Abstract
We introduce a new twosided stable matching problem that describes the summer internship matching practice of an Australian university. The model is a case between two models of Kamada and Kojima on matchings with distributional constraints. We study three solution concepts, the strong and weak stability concepts proposed by Kamada and Kojima, and a new one in between the two, called cutoff stability. Kamada and Kojima showed that a strongly stable matching may not exist in their most restricted model with disjoint regional quotas. Our first result is that checking its existence is NPhard. We then show that a cutoff stable matching exists not just for the summer internship problem but also for the general matching model with arbitrary heredity constraints. We present an algorithm to compute a cutoff stable matching and show that it runs in polynomial time in our special case of summer internship model. However, we also show that finding a maximum size cutoff stable matching is NPhard, but we provide a Mixed Integer Linear Program formulation for this optimisation problem.
1 Introduction
Centralized twosided matching market algorithms have received immense success in several application domains, including matching students to schools, residents to hospitals, and projects to workers.^{Footnote 1} We present a novel matching market model that we refer to as the summer internship matching market. The model captures the matching of student applicants to projects proposed by supervisors in an internship program. A distinctive feature of the model is that in order for an applicant to be assigned to any project, a certain amount of money needs to be contributed from the project supervisors’ funds.
Our problem is inspired by summer intern research programs in Australia. It is common for undergraduate students to undertake research projects over the summer. Each project is supervised by one or more members of the faculty, with many offering multiple projects. Even though the projects may be discounted by contributions from the faculty, supervisors are often required to contribute to the funding of these positions, from their personal research budget. Alternatively, they could be constrained by the amount of time they can allocate to supervision. These supervisorside constraints mean that not all projects can be funded.
Just as the standard hospital resident matching models do not just apply to the matching of doctors to hospitals [48], our model also does not just apply to matching of student interns. It applies to any twosided matching model in which very widely applicable budget requirements and budget constraints are involved. For example, our problem also models hiring scenario in which different teams have their own budgets and they want to hire employees. Certain employee roles could sit across various teams. In that case, multiple teams can pool in their money to fund joint positions.
The budget constraints that we consider lead to interesting research challenges. Firstly, applying standard matching algorithms such as the Deferred Acceptance Algorithm does not work as it do not deal with complex feasibility constraints. More critically, as we will discuss, there may exist no feasible matching that satisfies stability as considered in seminal papers on matching (see, e.g., Gale and Shapley [28] and Abdulkadiroğlu and Sönmez [1]).
1.1 Contributions
In this paper, we formalize the summer internship problem with budgets, abbreviated as SIP. It falls within the class of models that matches applicants, projects, and supervisors. The main characteristic of our problem is that supervisors have budgets that they can spread across their projects. Our model is more general than the widelyused hospitalresident matching model in which the hospitals are partitioned into regions and regions have upper capacities [35], a model that we abbreviate as REG. On the other hand, our model is a special case of the matching model under distributional constraints of Kamada and Kojima [37], where the feasibility of a matching is monotone in the number of applicants matched, the property called heredity by Goto et al. [30], that we abbreviate as HER.
First, we study the concept of strong stability proposed in Kamada and Kojima [35], where the authors showed that such solution may not exist for REG. Here we prove that the problem of checking the existence of a strongly stable matching is NPhard for REG (and thus also for any settings that contain REG as a special case, such as SIP and HER).
Then we study weak stability, also introduced in Kamada and Kojima [35] for REG and then studied in Kamada and Kojima [37] for HER. In the conference version of our paper [11] we provided a strongly polynomial algorithm for computing a weakly stable matching for our intermediate SIP model, based on the algorithm of Kamada and Kojima [37] for HER. Our algorithm uses as an oracle an algorithm based on network flows to repeatedly check whether a given matching is feasible or not. In this extended version of our conference paper, we strengthen this result by presenting a general algorithm that returns a matching satisfying cutoff stability. Fair matchings are exactly those matchings that can be induced by a set of cutoff scores. Meanwhile, cutoff stable matchings are those that are induced by minimal cutoffs, i.e., where the decrease of any cutoff would make the induced matching infeasible. We show that cutoff stability is an intermediate notion between weak and strong stability, and the computation of a cutoff stable matching is always possible with our algorithm for HER. Since HER covers many wellstudied settings including refugee matching (see, e.g., [9, 20]), our result have wideapplicability.^{Footnote 2} We apply the algorithm in the context of SIP and show that it can be implemented to run in polynomial time. However, we also show that finding a maximum size cutoff stable matching is NPhard even for REG (implying that is also NPhard for SIP and HER). However, we formulate a Mixed Integer Linear Program (MILP) for finding such an optimal solution for SIP. Many of our general structural results for the HER model. Unless specified otherwise, we will assume that our formal statements apply to HER.
Finally, we provide a normative criterion for egalitarian expost allocation of supervisor funding among projects, and a polynomialtime algorithm to find an egalitarian allocation. Combined with our polynomialtime algorithm for finding a weakly stable matching, we present a compelling approach for finding a desirable solution for the summer internship problem that appeals to both stability and fairness requirements.
1.2 Related work
