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An Algorithm for Strong Stability in the Student-Project Allocation Problem with Ties

  • Sofiat OlaosebikanEmail author
  • David Manlove
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 12016)

Abstract

We study a variant of the Student-Project Allocation problem with lecturer preferences over Students where ties are allowed in the preference lists of students and lecturers (spa-st). We investigate the concept of strong stability in this context. Informally, a matching is strongly stable if there is no student and lecturer l such that if they decide to form a private arrangement outside of the matching via one of l’s proposed projects, then neither party would be worse off and at least one of them would strictly improve. We describe the first polynomial-time algorithm to find a strongly stable matching or report that no such matching exists, given an instance of spa-st. Our algorithm runs in \(O(m^2)\) time, where m is the total length of the students’ preference lists.

1 Introduction

Matching problems, which generally involve the assignment of a set of agents to another set of agents based on preferences, have wide applications in many real-world settings, including, for example, allocating junior doctors to hospitals [25] and assigning students to projects [15]. In the context of assigning students to projects, each project is proposed by one lecturer and each student is required to provide a strictly-ordered preference list over the available projects that she finds acceptable. Also, lecturers may provide strictly-ordered preference lists over the students that find their projects acceptable, and/or over the projects that they propose. Typically, each project and lecturer have a specific capacity denoting the maximum number of students that they can accommodate. The goal is to find a matching, i.e., an assignment of students to projects that respects the stated preferences, such that each student is assigned at most one project, and the capacity constraints on projects and lecturers are not violated—the so-called Student-Project Allocation problem (spa) [1, 6, 19].

Two major models of spa exist in the literature: one permits preferences only from the students [15], while the other permits preferences from the students and lecturers [14, 19]. In the latter case, three different variants have been studied based on the nature of the lecturers’ preference lists. These include SPA with lecturer preferences over (i) students [1], (ii) projects [12, 21, 22], and (iii) (student, project) pairs [2]. Outwith assigning students to projects, applications of each of these three variants can be seen in multi-cell networks where the goal is to find a stable association of users to channels at base-stations [3, 4, 5].

In this work, we will concern ourselves with variant (i), i.e., the Student-Project Allocation problem with lecturer preferences over Students (spa-s). In this context, it has been argued in [25] that a natural property for a matching to satisfy is that of stability. Informally, a stable matching ensures that no student and lecturer would have an incentive to deviate from their current assignment. Abraham et al. [1] described two linear-time algorithms to find a stable matching in an instance of spa-s where the preference lists are strictly ordered. In their paper, they also proposed an extension of spa-s where the preference lists may include ties, known as the Student-Project Allocation problem with lecturer preferences over Students with Ties (spa-st) [23].

If we allow ties in the preference lists of students and lecturers, three stability definitions are possible, namely weak stability, strong stability and super-stability [8, 9, 10]. We give an informal definition in what follows. Suppose M is a matching in an instance of spa-st. Then M is (i) weakly stable, (ii) strongly stable, or (iii) super-stable, if there is no student and lecturer l such that if they decide to become assigned outside of M via one of l’s proposed projects, respectively,
  1. (i)

    both of them would strictly improve,

     
  2. (ii)

    one of them would strictly improve and the other would not be worse off,

     
  3. (iii)

    neither of them would be worse off.

     
Existing Results for spa-st. Manlove et al. [20] showed that every instance of spa-st admits a weakly stable matching, which could be of different sizes. Moreover, the problem of finding a maximum size weakly stable matching (max-spa-st) is NP-hard [11, 20], even for the Stable Marriage problem with Ties and Incomplete lists (smti). Cooper and Manlove [7] described a \(\frac{3}{2}\)-approximation algorithm for max-spa-st. On the other hand, Irving et al. argued in [9] that super-stability is a natural and most robust solution concept to seek in cases where agents have incomplete information. Recently, Olaosebikan and Manlove [23] showed that if an instance of spa-st admits a super-stable matching M, then all weakly stable matchings in the instance are of the same size (equal to the size of M), and match exactly the same set of students. The main result of their paper was a polynomial-time algorithm to find a super-stable matching or report that no such matching exists, given an instance of spa-st. Their algorithm runs in O(L) time, where L is the total length of all the preference lists.

Motivation for Strong Stability. It was motivated in [10] that weakly stable matching may be undermined by bribery or persuasion, in practical applications of the Hospitals-Residents problem with Ties (hrt). In what follows, we give a corresponding argument for an instance I of spa-st. Suppose that M is a weakly stable matching in I, and suppose that a student \(s_i\) prefers a project \(p_j\) (where \(p_j\) is offered by lecturer \(l_k\)) to her assigned project in M, say \(p_{j'}\) (where \(p_{j'}\) is offered by a lecturer different from \(l_k\)). Suppose further that \(p_j\) is full and \(l_k\) is indifferent between \(s_i\) and one of the worst student/s assigned to \(p_j\) in M, say \(s_{i'}\). Clearly, the pair \((s_i, p_j)\) does not constitute a blocking pair for the weakly stable matching M, as \(l_k\) would not improve by taking on \(s_i\) in the place of \(s_{i'}\). However, \(s_i\) might be overly invested in \(p_j\) that she is ready to persuade or bribe \(l_k\) to reject \(s_{i'}\) and accept her instead; \(l_k\) being indifferent between \(s_i\) and \(s_{i'}\) may decide to accept \(s_i\)’s proposal. We can reach a similar argument if the roles are reversed. However, if M is strongly stable, it cannot be potentially undermined by this type of (student, project) pair.

