Stable matchings of teachers to schools

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.

💡 Research Summary

The paper investigates the computational foundations of a centralized matching scheme for assigning teachers to schools, a problem motivated by the fact that many teachers specialize in two subjects. The authors formalize the Teacher Assignment Problem (TAP) with a set of teachers, a set of schools, and a set of subjects. Each teacher is characterized by a pair of distinct subjects, and each school has a capacity vector specifying how many teachers of each subject it can accept. Both teachers and schools submit strict preference lists over acceptable partners. A matching assigns each teacher to at most one school while respecting the subject‑specific capacities. Stability is defined via blocking pairs: a teacher and a school form a blocking pair if the teacher prefers the school to her current assignment (or is unassigned) and the school can accommodate the teacher without violating capacities, possibly by replacing other teachers according to four detailed cases that capture the two‑subject nature of teachers.

Two variants of school preferences are considered. In the first variant, a school’s ranking of applicants is independent of the teachers’ subjects; in the second, a school may rank teachers differently for each of its subject positions. For the first variant the authors prove that deciding whether a stable matching exists is NP‑complete, even when there are only three subjects, each school’s subject capacity is at most two, and each teacher’s preference list contains at most three schools. The reduction is from a restricted version of 3‑SAT called (2,2)‑E3‑SAT. However, if either the schools’ preferences or the teachers’ preferences are derived from a common master list (a global ordering of all participants), then a unique stable matching always exists and can be found in polynomial time by extending the classic serial dictatorship algorithm of Gale‑Shapley.

For the second variant, where schools have subject‑specific rankings, the problem remains NP‑complete even when master lists exist for each subject and a master list of schools is also given. Thus, allowing subject‑specific school preferences dramatically increases computational difficulty.

The paper also studies the optimization version of minimizing the number of blocking pairs. It establishes strong inapproximability results: for both stability definitions, no polynomial‑time algorithm can achieve any constant‑factor approximation unless P = NP, and even PTASs are ruled out. This shows that finding a “nearly stable” matching is computationally infeasible.

Overall, the work provides a thorough theoretical analysis of teacher‑school matching, highlighting how the structure of preference lists and the presence of master lists affect both the existence of stable outcomes and the tractability of finding them. The results have practical implications for the design of real‑world matching mechanisms in education systems.