# Assigning Candidate Tutors to Modules: A Preference Adjustment Matching Algorithm (PAMA)
## 2 Matching Problems and the Teacher Assignment Problem (TAP): A Review of Related Work
The new operators include a reinitialization operator that monitors population convergence and reinitializes the population when diversity is lost and a diversity-preserving operator that identifies when several quantum individuals converge to the same local optimum and reinitializes some of them for exploration of other regions of the search space. Matching theory and game theory principles provide another important methodology beyond these optimization-based approaches. Though originally designed for the college admission problem, the Gale-Shapley algorithm [9] has been broadly adapted to a wide array of matching problems and is one of the most well-known techniques for solving one-to-one matching problems. It guarantees stability, which means that each element is matched to a single element and that there are no blocking pairs; that is, no two elements favor one another over their current matches. The understanding of stable matchings was further enhanced by [PERSON]'s analysis [10], which added ideas that have been used to solve a variety of issues, including matching in market design. In this direction, [PERSON] et al. [11] addressed the teacher-school matching problem. They extended the definition of stability to cover special requirements such as different preferences for specialties and school capacity constraints. Their study proves that finding stable matching is NP-completein some cases. [PERSON] et al. (2012) continued in this research line by investigating how central placement mechanisms for teachers can be best designed such that teacher allocations among regions/schools would be improved. They developed new mechanisms eliminating strategic manipulation as well as complementing existing ones. Their approach balanced efficiency considerations with fairness properties, and empirical analysis showed significant improvements in reducing experience gaps between different regions while enhancing overall teacher satisfaction. While stable matching algorithms have been more influential, there have also been several applications of non-stable matching algorithms for assignment problems. For instance, the Hungarian algorithm ([PERSON], 2013) is an optimization algorithm introduced by [PERSON] that solves the problem of assignment in polynomial time. Shortly afterward, [PERSON] (2014) improved [PERSON]'s original algorithm by simplifying its steps, giving it a stricter termination, and extending its application to the transfer problem, as opposed to 1-to-1 matching. The Hopcroft-Karp algorithm ([PERSON], 2015) provided an efficient solution for maximum matchings in bipartite graphs. Furthermore, [PERSON] blossom algorithm ([PERSON], 2016) extended maximum matching solutions to general graphs, beyond just bipartite ones. Its novelty lies in the way it handles circles of odd length, called \"blossoms\". His method shrinks these circles to single nodes, searches for augmenting paths in the simplified graph, and then regrows the circles to construct the final solution. Where stability is not the paramount issue, these algorithms provide ways to solve complex matching problems. A different perspective on the TAP comes from [PERSON] et al. (2017), who developed a Shapley-value-based methodology for faculty assignments in higher education. In their approach, every faculty is viewed as a player in a cooperative game, with expertise over different cognitive subjects viewed as a contribution. Then, the Shapley value calculates a fair distribution of teaching assignments based on the individual contribution to possible teaching coalitions. This method turns out to be highly effective in dealing with multidimensional modules, where the faculty members need to be competent in many subjects. It also provides a mathematical framework for balancing individual competencies with collective teaching effectiveness.