Henceforth, if a spa-st instance admits a strongly stable matching, we say that such instance is solvable. Unfortunately not every instance of spa-st is solvable. To see this, consider the case where there are two students, two projects and two lecturers, the capacity of each project and lecturer is 1, the students have exactly the same strictly-ordered preference list of length 2, and each of the lecturers preference list is a single tie of length 2 (any matching will be undermined by a student and lecturer that are not assigned together). However, it should be clear from the discussions above that in cases where a strongly stable matching exists, it should be preferred over a matching that is merely weakly stable. Previous results for strong stability in the literature include [8, 10, 13, 16, 18].

Our Contribution. We present the first polynomial-time algorithm to find a strongly stable matching or report that no such matching exists, given an instance of spa-st—thus solving an open problem given in [1, 23]. Our algorithm is student-oriented, which implies that if the given instance is solvable then our algorithm will output a solution in which each student has at least as good a project as she could obtain in any strongly stable matching. We note that our algorithm is a non-trivial extension of the strong stability algorithms for smt (Stable Marriage problem with Ties), smti and hrt described in [8, 10, 18] (we discuss this further in [24, Sect. 4.3]).

The remainder of this paper is structured as follows. We give a formal definition of the spa-s problem, the spa-st variant, and the three stability concepts in Sect. 2. We describe our algorithm for spa-st under strong stability in Sect. 3. Further, in Sect. 3, we illustrate an execution of our algorithm with respect to an instance of spa-st before moving on to present the algorithm’s correctness and complexity results (all omitted proofs can be found in [24, Sect. 4.5]). Finally, we present some potential directions for future work in Sect. 4.

2 Preliminary Definitions

In this section, we give a formal definition of spa-s as described in the literature [1, 23]. We also give a formal definition of spa-st as described in [23], which is a generalisation of spa-s in which preference lists can include ties.

2.1 Formal Definition of Spa-S

An instance I of spa-s involves a set \(\mathcal {S} = \{s_1 , s_2, \ldots , s_{n_1}\}\) of students, a set \(\mathcal {P} = \{p_1 , p_2, \ldots , p_{n_2}\}\) of projects and a set \(\mathcal {L} = \{l_1 , l_2, \ldots , l_{n_3}\}\) of lecturers. Each student \(s_i\) ranks a subset of \(\mathcal {P}\) in strict order, which forms her preference list. We say that \(s_i\) finds \(p_j\) acceptable if \(p_j\) appears on \(s_i\)’s preference list. We denote by \(A_i\) the set of projects that \(s_i\) finds acceptable.

Each lecturer \(l_k \in \mathcal {L}\) offers a non-empty set of projects \(P_k\), where \(P_1, P_2, \ldots ,\) \(P_{n_3}\) partitions \(\mathcal {P}\), and \(l_k\) provides a preference list, denoted by \(\mathcal {L}_k\), ranking in strict order of preference those students who find at least one project in \(P_k\) acceptable. Also \(l_k\) has a capacity \(d_k \in \mathbb {Z}^+\), indicating the maximum number of students she is willing to supervise. Similarly each project \(p_j \in \mathcal {P}\) has a capacity \(c_j \in \mathbb {Z}^+\) indicating the maximum number of students that it can accommodate.

We assume that for any lecturer \(l_k\), \(\max \{c_j: p_j \in P_k\} \le d_k \le \sum \{c_j: p_j \in P_k\}\) (i.e., the capacity of \(l_k\) is (i) at least the highest capacity of the projects offered by \(l_k\), and (ii) at most the sum of the capacities of all the projects \(l_k\) is offering). We denote by \(\mathcal {L}_k^j\), the projected preference list of lecturer \(l_k\) for \(p_j\), which can be obtained from \(\mathcal {L}_k\) by removing those students that do not find \(p_j\) acceptable (thereby retaining the order of the remaining students from \(\mathcal {L}_k\)).

