Matching with sizes (or scheduling with processing set restrictions)

Bíró, Péter; McDermid, Eric

doi:10.1016/j.endm.2010.05.043

Cited by 15 publications

(19 citation statements)

References 12 publications

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“…1 For the third case with corequisites (i.e., an applicant can attend a course if and only if she attends all its corequisites) we propose another modification of the sequential allocation mechanism for finding Pareto optimal matchings. If the corequisites for all the applicants are the same, the model is closely related to matchings with sizes [7] or many-tomany matchings with price-budget constraints [13], and we strengthen the existing results by showing that the problem of finding a maximum size Pareto optimal matching is not approximable within N 1−ε , for any ε > 0, unless P=NP, where N is the total capacity of the applicants.…”

Section: Our Contributionsupporting

confidence: 69%

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

Cechlárová

Klaus

Manlove

2018

Discrete Optimization

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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-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.

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Section: Our Contributionsupporting

confidence: 69%

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

Cechlárová

Klaus

Manlove

2018

Discrete Optimization

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“…If couples are involved, the problem becomes hard. More precisely, the decision version of this problem is NP-complete [13,3], even in the special case where each hospital has a capacity of 2, and the acceptable hospital pairs for a couple are always of the form (h, h) for some h ∈ H. However, if the number of couples is small, which is a reasonable assumption in many practical applications, Maximum Matching with Couples becomes tractable, as shown by Theorem 1. We will use the following lemma to solve a special case of the Maximum Matching with Couples problem.…”

Section: Fixed-parameter Tractabilitymentioning

confidence: 99%

“…This problem was proved to be NPcomplete even if p = 2 (see [13,3]), so investigating the parameterized complexity of this problem might be of practical importance.…”

Section: An Application In Schedulingmentioning

confidence: 99%

Stable Assignment with Couples: Parameterized Complexity and Local Search

Marx

Schlotter

2009

Parameterized and Exact Computation

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Abstract. We study the Hospitals/Residents with Couples problem, a variant of the classical Stable Marriage problem. This is the extension of the Hospitals/Residents problem where residents are allowed to form pairs and submit joint rankings over hospitals. We use the framework of parameterized complexity, considering the number of couples as a parameter. We also apply a local search approach, and examine the possibilities for giving FPT algorithms applicable in this context. Furthermore, we also investigate the matching problem containing couples that is the simplified version of the Hospitals/Residents with Couples problem modeling the case when no preferences are given.

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“…Moreover, the schools do not have preferences over applicants. An existing problem in the literature that most closely resembles this case is called Matching with Couples [7], which is essentially the same as the Hospitals / Residents problem with Couples where agents are indifferent between all members of their preference lists. However we emphasize that our model differs from all of those discussed so far due to the applicants' specialization in two subjects and the schools being allowed to have different capacities for different subjects.…”

Section: Related Workmentioning

confidence: 99%

Modelling practical placement of trainee teachers to schools

Cechlárová

Fleiner

Manlove

et al. 2014

Cent Eur J Oper Res

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Several countries successfully use centralized matching schemes for assigning students to study places or fresh graduates to their first positions. In this paper we explore the computational aspects of a possible similar scheme for assigning trainee teachers to schools. Our model is motivated by the situation characteristic for Slovak and Czech education system where each pre-service teacher specializes in two subjects. We show that if the two subjects can be performed independently, then a feasible assignment can be found efficiently by employing network flow techniques, while the requirement to perform both subjects at the same school leads to intractable problems even under several strict restrictions concerning the total number of subjects, partial capacities of schools and the number of acceptable schools each teacher is allowed to list. Finally, we report on an integer programming model for solving the teachers assignment problem and the results of its application to real data.

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Matching with sizes (or scheduling with processing set restrictions)

Cited by 15 publications

References 12 publications

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

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

Stable Assignment with Couples: Parameterized Complexity and Local Search

Modelling practical placement of trainee teachers to schools

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