Other SPA models and approaches

The SPA variants that have been considered in the previous sections allow some form of

preference from the lecturers over the students that finds their projects acceptable and/or the projects that they offer. However, some practical applications of SPA consider lecturer preferences to be undesirable or unnecessary [31, 76]. In these applications, preferences are only allowed from the students; thus the underlying SPAmodel falls under the category of bipartite matching problems with one-sided preferences. The reader may recall that the fundamental matching problem in this category is the House Allocation problem (HA) [12,

123].

Informally, an instance of HAconsists of a set A of applicants and a set H of houses. Each applicant has a strictly-ordered preference list over a subset of the houses that she finds acceptable. In this setting, houses do not have preferences over applicants. A matching M in this context is a set of (applicant, house) pairs such that a house is paired with an applicant in M only if the applicant finds the house acceptable; and, each applicant and house is involved in at most one pair. Clearly, HA is similar toSMI with the applicants representing the men

and the houses representing the women, except that women have no preferences over men in theHAsetting. Extensions ofHAthat have been studied in the literature [83] include the

House Allocation problem with Ties(HAT) in which applicants’ preference lists may include

ties, the Capacitated House Allocation problem (CHA) in which houses can accommodate more than one applicant up to a fixed capacity, and a hybrid of HAT andCHA which is the Capacitated House Allocation problem with Ties(CHAT).

TheHAmodel and its extensions arise in several practical applications for which allocating

students to projects is one of them (see [83] for more applications). For bipartite matching problems where preference lists are restricted to agents in one set, the notion of stability as a desired solution concept becomes irrelevant. Other optimality criteria that have been considered in the literature include: Pareto optimality, where a matching is Pareto optimal if no agent can be better off without requiring another agent to be worse off; popularity, where a matching is popular if there is no other matching that is preferred by a majority of the agents; and profile-based optimality, where the profile of a matching is a vector indicating the number of agents who are assigned in the matching to their first choice, second choice, third choice, and so on. In terms of optimising the profile of a matching, several types of

2.3. The Student-Project Allocation Problem: SPA 29

optimal matching have been considered, including rank maximal matching, greedy maximum matching, and generous maximum matching.

Informally, a rank maximal matching is a matching that has lexicographically maximum profile, i.e., the maximum number of agents are assigned to their first choice and subject to this, the maximum number of agents are assigned to their second choice, and so on. A greedy maximum matchingis a matching of maximum cardinality that has lexicographically maximum profile. A generous maximum matching is a matching of maximum cardinality whose reverse profile is lexicographically minimum, i.e., the minimum number of agents are assigned to their R-th choice (where R is the maximum length of the preference lists taken over all the agents) and subject to this, the minimum number of agents are assigned to their (R − 1)-th choice, and so on. For a formal definition of these optimality criteria, and for algorithmic results under theHAmodel and its extensions, we refer the interested reader to [83].

For a given instance I ofSPA, Kwanashie et al. [76] described an algorithm to find a greedy

maximum and generous maximum matching in I, both with time complexity O(n2 1Rm), where n1 is the number of students, R is the maximum length of the preference lists, and m is the total length of the students’ preference lists.

We now move on to another variant ofSPAwith one-sided preferences. We denote by SPA-

LQP an instance of SPA where each project, in addition to its capacity, has a lower quota, which is the minimum number of students that must be assigned to the project. Let I be an arbitrary instance ofSPA-LQP, two different variants of I exist in the literature [75] with respect to the definition of a feasible solution. We give the two definitions as follows.

1. A matching M in I is a feasible solution if for each project pj, either pjmeets its lower quota in M or pj is not assigned to any student in M , i.e., pj remains closed.

2. A matching M in I is a feasible solution if for each project pj, pjmeets its lower quota in M , i.e., pj can still run even if the number of students assigned to pj in M is less than pj’s lower quota.

Kwanashie [75] showed that for the first definition, the problem of finding a greedy max- imum or generous maximum matching in I is NP-hard, and for the second definition, a feasible solution need not exist in I.

Several other variants ofSPAhave been studied in the literature, many of which take account

of specific problem-constraints arising within the application context. For example, Chiaran- dini et al. [31] studied the allocation of students to projects which exists at the Faculty of Science, University of Southern Denmark. In their model, students have preferences over projects, and students who want to be in the same team can register together as a group.

Also, each lecturer provides lower and upper bounds on the sizes of each team, as well as a capacity constraint on the number of teams of students she can supervise for each project. To find an allocation of students to projects, the authors [31] described a Mixed Integer Linear Programming (MILP) formulation which computes allocations that are Pareto optimal, fair, envy-free and stable.

Other centralised matching schemes that are based on different formulations ofSPAinclude student-project allocation mechanisms at the Department of Civil and Environmental Engi- neering, University of Southampton [17, 48], and the School of Electrical and Electronic En- gineering, Nanyang Technological University, Singapore [116]. To find optimal allocations, techniques that have been used in the literature include constraint programming [36, 118], evolutionary algorithm [115], genetic algorithm [48], goal programming [101], integer pro- gramming [17, 65, 103, 112, 119], and local search [41]. See [31] for a recent survey.

31

Chapter 3

Structural Result for

SPA

-

S

3.1

Introduction

In this chapter we consider the variant ofSPAwhere students have preferences over projects,

lecturers have preferences over students, and the preference lists of students and lecturers are strictly ordered. We introduced this variant as the Student-Project Allocation problem with lecturer preferences over Students(SPA-S) in Section 2.3.2; see that section for the problem definition along with key algorithmic results. We recall that an arbitrary instance of SPA-

S can have many stable matchings, similar to the SM and HRT settings [53]. Our goal is to characterize the structure of the set of stable matchings for an instance of SPA-S under the restriction that for each student, all of the projects in her preference list are offered by different lecturers. In the remainder of this chapter, it should be clear that any usage ofSPA-S

assumes this restriction.

To achieve our goal, we show that the set of stable matchings in an instance of SPA-Sforms a distributive lattice with respect to a dominance relation that we will define. A similar structure holds for stable matchings in the SM and HRT settings as mentioned in Sections 2.1.2 and 2.2.3 respectively.