Student Course Allocation with Constraints
24 Jun 2019-pp 51-68
TL;DR: This work empirically evaluates two many-to-one matching algorithms with synthetic data modeled on real-world instances and presents the evaluation of these two algorithms on different metrics including MEAR scores, matching size and number of unstable pairs.
Abstract: Real-world matching scenarios, like the matching of students to courses in a university setting, involve complex downward-feasible constraints like credit limits, time-slot constraints for courses, basket constraints (say, at most one humanities elective for a student), in addition to the preferences of students over courses and vice versa, and class capacities. We model this problem as a many-to-many bipartite matching problem where both students and courses specify preferences over each other and students have a set of downward-feasible constraints. We propose an Iterative Algorithm Framework that uses a many-to-one matching algorithm and outputs a many-to-many matching that satisfies all the constraints. We prove that the output of such an algorithm is Pareto-optimal from the student-side if the many-to-one algorithm used is Pareto-optimal from the student side. For a given matching, we propose a new metric called the Mean Effective Average Rank (MEAR), which quantifies the goodness of allotment from the side of the students or the courses. We empirically evaluate two many-to-one matching algorithms with synthetic data modeled on real-world instances and present the evaluation of these two algorithms on different metrics including MEAR scores, matching size and number of unstable pairs.
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TL;DR: In this article, the authors considered the min bp hrc problem, where the goal is to find a matching that admits the minimum number of blocking pairs (i.e., is "as stable as possible").
Abstract: The Hospitals / Residents problem with Couples (hrc) models the allocation of intending junior doctors to hospitals where couples are allowed to submit joint preference lists over pairs of (typically geographically close) hospitals. It is known that a stable matching need not exist, so we consider min bp hrc, the problem of finding a matching that admits the minimum number of blocking pairs (i.e., is “as stable as possible”). We show that this problem is NP-hard and difficult to approximate even if each couple finds only one hospital pair acceptable. However if we further assume that the preference list of each single resident and hospital is of length at most 2, we give a polynomial-time algorithm for this case. We then show how to adapt an earlier Integer Programming (IP) model for hrc to yield an IP formulation for min bp hrc. Finally, we discuss an empirical evaluation of the IP model applied to randomly-generated instances of min bp hrc. Our main finding is that the number of blocking pairs admitted by a solution is very small, i.e., usually at most 1, and never more than 2, for the (28,000) instances considered.
25 citations
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01 Jan 2018TL;DR: In this paper, the problem of assigning students to courses at a university is considered, where each applicant has a subset of acceptable courses that she ranks in strict order of preference. And the problem is solved in polynomial time for additive and lexicographic preferences.
Abstract: We consider the problem of allocating applicants to courses, where each applicant has a subset of acceptable courses that she ranks in strict order of preference. Each applicant and course has a capacity, indicating the maximum number of courses and applicants they can be assigned to, respectively. We thus essentially have a many-tomany bipartite matching problem with one-sided preferences, which has applications to the assignment of students to optional courses at a university. We consider additive preferences and lexicographic preferences as two means of extending preferences over individual courses to preferences over bundles of courses. We additionally focus on the case that courses have prerequisite constraints: we will mainly treat these constraints as compulsory, but we also allow alternative prerequisites. We further study the case where courses may be corequisites. For these extensions to the basic problem, we present the following algorithmic results, which are mainly concerned with the computation of Pareto optimal matchings (POMs). Firstly, we consider compulsory prerequisites. For additive preferences, we show that the problem of finding a POM is NP-hard. On the other hand, in the case of lexicographic preferences we give a polynomial-time algorithm for finding a POM, based on the well-known sequential mechanism. However we show that the problem of deciding whether a given matching is Pareto optimal is co-NP-complete. We further prove that finding a maximum cardinality (Pareto optimal) matching is NP-hard. Under alternative prerequisites, we show that finding a POM is NP-hardfor either additive or lexicographic preferences. Finally we consider corequisites. We prove that, as in the case of compulsory prerequisites, finding a POM is NP-hard for additive preferences, though solvable in polynomial time for lexicographic preferences. In the latter case, the problem of finding a maximum cardinality POM is NP-hard and very difficult to approximate.