Given a pair \((s_i, p_j) \in \mathcal {S} \times \mathcal {P}\), where \(p_j\) is offered by \(l_k\), we refer to \((s_i, p_j)\) as an acceptable pair if \(p_j \in A_i\) and \(s_i \in \mathcal {L}_k\). An assignment M is a collection of acceptable pairs in \(\mathcal {S} \times \mathcal {P}\). If \((s_i, p_j) \in M\), we say that \(s_i\) is assigned to \(p_j\), and \(p_j\) is assigned \(s_i\). For convenience, if \(s_i\) is assigned in M to \(p_j\), where \(p_j\) is offered by \(l_k\), we may also say that \(s_i\) is assigned to \(l_k\), and \(l_k\) is assigned \(s_i\). For any project \(p_j \in \mathcal {P}\), we denote by \(M(p_j)\) the set of students assigned to \(p_j\) in M. Project \(p_j\) is undersubscribed, full or oversubscribed according as \(|M(p_j)|\) is less than, equal to, or greater than \(c_j\), respectively. Similarly, for any lecturer \(l_k \in \mathcal {L}\), we denote by \(M(l_k)\) the set of students assigned to \(l_k\) in M. Lecturer \(l_k\) is undersubscribed, full or oversubscribed according as \(|M(l_k)|\) is less than, equal to, or greater than \(d_k\), respectively. A matching M is an assignment such that \(|M(s_i)|\le 1\), \(|M(p_j)|\le c_j\) and \(|M(l_k)|\le d_k\). If \(s_i\) is assigned to some project in M, we let \(M(s_i)\) denote that project; otherwise \(M(s_i)\) is undefined.

2.2 Ties in the Preference Lists

We now give a formal definition, similar to the one given in [23], for the generalisation of spa-s in which the preference lists can include ties. In the preference list of lecturer \(l_k\in \mathcal L\), a set T of r students forms a tie of length r if \(l_k\) does not prefer \(s_i\) to \(s_{i'}\) for any \(s_i, s_{i'} \in T\) (i.e., \(l_k\) is indifferent between \(s_i\) and \(s_{i'}\)). A tie in a student’s preference list is defined similarly. For convenience, in what follows we consider a non-tied entry in a preference list as a tie of length one. We denote by spa-st the generalisation of spa-s in which the preference list of each student (respectively lecturer) comprises a strict ranking of ties, each comprising one or more projects (respectively students). An example spa-st instance \(I_1\) is given in Fig. 1, which involves the set of students \(\mathcal {S} = \{s_1, s_2, s_3\}\), the set of projects \(\mathcal {P} = \{p_1, p_2, p_3\}\) and the set of lecturers \(\mathcal {L} = \{l_1, l_2\}\). Ties in the preference lists are indicated by round brackets.
Fig. 1.

An example spa-st instance \(I_1\).

In the context of spa-st, we assume that all notation and terminology carries over from spa-s with the exception of stability, which we now define. When ties appear in the preference lists, three types of stability arise, namely weak stability, strong stability and super-stability [8, 9, 10]. For our purpose in this paper, we only give a formal definition of strong stability in the context of spa-st. Henceforth, I is an instance of spa-st, \((s_i, p_j)\) is an acceptable pair in I and \(l_k\) is the lecturer who offers \(p_j\).

Definition 1

(Strong stability). We say that M is strongly stable in I if it admits no blocking pair, where a blocking pair for M is an acceptable pair \((s_i, p_j) \in (\mathcal {S} \times \mathcal {P}) \setminus M\) such that either (1a and 1b) or (2a and 2b) holds as follows:
  1. (1a)

    either \(s_i\) is unassigned in M, or \(s_i\) prefers \(p_j\) to \(M(s_i)\);

     
  2. (1b)
    either (i), (ii), or (iii) holds as follows:
    1. (i)

      \(p_j\) is undersubscribed and \(l_k\) is undersubscribed;

       
    2. (ii)

      \(p_j\) is undersubscribed, \(l_k\) is full, and either \(s_i \in M(l_k)\) or \(l_k\) prefers \(s_i\) to the worst student/s in \(M(l_k)\) or is indifferent between them;

       
    3. (iii)

      \(p_j\) is full and \(l_k\) prefers \(s_i\) to the worst student/s in \(M(p_j)\) or is indifferent between them.

       
     
  3. (2a)

    \(s_i\) is indifferent between \(p_j\) and \(M(s_i)\);

     
  4. (2b)
    either (i), (ii), or (iii) holds as follows:
    1. (i)

      \(p_j\) is undersubscribed, \(l_k\) is undersubscribed and \(s_i \notin M(l_k)\);

       
    2. (ii)

      \(p_j\) is undersubscribed, \(l_k\) is full, \(s_i \notin M(l_k)\), and \(l_k\) prefers \(s_i\) to the worst student/s in \(M(l_k)\);

       
    3. (iii)

      \(p_j\) is full and \(l_k\) prefers \(s_i\) to the worst student/s in \(M(p_j)\).

       
     

Some intuition for the strong stability definition is given in [24, Sect. 3]. In the remainder of this paper, any usage of the term blocking pair refers to the version of this term for strong stability as defined above.

3 An Algorithm for Spa-St under strong stability

In this section we present our algorithm for spa-st under strong stability, which we will refer to as Algorithm SPA-ST-strong. In Sect. 3.1, we give some definitions relating to the algorithm. In Sect. 3.2, we give a description of our algorithm and present it in pseudocode form. We illustrate an execution of our algorithm with respect to a spa-st instance in Sect. 3.3. Finally, we present the algorithm’s correctness and complexity results in Sect. 3.4.