The literature on twosided matching was inspired by the seminal paper of Gale and Shapley [27] who considered matching markets that match students to schools and hospitals to residents. The paper has spawned richer matching models and resulted in new algorithmic work (see, e.g., Manlove [45]). Our work is an extension of these models and falls in the general umbrella of matching markets with various kinds of distributions constraints (see, e.g., Aziz et al. [9, 12, 24, 25, 36, 42]).
Our concept of budgetfeasibility is a type of feasibility constraint, as defined by Kamada and Kojima [37] and thus our problem falls under the umbrella of matching under feasibility constraints. Thus, the notions of strong and weak stability studied by Kamada and Kojima [37] also apply in our model. In our paper, we focus on computational results such as establishing NPcompleteness or polynomialtime solvability of stable matchings. Our model also has the additional dimension of budget allocations for which we explore fairness concepts as well as algorithms to divide the budget in an egalitarian manner. We also propose an intermediate concept called cutoff stability and prove that it can be achieved for the general matching model with arbitrary heredity constraints. Cutoff stability is a new notion for the HER model, though similar notions have been studied for different stable matching models, such as the choice function model of Fleiner and Jankó [23]. Our concept of cutoff stability defined for general feasibility constraints is similar in spirit to the withintype envyfreeness concept of Echenique and Yenmez [22] in the context of school choice with affirmative action.
Goto et al. [30] also considered twosided matching under general feasibility constraints that satisfy the heredity property. Their main contribution is proposing an algorithm called Adaptive Deferred Acceptance that satisfies strategyproofness, nonwastefulness, and a fairness property (that is weaker than the weak stability concept of Kamada and Kojima [37] and hence also cutoff stability). Kamada and Kojima [39] also consider heredity constraints that apply to individual schools/hospitals.
Our model bears some similarities with the hospitalresident matching problem with regional constraints [10, 17, 29, 35, 38]. In these regionbased problems, at most a certain number of students can be selected from given regions. On the other hand, in the summer internship problem, a supervisor’s budget is divisible and can be spread partially over all of her projects. If the regions are disjoint (as studied by Kamada and Kojima [35]), then the regionbased model is a special case of our model. The general setting with region constraints has not seen many positive results, and the more constrained hierarchical regions as studied by Goto et al. [29] and Kamada and Kojima [38] can neither replicate, nor be replicated by a set of supervisors. In particular, the concept of stability with regional priorities proposed by Kamada and Kojima [38] cannot be directly applied to our setting. Our concept of cutoff stability is an alternative intermediate notion of stability and it can be applied to a wider range of settings (those subsumed under the HER model).
Abraham et al. [3] considered a different model for studentproject allocation. Notable differences in their model include: (1) no project has multiple supervisors, (2) each supervisor has a universal priority list over students which is not project specific, and (3) a supervisor has a rigid capacity constraint for the number of projects to supervise. Our model allows supervisors to explore more efficient outcomes by pooling in their budgets to host a student. The universal priority list of each supervisor makes the model of Abraham et al. [3] much more restricted and different from our model.
Two other recent models are similar to our setting. Goto et al. [30] introduce the StudentProjectRoom matching problem. Rooms are indivisible, and at most one room can be allocated to each project. Ismaili et al. [33] extend this to a more general StudentProjectResource allocation problem. Resources are still indivisible, but there is no longer a restriction imposed on the number of resources that can be allocated to each project. Our model is distinct from both of these, as it allows the resources (in our case, supervisor budgets) to be divisible. This divisibility allows for better computational results. For instance, verifying the feasibility of a matching, and finding a weakly stable (and thus nonwasteful) matching can be done in polynomial time.
There is also work on matching with budget constraints (see, e.g., Kawase and Iwasaki [40, 41], Ismaili et al. [34]). The models considered in these papers are different in several respects. For example, hospitals have additive utilities and each hospital gives monetary compensation to doctors.
Regarding the LP descriptions and IP techniques for twosided stable matching problems, Baïou and Balinski [13] gave the first description on the stable admissions polytope. Integer programming techniques have been used later for college admissions with special features [4], stable project allocation under distributional constraints [5], the hospital–resident problem with couples [18], and ties [21, 43].
2 Preliminaries
First, we introduce our summer internship matching model (SIP), we show that checking the feasibility of a matching is polynomial time tractable, and finally we also show the relation of our model to the REG and HER models of Kamada and Kojima described in Kamada and Kojima [35] and Kamada and Kojima [37], respectively.
2.1 Model
Let A be a finite set of applicants, and P a finite set of projects. Each applicant \(a\in A\) has a strict preference list \(\succ _a\) that ranks the projects that a finds acceptable. Each project \(p\in P\) has a preference list \(\succ _p\) over the subset of applicants that p finds acceptable, and a maximum capacity \(c_p\).
Furthermore, let S denote the set of project supervisors. Each supervisor \(s\in S\) has a list of projects \(P_s\) that they supervise, and is endowed with a budget (e.g. quantity of funds) \(q_s\) that they can allocate among those projects. We assume that these budgets are infinitely divisible and that each applicant requires one unit of funding. Further, we assume that these endowments are publicly known, and thus supervisors cannot strategise by misreporting their budgets. Additionally, denote the list of supervisors for project p by \(S_p\).
We say that an applicant \(a\in A\) is matched to project \(p\in P\) if \((a, p)\in M\). A matching M is a subset of \(A\times P\) that satisfies the following conditions:

