem, such as the deferred acceptance algorithm, offer efficient performance for the strict problem setting, adaptation to alternative settings often requires careful customization. Optimization-based approaches are free of the need to create customized algorithms for each unique context and can readily accommodate such extensions as (incomplete) preference lists with ties, alternative and nontraditional objective functions, and side constraints including those that ensure stable matching outcomes free of waste. We explore the flexibility of optimization-based approaches in several ways. First, we introduce four new constraint sets that prevent justified envy and a new system of constraints that prevents waste; taken together, they jointly ensure stable matching outcomes. Second, we create two algorithms to accelerate the generation of our proposed constraints. Third, we construct aggregate objective functions to reflect multiple hierarchical emphases by imposing a strict lexicographical order on the individual components. Fourth, we conduct comprehensive experiments to study the computational performance of our proposed optimization models and compare them with models from the extant literature under a variety of problem attributes. Our experiments reveal the circumstances under which each stability representation excels in terms of optimality criteria and computational efficiency on a variety of real and synthetic data sets. One such setting in which our proposed stability representations excel includes the important context of when sufficient seats exist for applicants, such as school choice problems and hospital residency matching.