3.1 Definitions Relating to the Algorithm

Given a pair \((s_i, p_j) \in M\), for some strongly stable matching M in I, we call \((s_i, p_j)\) a strongly stable pair. During the execution of the algorithm, students become provisionally assigned to projects (and implicitly to lecturers), and it is possible for a project (and lecturer) to be provisionally assigned a number of students that exceeds its capacity. We describe a project (respectively lecturer) as replete if at any time during the execution of the algorithm it has been full or oversubscribed. We say that a project (respectively lecturer) is non-replete if it is not replete.

The provisional assignment graph is an undirected bipartite graph \(G = (S \cup P, E)\), with \(S \subseteq \mathcal {S}\) and \(P \subseteq \mathcal {P}\) such that there is an edge \((s_i, p_j) \in E\) if and only if \(s_i\) is provisionally assigned to \(p_j\). During the execution of the algorithm, it is possible for a student to be adjacent to more than one project in G. Thus, we denote by \(G(s_i)\) the set of projects that are adjacent to \(s_i\) in G. Given a project \(p_j \in P\), we denote by \(G(p_j)\) the set of students who are provisionally assigned to \(p_j\) in G and we let \(d_G(p_j) = |G(p_j)|\). Similarly, we denote by \(G(l_k)\) the set of students who are provisionally assigned to a project offered by \(l_k\) in G and we let \(d_G(l_k) = |G(l_k)|\).

As stated earlier, for a project \(p_j\), it is possible that \(d_G(p_j) > c_j\) at some point during the algorithm’s execution. Thus, we denote by \(q_j = \min \{c_j, d_G(p_j)\}\) the quota of \(p_j\) in G, which is the minimum between \(p_j\)’s capacity and the number of students who are provisionally assigned to \(p_j\) in G. Similarly, for a lecturer \(l_k\), it is possible that \(d_G(l_k) > d_k\) at some point during the algorithm’s execution. At this point, we denote by \(\alpha _k = \sum \{q_j: p_j \in P_k \cap P\}\) the total quota of projects offered by \(l_k\) that is provisionally assigned to students in G and we denote by \(q_k = \min \{d_k, d_G(l_k), \alpha _k\}\) the quota of \(l_k\) in G.

The algorithm proceeds by deleting from the preference lists certain \((s_i, p_j)\) pairs that are not strongly stable. By the term delete \((s_i, p_j)\), we mean the removal of \(p_j\) from \(s_i\)’s preference list and the removal of \(s_i\) from \(\mathcal {L}_k^j\) (the projected preference list of lecturer \(l_k\) for \(p_j\)); in addition, if \((s_i, p_j) \in E\) we delete the edge from G. By the head and tail of a preference list at a given point we mean the first and last tie respectively on that list after any deletions might have occurred (recalling that a tie can be of length 1). Given a project \(p_j\), we say that a student \(s_i\) is dominated in \(\mathcal {L}_k^j\) if \(s_i\) is worse than at least \(c_j\) students who are provisionally assigned to \(p_j\). The concept of a student becoming dominated in a lecturer’s preference list is defined in a slightly different manner.

Definition 2

(Dominated in \(\mathcal {L}_k\)). At a given point during the algorithm’s execution, let \(\alpha _k\) and \(d_G(l_k)\) be as defined above. We say that a student \(s_i\) is dominated in \(\mathcal {L}_k\) if \(\min \{d_G(l_k), \alpha _k\} \ge d_k\), and \(s_i\) is worse than at least \(d_k\) students who are provisionally assigned in G to a project offered by \(l_k\).

Definition 3

(Lower rank edge). We define an edge \((s_i, p_j) \in E\) as a lower rank edge if \(s_i\) is in the tail of \(\mathcal {L}_k\) and \(\min \{d_G(l_k), \alpha _k\} > d_k\).

Definition 4

(Bound). Given an edge \((s_i, p_j) \in E\), we say that \(s_i\) is bound to \(p_j\) if (i) \(p_j\) is not oversubscribed or \(s_i\) is not in the tail of \(\mathcal {L}_k^j\) (or both), and (ii) \((s_i, p_j)\) is not a lower rank edge or \(s_i\) is not in the tail of \(\mathcal {L}_k\) (or both). If \(s_i\) is bound to \(p_j\), we may also say that \((s_i, p_j)\) is a bound edge. Otherwise, we refer to it as an unbound edge.1