Each applicant is matched to at most one project (for all \(a\in A\), \(\{(a, p)\in M\ \mathbin {:}\ p\in P\}\le 1\)), and a finds the project they are matched to acceptable.

The number of applicants matched to any project does not exceed that project’s capacity (for all \(p\in P\), \(\{(a, p)\in M\ \mathbin {:}\ a\in A\}\le c_p\)), and p finds all applicants matched to it acceptable.
We use M(a) to refer to the project that applicant a is matched to (\(M(a)=\emptyset \) if a is unmatched). Meanwhile, M(p) denotes the set of applicants matched to p.
Let \(x_{s,p}\) be the amount of funds a supervisor s allocates to project p. We call a matching M feasible (or supervisorfeasible) if there exists a set \(\{x_{s,p}\}_{s\in S, p \in P_s}\) that satisfies the following conditions:

\(x_{s,p}\ge 0\) for all \(s\in S,~p\in P_s\)

Every project receives one unit of funding for each applicant matched to it: \(\sum _{s\in S_p} x_{s,p} = M(p) \text { for all } p\in P\)

Supervisors do not exceed their endowment: \(\sum _{p\in P_s} x_{s,p} \le q_{s} \text { for all } s\in S\)
We call any set \(\{x_{s,p}\}_{s\in S, p \in P_s}\) that satisfies the above conditions for matching M a feasible funding allocation. The mathematical model presented exactly captures the student research internship program in our university: each supervisor can be part of multiple internship project proposals but does not necessarily have the funding to contribute to all of them.
Example 1
(Summer Internship Problem) Consider the following instance of the summer internship problem with 2 applicants 2 supervisors, and 2 projects.
The only three feasible matchings are the empty matching and the two matchings in which some applicant is matched to project \(p_2\). The reason no one can be matched to project \(p_1\) is that \(p_1\) has a sole supervisor \(s_1\) who does not have sufficient funding to fund \(p_1\). On the other hand, the combined funding of \(s_1\) and \(s_2\) is more than 1 for project \(p_2\) so \(p_2\) can be funded. One possible feasible funding allocation x is where \(s_1\) contributed half of the budget of project \(p_2\) and \(s_2\) contributes the rest: \(x_{s_1,p_2}=0.5\), \(x_{s_2,p_2}=0.5\).
2.2 Checking feasibility of a matching
We show that checking feasibility of a matching in our model can be done efficiently by reducing the question to a network flow problem.^{Footnote 3} Define the funding flow graph \(G_M\) associated with a matching M as follows:

\(V(G_M) =\{s^*\}\cup S\cup P\cup \{t^*\}\), where \(s^*\) is the source, and \(t^*\) is the sink