7 citations
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TL;DR: In this article, the authors studied the relationship between college admission and the stability of marriage in the United States, and found that college admission is correlated with the number of stable marriages.
Abstract: (2013). College Admissions and the Stability of Marriage. The American Mathematical Monthly: Vol. 120, No. 5, pp. 386-391.
5,655 citations
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TL;DR: The design of the new clearinghouse adopted by the National Resident Matching Program, which annually fills approximately 20,000 jobs for new physicians, is reported, finding the set of stable matchings, and the opportunities for strategic manipulation, are surprisingly small.
Abstract: We report on the design of the new clearinghouse adopted by the National Resident Matching Program, which annually fills approximately 20,000 jobs for new physicians in the United States. Because that market exhibits many complementarities between applicants and between positions, the theory of simple matching markets does not apply directly. However, computational experiments reveal that the theory provides a good approximation, and furthermore the set of stable matchings, and the opportunities for strategic manipulation, are surprisingly small. A new kind of core convergence' result is presented to explain this; the fact that each applicant can interview for only a small fraction of available positions is important. We also describe in detail engineering aspects of the design process.
693 citations
TL;DR: The National Resident Matching Program (nrmp) as mentioned in this paper is a clearinghouse used by the US Department of Health and Human Services (HHS) to match new physicians to residency programs.
Abstract: We report on the design of the new clearinghouse adopted by the National Resident Matching Program, which annually fills approximately 20,000 jobs for new physicians. Because the market has complementarities between applicants and between positions, the theory of simple matching markets does not apply directly. However, computational experiments show the theory provides good approximations. Furthermore, the set of stable matchings, and the opportunities for strategic manipulation, are surprisingly small. A new kind of “core convergence” result explains this; that each applicant interviews only a small fraction of available positions is important. We also describe engineering aspects of the design process. The entry level labor market for new physicians in the United States is organized via a centralized clearinghouse called the National Resident Matching Program (nrmp). Each year, approximately 20,000 jobs are filled in a process in which graduating physicians and other applicants interview at residency programs throughout the country, and then compose and submit Rank Order Lists (rols) to the nrmp, each indicating an applicant’s preference ordering among the positions for which she has interviewed. Similarly, the residency programs submit rols of the applicants they have interviewed, along with the number of positions they wish to fill. The nrmp processes these rols and capacities to produce a matching of applicants to residency programs. The clearinghouse used in this market dates from the early 1950’s. It replaced a decentralized process that suffered a market failure when residency programs and applicants started to seek each other out individually through informal channels, earlier and earlier in advance of employment, rather than ∗ We thank Aljosa Feldin for able assistance with the theoretical computations reported in section VI. Parts of this work were financed by the National Resident Matching Program, and parts by the National Science Foundation.
621 citations
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TL;DR: The histories of seven regional markets for new physicians and surgeons in the United Kingdom are considered and it is shown that centralized markets that produced unstable matches in environments in which agents could act upon instabilities fared no better than the decentralized markets they replaced.
Abstract: The histories of seven regional markets for new physicians and surgeons in the United Kingdom are considered. Like the American market, these markets have experienced failures that led to the adoption of centralized market mechanisms. Because different regions employ different centralized mechanisms, these markets provide a test of the hypothesis that the success of the American market is related to the fact that it produces matches which are stable in the sense that no two agents mutually prefer to be matched to one another than to their assigned partners. Even in the more complex U.K. markets, this kind of stability plays an important role. Centralized markets that produced unstable matches in environments in which agents could act upon instabilities fared no better than the decentralized markets they replaced.
401 citations
TL;DR: In this article, the authors describe polynomial-time algorithms that will establish, in either of these two cases, whether a matching of the appropriate kind exists, and if so will find such a matching.
281 citations