We form a reduced assignment graph \(G_r = (S_r, P_r, E_r)\) from a provisional assignment graph G as follows. For each edge \((s_i, p_j) \in E\) such that \(s_i\) is bound to \(p_j\), we remove the edge \((s_i, p_j)\) from \(G_r\) and we reduce the quota of \(p_j\) in \(G_r\) (and implicitly \(l_k\)2) by one. Further, we remove all other unbound edges incident to \(s_i\) in \(G_r\). Each isolated student vertex is then removed from \(G_r\). Finally, if the quota of any project is reduced to 0, or \(p_j\) becomes an isolated vertex, then \(p_j\) is removed from \(G_r\). For each surviving \(p_j\) in \(G_r\), we denote by \(q_j^*\) the revised quota of \(p_j\), where \(q_j^*\) is the difference between \(p_j\)’s quota in G (i.e., \(q_j\)) and the number of students that are bound to \(p_j\). Similarly, we denote by \(q_k^*\) the revised quota of \(l_k\) in \(G_r\), where \(q_k^*\) is the difference between \(l_k\)’s quota in G (i.e., \(q_k\)) and the number of students that are bound to a project offered by \(l_k\). Further, for each \(l_k\) who offers at least one project in \(G_r\), we let \(n = \sum \{q_j^* : p_j \in P_k \cap P_r\} - q_k^*\), where n is the difference between the total revised quota of projects in \(G_r\) that are offered by \(l_k\) and the revised quota of \(l_k\) in \(G_r\). Now, if \(n \le 0\), we do nothing; otherwise, we extend \(G_r\) as follows. We add n dummy student vertices to \(S_r\). For each of these dummy vertices, say \(s_{d_i}\), and for each project \(p_j \in P_k \cap P_r\) that is adjacent to a student vertex in \(S_r\) via a lower rank edge, we add the edge \((s_{d_i}, p_j)\) to \(E_r\).3

Given a set \(X \subseteq S_r\) of students, define \(\mathcal {N}(X)\), the neighbourhood of X, to be the set of project vertices adjacent in \(G_r\) to a student in X. If for all subsets X of \(S_r\), each student in X can be assigned to one project in \(\mathcal {N}(X)\), without exceeding the revised quota of each project in \(\mathcal {N}(X)\) (i.e., \(|X| \le \sum \{q_j^*: p_j \in \mathcal {N}(X)\}\) for all \(X \subseteq S_r\)); then we say that \(G_r\) admits a perfect matching that saturates \(S_r\).

Definition 5

(Critical set). It is well known in the literature [17] that if \(G_r\) does not admit a perfect matching that saturates \(S_r\), then there must exist a deficient subset \(Z \subseteq S_r\) such that \(|Z| > \sum \{q_j^*: p_{j} \in \mathcal {N}(Z)\}\). To be precise, the deficiency of Z is defined by \(\delta (Z) = |Z| - \sum \{q_j^*: p_{j} \in \mathcal {N}(Z)\}\). The deficiency of \(G_r\), denoted \(\delta (G_r)\), is the maximum deficiency taken over all subsets of \(S_r\). Thus, if \(\delta (Z) = \delta (G_r)\), we say that Z is a maximally deficient subset of \(S_r\), and we refer to Z as a critical set.

We denote by \(P_R\) the set of replete projects in G and we denote by \(P_R^*\) a subset of projects in \(P_R\) which is obtained as follows. For each project \(p_j \in P_R\), let \(l_k\) be the lecturer who offers \(p_j\). For each student \(s_i\) such that \((s_i, p_j)\) has been deleted, we add \(p_j\) to \(P_R^*\) if (i) and (ii) holds as follows:
  1. (i)

    either \(s_i\) is unassigned in G, or \((s_i, p_{j'}) \in G\) where \(s_i\) prefers \(p_j\) to \(p_{j'}\), or \((s_i, p_{j'}) \in G\) and \(s_i\) is indifferent between \(p_j\) and \(p_{j'}\) where \(p_{j'} \notin P_k\);

     
  2. (ii)

    either \(l_k\) is undersubscribed in G, or \(l_k\) is full in G and either \(s_i \in G(l_k)\) or \(l_k\) prefers \(s_i\) to some student assigned to \(l_k\) in G.

     

Definition 6

(Feasible matching). A feasible matching in the final provisional assignment graph G is a matching M obtained as follows:
  1. 1.

    Let \(G^*\) be the subgraph of G induced by the students who are adjacent to a project in \(P_R^*\). First, find a maximum matching \(M^*\) in \(G^*\);

     
  2. 2.

    Using \(M^*\) as an initial solution, find a maximum matching M in G.

     

3.2 Description of the Algorithm

Algorithm SPA-ST-strong, described in Algorithm 1, begins by initialising an empty bipartite graph G which will contain the provisional assignments of students to projects (and implicitly to lecturers). We remark that such assignments (i.e., edges in G) can subsequently be broken during the algorithm’s execution.