Arcs (s, p), for all supervisorproject pairs where \(p\in P_s\) with capacity \(\infty \)

Arcs \((s^*,s)\), for all \(s\in S\), each with capacity \(q_s\)

Arcs \((p,t^*)\), for all \(p\in P\), each with capacity M(p)
Theorem 1
The feasibility of a matching can be checked in polynomial time \(O(\max \{S,P\}^3)\) for the summerinternship problem.
Proof
Our first claim is that a matching M is feasible if and only if \(G_M\) admits a feasible \(s^*\)\(t^*\) flow of size M. Note that by construction the value of an \(s^*\)\(t^*\) flow is at most M.
Suppose M is feasible. Define a flow f on \(G_M\) as follows:

\(f(s,p)=x_{s,p}\), \(\forall s\in S, p\in P_s\)

\(f(s^*,s)=\sum _{p\in P_s} x_{s,p}\), \(\forall s\in S\)

\(f(p,t^*)=\sum _{s\in S_p} x_{s,p}\), \(\forall p\in P\)
It is easy to see that this is a feasible flow of size M. Now suppose that \(G_M\) admits a feasible flow f of size M. Set \(x_{s,p}=f(s,p)\), \(\forall s\in S, p\in P_s\). We can then show that this \(\{x_{s,p}\}\) satisfies the conditions of feasibility.
Now that we have established the claim, we use the fact that the maximum flow problem can be solved in \(O(V^3)\) time (where V is the set of vertices), using for instance, the algorithm proposed by Malhotra et al. [44]. \(V(G_M)=S+P+2\) and, given a matching M, \(G_M\) can be constructed in \(O((S+P)^2)\) time. We can check whether M is feasible in \(O(\max \{S,P\}^3)\) time by computing a maximum flow and verifying whether it equals M. \(\square \)
Note that by the integer property of the network flow problem, if all the capacities of the supervisors are integer and the flow is feasible then an integer funding allocation exists.
2.3 Connection with the models by Kamada and Kojima
Our model satisfies the heredity property of Kamada and Kojima [37] and Kamada and Kojima [30], that says that if a matching is feasible then it remains feasible if the numbers of applicants matched to each project decreases or remains the same. Let \({\mathcal {M}}\) denote all the subsets of \(A\times P\), that we call as generalized matchings. Their matching model under distributional constraints that satisfy heredity (HER), can be represented by a feasibility function \(f:{\mathcal {M}}\rightarrow \{0,1\}\) that satisfies the following condition:

\(f(\emptyset )=1\) and, for any two matchings M and \(M'\), if \(M'(p)\le M(p) \ \forall p\in P\) then \(f(M)=1 \Rightarrow f(M')=1\).
A generalized matching M is feasible if and only if \(f(M)=1\). Note that the feasibility constraints are ‘anonymous’ in the sense that they do not depend on the identity of the applicants matched but only on their quantity. When considering the feasibility of a generalized matching in the context of SIP we relax the condition that every applicant can be matched to at most one project, but we obey the further requirements, namely the projectcapacity and budget conditions. These conditions can be still checked efficiently by the very same reduction to the maximum flow problem.
Kamada andKojima [35] studied a basic model with regional upper quotas, where the regions form a partition of the set of hospitals (REG). We can represent this in our model by replacing each region with a supervisor, endowed with a budget equal to that region’s capacity, and supervising a set of projects that correspond to the hospitals in that region.
Thus, REG is a special case of SIP, and SIP is a special case of HER. Therefore, every hardness result for REG implies the same hardness result for SIP and HER, and any easiness result for HER implies the same result for SIP, and an easiness result for SIP also holds for REG.
2.4 Fair matchings and cutoff scores
We say that matching M is fair, if for every pair \((a, p)\notin M\), \(p\succ _aM(a)\) implies that \(a' \succ _p a\) for every \(a'\in M(p)\).^{Footnote 4} This is a basic property for both strong and weak stability by Kamada and Kojima [37], and so also for our intermediate notion of cutoff stability.
In the classical Gale–Shapley model [27] a matching is stable if and only if it is fair and nonwasteful, where nonwastefulness means that there exists no project p with unfilled capacity and applicant a where \(p\succ _aM(a)\). In our context we can define different nonwastefulness notions leading to weak and strong stability and a new intermediate stability concept, called cutoff stability. We will define and analyze these stability concepts in Sects. 3, 4 and 5, respectively.
The concept of cutoff scores is closely related to fair matchings. Let \(d:P\rightarrow [0,1,\dots ,A+1]\) be the cutoff score function, where d(p) is the cutoff at project p. Without loss of generality we assume that each project p assigns a score to each applicant a in accordance with its preference list, that is a has score \(Ak+1\) if she is ranked kth by project p. Given cutoff scores d, we say that applicant a is admissible to project p if her score achieves the cutoff. Cutoff scores d induce matching M, if every applicant is matched to the best project of her preference where she is admissible.^{Footnote 5} The following observation is wellknown (see, e.g. Lemma 3 in Fleiner and Jankó [23]), for completeness we give a short proof.
Proposition 1
A matching is fair if and only if it is induced by some cutoff scores.
Proof
A matching induced by cutoffs is always fair by definition, since \(p\succ _aM(a)\) implies that a could not reach the cutoff score at p so everyone assigned to p has a higher rank than her. In the other direction let M be a fair matching. For every project p we set the cutoff d(p) to be equal to the score of the lowest ranked applicant in M(p). It is easy to see that d induces M. \(\square \)
In the following sections we will consider the strong and weak stability concepts of Kamada and Kojima [37], and subsequently introduce our new solution concept of cutoff stability.
3 Strong stability
An applicantproject pair (a, p) is a blocking pair for matching M if:

applicant a prefers project p to the project they are currently matched to: \(p \succ _a M(a)\), and

either:

p is under capacity and finds a acceptable: \(M(p)<c_p\) and \(a \succ _p \emptyset \); or

p prefers a to one of its currently matched applicants: \(\exists \ a'\in M(p)\) such that \(a \succ _p a'\)

Definition 1
(Strong Stability) We call a matching M strongly stable if for any blocking pair (a, p) for matching M, the following two conditions are satisfied:

\(a'\succ _{p}a\) for all applicants \(a'\in M(p)\)

The matching \(M'=(M\cup \{(a, p)\}) {\setminus } \{(a,M(a))\}\) is not feasible.
The first condition implies that we only allow the existence of blocking pairs involving a project p that is under its maximum capacity \(c_p\). The second condition implies that even if slot is free at project p, adding applicant a to project p will result in a distributional constraint being violated so that \(M'=(M\cup \{(a, p)\}) {\setminus } \{(a,M(a))\}\) is not feasible. Thus, we allow blocking pairs that cannot be satisfied without violating our feasibility constraint to exist in a strongly stable matching.
Alternatively, we can define strong stability in terms of fairness and strong nonwastefulness, as follows. We say that matching M is strongly nonwasteful if for every pair \((a, p)\notin M\), \(p\succ _aM(a)\) implies that \((M\cup \{(a, p)\}){\setminus } \{(a,M(a))\}\) is not feasible or project p is at capacity. It is easy to see that M is strongly stable if and only if it is fair and strongly nonwasteful.
Our first observation is that for our problem, a strongly stable matching does not exist. This follows from the observation that our model is more general than the hospital resident setting with disjoint regions [35]. We provide an adaptation of an example (Example 1) of Kamada and Kojima [37] for the sake of completeness.
Example 2
(A strongly stable matching does not necessarily exist for the summer internship problem)
Consider the following instance of the summer internship problem:
It is easy to see that \(M\le 1\). If both applicants are unmatched then \((a_1,p_1)\) forms a blocking pair. Suppose without loss of generality that \(a_1\) is matched in the feasible matching. If \(M=\{(a_1,p_1)\}\) then \((a_1,p_2)\) is a blocking pair that does not satisfy the second condition of strong stability. If \(M=\{(a_1,p_2)\}\) then \((a_2,p_2)\) is a blocking pair that does not satisfy the first condition of strong stability. Thus, every feasible matching admits a blocking pair that is not permitted under strong stability, and is therefore not strongly stable.
Below we show that the problem of deciding the existence of a strongly stable matching is NPcomplete. We reduce for Restricted MAXSMTI, the problem of deciding whether there exists a complete stable matching for the stable marriage problem with incomplete lists and ties under the restriction that the preferences of the men are strict, and the preference list of each woman is either strict or consists solely of a tie of length two [46]. A matching is said to be weakly stable if it is not blocked by a pair where both parties strictly prefer each other to their current partners.
First, we introduce an instance that will serve as the core of the construction imitating an indifferent woman.
Example 3
(An instance with two strongly stable matchings, covering different agents) The following instance is the simplest possible to demonstrate the above mentioned property.
One can easily check that there are two strongly stable matchings: \(M_1=\{(a_1,p_1)\}\) and \(M_2=\{(a_2,p_2)\}\).
Note that this example also shows that the Rural Hospitals’ Theorem of Roth [47] does not hold for strongly stable matchings, which says that always the same applicants are matched in every stable solution in a manytoone college admissions problem. Actually, the same two matchings are also the only weakly stable matchings, and thus the only cutoff stable matchings, by the same argument.
Using the example as a gadget, we will show that deciding whether an instance of our problem has a strongly stable solution is an NPcomplete problem.
Theorem 2
Checking the existence of a strongly stable matching is NPcomplete even for REG.
Proof
We describe the proof in the context of SIP, where all the supervisors have capacity one and each is responsible for at most two distinct projects. Given a solution, we can check whether it is strongly stable by considering each potential blocking pair in polynomial time, so the problem is in NP. For proving NPhardness, we reduce again from Restricted MAXSMTI problem [46] in two steps.
