Stable matchings of teachers to schools

TL;DR

This paper investigates the computational complexity of finding stable teacher-school matchings in centralized assignment systems, revealing NP-completeness results and inapproximability under various preference models.

Contribution

It introduces two stability definitions for teacher-school matchings with teachers specializing in two subjects and analyzes their computational complexity.

Findings

Deciding stable matchings is NP-complete under various preference models.

NP-completeness persists even with limited subjects or master lists.

Finding matchings with minimal blocking pairs is strongly inapproximable.

Abstract

Several countries successfully use centralized matching schemes for school or higher education assignment, or for entry-level labour markets. In this paper we explore the computational aspects of a possible similar scheme for assigning teachers to schools. Our model is motivated by a particular characteristic of the education system in many countries where each teacher specializes in two subjects. We seek stable matchings, which ensure that no teacher and school have the incentive to deviate from their assignments. Indeed we propose two stability definitions depending on the precise format of schools' preferences. If the schools' ranking of applicants is independent of their subjects of specialism, we show that the problem of deciding whether a stable matching exists is NP-complete, even if there are only three subjects, unless there are master lists of applicants or of schools. By contrast, if the schools may order applicants differently in each of their specialization subjects, the problem of deciding whether a stable matching exists is NP-complete even in the presence of subject-specific master lists plus a master list of schools. Finally, we prove a strong inapproximability result for the problem of finding a matching with the minimum number of blocking pairs with respect to both stability definitions.