The while loop of the algorithm involves each student \(s_i\) who is not adjacent to any project in G and who has a non-empty list applying in turn to each project \(p_j\) at the head of her list. Immediately, \(s_i\) becomes provisionally assigned to \(p_j\) in G (and to \(l_k\)). If, by gaining a new provisional assignee, project \(p_j\) becomes full or oversubscribed then we set \(p_j\) as replete. Further, for each student \(s_t\) in \(\mathcal {L}_k^j\), such that \(s_t\) is dominated in \(\mathcal {L}_k^j\), we delete the pair \((s_t, p_j)\). As we will prove later, such pairs cannot belong to any strongly stable matching. Similarly, if by gaining a new provisional assignee, \(l_k\) becomes full or oversubscribed then we set \(l_k\) as replete. For each student \(s_t\) in \(\mathcal {L}_k\), such that \(s_t\) is dominated in \(\mathcal {L}_k\) and for each project \(p_u \in P_k\) that \(s_t\) finds acceptable, we delete the pair \((s_t, p_u)\). This continues until every student is provisionally assigned to one or more projects or has an empty list. At the point where the while loop terminates, we form the reduced assignment graph \(G_r\) and we find the critical set Z of students in \(G_r\) (we describe how to find Z on Page 9). As we will see later, no project \(p_j \in \mathcal {N}(Z)\) can be assigned to any student in the tail of \(\mathcal {L}_k^j\) in any strongly stable matching, so all such pairs are deleted.

At the termination of the inner repeat-until loop in line 21, i.e., when Z is empty, if some project \(p_j\) that is replete ends up undersubscribed, we carry out some certain deletions4. We let \(s_r\) be any one of the most preferred students (according to \(\mathcal {L}_k^j\)) who was provisionally assigned to \(p_j\) during some iteration of the algorithm but is not assigned to \(p_j\) at this point (for convenience, we henceforth refer to such \(s_r\) as the most preferred student rejected from \(p_j\) according to \(\mathcal {L}_k^j\)). If the students at the tail of \(\mathcal {L}_k\) (recalling that the tail of \(\mathcal {L}_k\) is the least-preferred tie in \(\mathcal {L}_k\) after any deletions might have occurred) are no better than \(s_r\), it turns out that none of these students \(s_t\) can be assigned to any project offered by \(l_k\) in any strongly stable matching – such pairs \((s_t, p_u)\), for each project \(p_u \in P_k\) that \(s_t\) finds acceptable, are deleted. The repeat-until loop is then potentially reactivated, and the entire process continues until every student is provisionally assigned to a project or has an empty list.

At the termination of the outer repeat-until loop in line 30, if a student is adjacent in G to a project \(p_j\) via a bound edge, then we may potentially carry out extra deletions. First, we let \(l_k\) be the lecturer who offers \(p_j\) and we let U be the set of projects that are adjacent to \(s_i\) in G via an unbound edge. For each project \(p_u \in U \setminus P_k\), it turns out that the pair \((s_i, p_u)\) cannot belong to any strongly stable matching, thus we delete all such pairs. Finally, we let M be any feasible matching in the provisional assignment graph G. If M is strongly stable relative to the given instance I then M is output as a strongly stable matching in I. Otherwise, the algorithm reports that no strongly stable matching exists in I. We present Algorithm SPA-ST-strong in pseudocode form in Algorithm 1.
Finding the Critical Set. Consider the reduced assignment graph \(G_r = (S_r, P_r,E_r)\) formed from G at a given point during the algorithm’s execution (at line 15). To find the critical set of students in \(G_r\), first we need to construct a maximum matching \(M_r\) in \(G_r\), with respect to the revised quota \(q_j^*\), for each \(p_j \in P_r\). In this context, a matching \(M_r \subseteq E_r\) is such that \(|M_r(s_i)| \le 1\) for all \(s_i \in S_r\), and \(|M_r(p_j)| \le q_j^*\) for all \(p_j \in P_r\). We describe how to construct \(M_r\) as follows:
  1. 1.

    Let \(G_r'\) be the subgraph of \(G_r\) induced by the dummy students adjacent to a project in \(G_r\). First, find a maximum matching \(M_r'\) in \(G_r'\).

     
  2. 2.

    Using \(M_r'\) as an initial solution, find a maximum matching \(M_r\) in \(G_r\).5

     

Given a maximum matching \(M_r\) in the reduced assignment graph \(G_r\), the critical set Z consists of the set U of unassigned students together with the set \(U'\) of students reachable from a student in U via an alternating path (see [24, Lemma 1] for a proof).

3.3 Example Algorithm Execution

In this section, we illustrate an execution of Algorithm SPA-ST-strong with respect to the spa-st instance \(I_3\) shown in Fig. 2 (Page 10), which involves the set of students \(\mathcal {S} = \{s_i: 1 \le i \le 8\}\), the set of projects \(\mathcal {P} = \{p_j: 1 \le j \le 6\}\) and the set of lecturers \(\mathcal {L} = \{l_k: 1 \le k \le 3\}\). The algorithm starts by initialising the bipartite graph \(G = \{\}\), which will contain the provisional assignment of students to projects. We assume that the students become provisionally assigned to each project at the head of their list in subscript order. Figures 3, 4 and 5 illustrate how this execution of Algorithm SPA-ST-strong proceeds with respect to \(I_3\).
Fig. 2.