Part i) Suppose that we have an instance I of MAXSMTI. First we will create an instance \(I'\) of SIP, such that there is a onetoone correspondence in between the weakly stable matchings in I and the strongly (and cutoff and weakly) stable matchings in \(I'\). For I, we use the same notation of \(W=W^s\cup W^t\) to denote the set of women with strict preferences and with a single tie, respectively, and let U denote the set of men. Every man in I will be replaced by an applicant in \(I'\) with essentially the same preferences. Now, every woman in \(w_j\in W^s\) will be replaced with a single project \(p_j\) with identical preferences. For every woman \(w_j\in W^t\) we create two projects, \(p_j^1\) and \(p_j^2\) that will correspond to the projects in Example 3. If \(u_i\) and \(u_k\) were the two men in the single tie of \(w_j\) in I then let the two corresponding applicants \(a_i\) and \(a_k\) complete the instance of Example 3. So the preference lists of the projects are \(\succ _{p_j^1}: a_i\) and \(\succ _{p_j^2}: a_k\), whilst \(a_i\) has only \(p_j^1\) in her preference list, and \(a_k\) has only \(p_j^2\) in her preference list. In the first part of the reduction we show that if M is a weakly stable matching in I then there is a corresponding strongly stable matching \(M'\) in \(I'\) with the same size, and vice versa. The correspondence between the matchings is as follows.

for every \(w_j\in W^s\) and \(u_i\in U\), \((u_i,w_j)\in M \iff (a_i,p_j)\in M'\)

for every \(w_j\in W^t\) and \(u_i\in U\), where \(u_i\) is the first man in the tie of \(w_j\), \((u_i,w_j)\in M \iff (a_i,p_j^1)\in M'\)

for every \(w_j\in W^t\) and \(u_k\in U\), where \(u_k\) is the second man in the tie of \(w_j\), \((u_k,w_j)\in M \iff (a_k,p_j^2)\in M'\)
We can observe that the sizes of M and \(M'\) are the same.
Part ii) In the second part of the reduction we create an extended instance \(I''\) of SIP by adding a gadget \(G^*\), which is a copy of the unsolvable instance in Example 2 consisting of two projects \(P^*=\{p_1^*,p_2^*\}\) and two applicants \(A^*=\{a_1^*,a_2^*\}\), together with an additional project \(p^*\). Let suppose that \(p^*\) accepts one of the applicants in \(G^*\), say \(a_1^*\). Let \(p^*\) be the most preferred project by \(a_1^*\), so including this project in \(G^*\) would turn this instance solvable by assigning \(p^*\) to \(a_1^*\) and \(p_1^*\) to \(a_2^*\). To link \(G^*\cup p^*\) with the rest of \(I'\) we put all the applicants in \(I'\) ahead of \(a_1^*\) in the preference list of \(p^*\) in a random order, and we also append \(p^*\) to the end of the preference list of each applicant in \(I'\).
We will show that I has a complete weakly stable matching if and only if \(I''\) has a strongly stable matching. Let us suppose first that M is a complete stable matching of I, we construct a strongly stable matching \(M''\) of \(I''\) as follows. First we create \(M'\) in \(I'\) that covers all the applicants in \(I'\), and then we create \(M''\) by adding \(\{(a_1^*,p^*),(a_2^*,p_1^*)\}\). In the other direction, if \(M''\) is a strongly stable matching in \(I''\) then \(p^*\) must be assigned to \(a_1^*\), as otherwise gadget \(G^*\) would not admit a strongly stable solution. Hence all the applicants in \(I'\) must be matched to projects in \(I'\), since any unmatched applicant would block with \(p^*\) otherwise. Therefore, we can create the corresponding complete matching M in I that must be weakly stable due to part i) of the reduction. \(\square \)
Our hardness result is strong because it holds for a very restricted setting of Kamada and Kojima [35] that concerns disjoint regions (REG), even if at most two hospitals belong to each region.^{Footnote 6} This case can also occur when one hospital has a common upper quota for two different types of jobs, e.g., daytime and night shifts, or surgical and medical internship positions. Another motivating example is the Hungarian college admission scheme, where students can be admitted to a programme under two contracts, statefunded and privatelyfunded, and there is a common upper bound on them [17].
4 Weak stability
In view of the nonexistence and NPcompleteness of checking the existence of strongly stable matchings, one can consider a weaker stability criterion. Kamada and Kojima [37] proposed a weak stability concept for a setting that does not concern budgets but which has an abstract feasibility indicator function for any given matching. We present the definition in our terminology of applicants and projects.
Definition 2
(Weak Stability) We call a matching M weakly stable if for any blocking pair (a, p) for matching M, the following two conditions are satisfied.

