Pareto optimal matchings of students to courses in the presence of prerequisites

Cechlárová, K., Klaus, B. and Manlove, D. F. (2018) Pareto optimal matchings of students to courses in the presence of prerequisites. Discrete Optimization, 29, pp. 174-195. (doi:10.1016/j.disopt.2018.04.004)

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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-to-many 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-hard for 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.

Item Type:Articles
Status:Published
Refereed:Yes
Glasgow Author(s) Enlighten ID:Manlove, Professor David
Authors: Cechlárová, K., Klaus, B., and Manlove, D. F.
College/School:College of Science and Engineering > School of Computing Science
Journal Name:Discrete Optimization
Publisher:Elsevier
ISSN:1572-5286
ISSN (Online):1873-636X
Published Online:27 July 2018
Copyright Holders:Copyright © 2018 The Authors
First Published:First published in Discrete Optimization 29:174-195
Publisher Policy:Reproduced under a Creative Commons License

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Project CodeAward NoProject NamePrincipal InvestigatorFunder's NameFunder RefLead Dept
3008080IP-MATCH: Integer Programming for Large and Complex Matching ProblemsDavid ManloveEngineering and Physical Sciences Research Council (EPSRC)EP/P028306/1Computing Science
607071Efficient Algorithms for Mechanism Design Without Monetary Transfer.David ManloveEngineering and Physical Sciences Research Council (EPSRC)EP/K010042/1COM - COMPUTING SCIENCE