An instance \(I_3\) of spa-st.

Fig. 3.

Iteration (1).

Iteration 1: At the termination of the while loop during the first iteration of the inner repeat-until loop, every student, except \(s_3\), \(s_6\) and \(s_7\), is provisionally assigned to every project in the first tie on their preference list. Edge \((s_3, p_4) \notin G^{(1)}\) because \((s_3, p_4)\) was deleted as a result of \(s_6\) becoming provisionally assigned to \(p_4\), causing \(s_3\) to be dominated in \(\mathcal {L}_2^4\). Also, edge \((s_6, p_2) \notin G^{(1)}\) because \((s_6, p_2)\) was deleted as a result of \(s_4\) becoming provisionally assigned to \(p_2\), causing \(s_6\) to be dominated in \(\mathcal {L}_1\) (at that point in the algorithm, \(\min \{d_G(l_1), \alpha _1\} = \min \{4,3\} = 3 = d_1\) and \(s_6\) is worse than at least \(d_1\) students who are provisionally assigned to \(l_1\)). Finally, edge \((s_7, p_3) \notin G^{(1)}\) because \((s_7, p_3)\) was deleted as a result of \(s_5\) becoming provisionally assigned to \(p_5\), causing \(s_7\) to be dominated in \(\mathcal {L}_2^3\).

To form \(G_r^{(1)}\), the bound edges \((s_5, p_3), (s_6, p_4), (s_7, p_1)\) and \((s_8, p_5)\) are removed from the graph. We can verify that edges \((s_4, p_2)\) and \((s_5, p_2)\) are unbound, since they are lower rank edges for \(l_1\). Also, since \(p_1\) is oversubscribed, and each of \(s_1, s_2\) and \(s_3\) is at the tail of \(\mathcal {L}_1^1\), edges \((s_1, p_1)\), \((s_2, p_1)\) and \((s_3, p_1)\) are unbound. Further, the revised quota of \(l_1\) in \(G_r^{(1)}\) is 2, and the total revised quota of projects offered by \(l_1\) (i.e., \(p_1\) and \(p_2\)) is 3. Thus, we add one dummy student vertex \(s_{d_1}\) to \(G_r^{1}\), and we add an edge between \(s_{d_1}\) and \(p_2\) (since \(p_2\) is the only project in \(G_r^{(1)}\) adjacent to a student in the tail of \(\mathcal {L}_1\) via a lower rank edge). With respect to the maximum matching \(M_r^{(1)}\), it is clear that the critical set \(Z^{(1)} =\{s_1, s_2, s_3\}\), thus we delete the edges \((s_1, p_1)\), \((s_2, p_1)\) and \((s_3, p_1)\) from \(G^{(1)}\); and the inner repeat-until loop is reactivated.
Fig. 4.

Iteration (2).

Iteartion 2: At the beginning of this iteration, each of \(s_1\) and \(s_2\) is unassigned and has a non-empty list; thus we add edges \((s_1, p_6)\) and \((s_2, p_2)\) to the provisional assignment graph obtained at the termination of iteration (1) to form \(G_r^{(2)}\). It can be verified that every edge in \(G_r^{(2)}\), except \((s_4, p_2)\) and \((s_5, p_2)\), is a bound edge. Clearly, the critical set \(Z^{(2)} = \emptyset \), thus the inner repeat-until loop terminates. At this point, project \(p_1\), which was replete during iteration (1), is undersubscribed in iteration (2). Moreover, the students at the tail of \(\mathcal {L}_1\) (i.e., \(s_4\) and \(s_5\)) are no better than \(s_3\), where \(s_3\) is one of the most preferred students rejected from \(p_1\) according to \(\mathcal {L}_1^1\); thus we delete edges \((s_4, p_2)\) and \((s_5, p_2)\). The outer repeat-until loop is then reactivated (since \(s_4\) is unassigned and has a non-empty list).
Fig. 5.

Iteration (3).

Iteration 3: At the beginning of this iteration, the only student that is unassigned and has a non-empty list is \(s_4\); thus we add edges \((s_4, p_5)\) and \((s_4, p_6)\) to the provisional assignment graph obtained at the termination of iteration (2) to form \(G_r^{(3)}\). The provisional assignment of \(s_4\) to \(p_5\) led to \(p_5\) becoming oversubscribed; thus \((s_8, p_5)\) is deleted (since \(s_8\) is dominated on \(\mathcal {L}_3^5\)). Further, \(s_8\) becomes provisionally assigned to \(p_1\). It can be verified that all the edges in \(G_r^{(3)}\) are bound edges. Moreover, the reduced assignment graph \(G_r^{(3)} = \emptyset \).