\(a'\succ _{p}a\) for all applicants \(a'\in M(p)\)

generalized matching \(M\cup \{(a, p)\}\) is not feasible.
Note the similarity in the definition of strong stability and weak stability. The only difference is that in the second condition, applicant a can have two contracts: one with project M(a) and another with the project p she is blocking with. One way to see this is that in order for applicant a to block with p, it must sign the contract with p before it opts to annul its match with project M(a). We call any blocking pair that satisfies these condition permitted under weak stability.
Alternatively, we can also define weak stability in terms of fairness and weak nonwastefulness. We say that matching M is weakly nonwasteful if for every pair \((a, p)\notin M\), \(p \succ _aM(a)\) implies that \(M\cup \{(a, p)\}\) is not feasible or project p is at capacity. It is easy to see that M is weakly stable if and only if it is fair and weakly nonwasteful, for a short proof see Proposition 1 in Kamada and Kojima [37].
We say that p is unconstrained for a feasible matching M if for any \((a, p)\notin M\), \(M\cup \{(a, p)\}\) remains feasible, i.e., when one more applicant can be added to project p by keeping the solution feasible.
Proposition 2
A matching M is weakly stable if and only if it is induced by cutoff scores d such that for every unconstrained project p, \(d(p)=0\).
Proof
Following Proposition 1, we only need to show that the condition of every unconstrained project having zero cutoff is equivalent to weak nonwastefulness. Suppose first that M is weakly nonwasteful. This means that there cannot exist an unconstrained project p and an applicant a such that \(p\succ _aM(a)\), so indeed for all unconstrained project we can set the cutoff to be zero, and for the constrained projects we just set the cutoff to be equal to the score of lowest ranked applicant assigned. In the other direction, if we have cutoff scores satisfying that every unconstrained project has cutoff score zero, then there cannot exist an applicant a such that \(p\succ _aM(a)\) for an unconstrained project p. \(\square \)
In the conference version of our paper [11] we presented a polynomialtime algorithm that always returns a weakly stable matching based on the algorithm by Kamada and Kojima [37] (Appendix B.3). In this extended version of our conference paper, we strengthen this result by showing that a socalled cutoff stable matching can also be computed efficiently for SIP, by designing an algorithm that also works for HER.
5 Cutoff stability
In this section, we discuss cutoff stability that applies to any matching problem with feasibility constraints.
We say that M is cutoff nonwasteful if M is nonwasteful and for every pair \((a, p)\notin M\), \(p\succ _a M(a)\) implies that either