Again, every unassigned students has an empty list. We also have that a project \(p_2\), which was replete in iteration (2), is undersubscribed in iteration (3). However, no further deletion is carried out in line 29 of the algorithm, since the student at the tail of \(\mathcal {L}_1\) (i.e., \(s_2\)) is better than \(s_4\) and \(s_5\), where \(s_4\) and \(s_5\) are the most preferred students rejected from \(p_2\) according to \(\mathcal {L}_1^2\). Hence, the repeat-until loop terminates. Also, no deletion is carried out in line 36 of the algorithm. We observe that \(P_R^* = \{p_5\}\), since \((s_8, p_5)\) has been deleted, \(s_8\) prefers \(p_5\) to her provisional assignment in G and \(l_3\) is undersubscribed. Thus we need to ensure \(p_5\) fills up in the feasible matching M constructed from G, so as to avoid \((s_8, p_5)\) from blocking M. Finally, the algorithm outputs the feasible matching \(M = \{(s_1, p_6), (s_2, p_2), (s_4, p_5), (s_5, p_3), (s_6, p_4), (s_7, p_1), (s_8, p_1)\}\) as a strongly stable matching in \(I_3\).

3.4 Correctness of the algorithm

The correctness and complexity of Algorithm SPA-ST-strong is established via a sequence of lemmas, namely Lemmas 4–14 in [24, Sect. 4.5]. These are omitted here for space reasons, but may be summarised as follows:
  1. 1.

    no strongly stable pair is deleted during the execution of the algorithm;

     
  2. 2.
    no strongly stable matching exists if some:
    1. (a)

      non-replete lecturer \(l_k\) has fewer assignees in the feasible matching M than provisional assignees in the final assignment graph G, or

       
    2. (b)

      replete lecturer is not full in M, or

       
    3. (c)

      student is bound to two or more projects that are offered by different lecturers, or

       
    4. (d)

      pair \((s_i, p_j)\) was deleted where \(p_j\) is offered by \(l_k\), each of \(p_j\) and \(l_k\) is undersubscribed in M, and for any \(p_{j'} \in P_k\) such that \(s_i\) is indifferent between \(p_j\) and \(p_{j'}\), \((s_i, p_{j'}) \notin M\);

       
     
  3. 3.

    if the algorithm outputs “no strongly stable matching exists” then at least one of the properties in (2) above must hold;

     
  4. 4.

    Algorithm SPA-ST-strong may be implemented to run in \(O(m^2)\) time, where m is the total length of the students’ preference lists.

     

The following theorem collects together Lemmas 4–14 in [24] and establishes the correctness and complexity of Algorithm SPA-ST-strong.

Theorem 1

For a given instance I of spa-st, Algorithm SPA-ST-strong determines in \(O(m^2)\) time whether or not a strongly stable matching exists in I. If such a matching does exist, all possible executions of the algorithm find one in which each assigned student is assigned at least as good a project as she could obtain in any strongly stable matching, and each unassigned student is unassigned in every strongly stable matchings.

Given the optimality property established by Theorem 1, we define the strongly stable matching found by Algorithm SPA-ST-strong to be student-optimal. For example, in the spa-st instance illustrated in Fig. 1, it may be verified that the student-optimal strongly stable matching is \(\{(s_1, p_1), (s_2, p_2), (s_3, p_3)\}\).

4 Conclusion

We leave open the formulation of a lecturer-oriented counterpart to Algorithm SPA-ST-strong. From an experimental perspective, an interesting direction would be to carry out an empirical analysis of Algorithm SPA-ST-strong, to investigate how various parameters (e.g., the density and position of ties in the preference lists, the length of the preference lists, or the popularity of some projects) affect the existence of a strongly stable matching, based on randomly generated and/or real instances of spa-st.

Footnotes

  1. 1.

    An edge \((s_i, p_j) \in E\) can change state from bound to unbound, but not vice versa.

  2. 2.

    If \(s_i\) is bound to more than one projects offered by \(l_k\), for all the bound edges involving \(s_i\) and these projects that we remove from \(G_r\), we only reduce \(l_k\)’s quota in \(G_r\) by one.

  3. 3.

    An intuition as to why we add dummy students to \(G_r\) is as follows. Given a lecturer \(l_k\) whose project is provisionally assigned to a student in \(G_r\). If \(q_k^* < \sum \{q_j^* : p_j \in P_k \cap P_r\}\), then we need n dummy students to offset the difference between \(\sum \{q_j^* : p_j \in P_k \cap P_r\}\) and \(q_k^*\), so that we do not oversubscribe \(l_k\) in any maximum matching obtained from \(G_r\).

  4. 4.

    This type of deletion was also carried out in Algorithm SPA-ST-super for super-stability [23].

  5. 5.

    By making sure that all the dummy students are matched in step 1, we are guaranteed that no lecturer is oversubscribed with non-dummy students in \(G_r\).

Notes

Acknowledgement

The authors would like to convey their sincere gratitude to Adam Kunysz for valuable discussions concerning Algorithm SPA-ST-strong. They would also like to thank the anonymous reviewers for their helpful suggestions.

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Copyright information

© Springer Nature Switzerland AG 2020

Authors and Affiliations

  1. 1.School of Computing ScienceUniversity of GlasgowGlasgowScotland

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