\((M\cup \{(a, p)\}){\setminus } \{(a,M(a))\}\) is not feasible, or

there exists another applicant \(a'\notin M(p)\), such that \(a'\succ _p a\), \(p\succ _{a'}M(a')\) and \((M\cup \{(a',p)\}){\setminus } \{(a',M(a'))\}\) is not feasible, or

project p is at capacity
We say a matching is cutoff stable if it is fair and cutoff nonwasteful. To get an intuition about the meaning of cutoff stability, and in particular about the second condition above, we first give an alternative characterization using minimal cutoff scores.
Let us define the notion of a matching induced by minimal cutoff scores, as explored by Fleiner and Jankó [23] in a model without distributional constraints, and by Biró and Kiselgof [16] for a college admission model with ties. Let \(d^{p}\) denote the cutoff scores after decreasing the cutoff of p by one, and keeping the other cutoffs the same, i.e., \(d^{p}(p)=d(p)1\), and \(d^{p}(p')=d(p')\) for every \(p'\ne p\). We say that cutoffs d are minimal if we cannot decrease the cutoff score of any project without making the induced matching infeasible. More formally for every project p, either \(d(p)=0\) or the matching induced by \(d^{p}\), which we call \(M^{d}\), is not feasible.
Proposition 3
A matching is cutoff stable if and only if it is induced by minimal cutoff scores.
Proof
Due to Proposition 1, we only need to show that the minimality of the cutoff score is equivalent to cutoff nonwastefulness. Suppose first that matching M is induced by minimal cutoff scores d, which means that for any project p, if \(d(p)>0\) then \(d^{p}\) is not feasible. Consider project p with \(d(p)>0\) and let a be the applicant who is assigned a score of \(d^{p}(p)\) by project p. \(M^{p}\), the matching induced \(d^{p}\), is not feasible which means \(M^{p}\ne M\). Therefore, \(M^{p}=(M\cup \{(a, p)\}){\setminus } \{(a,M(a))\}\) and thus \(p\succ _a M(a)\), which in turn means that a is the highest ranked applicant who forms a blocking pair with p. Thus, the induced matching M is cutoff nonwasteful. Now suppose that M is a cutoff stable matching and let d be some cutoff scores that induce M which are minimal in that sense that no smaller cutoff scores can induce M. Consider project p with \(d(p)>0\). The matching \(M^{p}\) induced by \(d^{p}\) must be different from M. Since M is cutoff nonwasteful, \(M^{p}\) must be infeasible for every project p with \(d(p)>0\) and thus cutoffs d are minimal. \(\square \)
Going back to the interpretation of the second condition in the first definition of cutoff stability, we can see that it is possible that a blocking pair (a, p) exists that would violate strong stability, but then there must be another blocking pair \((a',p)\), where \(a'\) is higher in the preference of p than a and blocking with this pair would make the matching infeasible.
We can show the natural correspondence in between the three solution concepts as follows.
Proposition 4
Every strongly stable matching is cutoff stable, and every cutoff stable matching is weakly stable.
Proof
By definition, strong nonwastefulness implies cutoff nonwastefulness, and cutoff nonwastefulness implies weak nonwastefulness. \(\square \)
Furthermore, these notions do not coincide, as illustrated in the following example.
Example 4
(An instance where the sets of strongly stable, cutoff stable and weakly stable matchings are distinct)
Due to the supervisor capacity every feasible matching has size at most two, and any matching of size fewer that two is weakly wasteful. Thus, all the relevant matchings for weak, cutoff and strong stability have size two, so we consider all the matchings of size two below. \(M_1=\{(a_1,p_1),(a_2,p_2)\}\) and \(M_2=\{(a_1,p_2),(a_2,p_1)\}\) are strongly stable.
\(M_3=\{(a_1,p_2),(a_3,p_3)\}\) is not strongly stable, due to blocking pair \((a_1,p_1)\). However, \(M_3\) is cutoff stable, since \(a_2\succ _{p_1} a_1\) and \(M_3\cup \{(a_2,p_1)\}\) is not feasible.
\(M_4=\{(a_1,p_3),(a_2,p_1)\}\) is not cutoff stable, due to blocking pairs \((a_1,p_2)\) and \((a_2,p_2)\). However, \(M_4\) is weakly stable as adding an applicant to \(a_2\) results in an infeasible matching. All other matchings of size two are not fair or not feasible.
Note that for the classical college admission model these notions are equivalent to stability as formulated by Gale and Shapley [27]. Further theoretical findings about cutoff scores for this basic model are discussed by Azevedo and Leshno [8].
5.1 Algorithm for computing a cutoff stable matching
We present an algorithm (Algorithm 1) that shows the existence of a cutoff stable matching for every instance of HER, and can compute a cutoff stable matching for an instance of SIP in strongly polynomialtime. Algorithm 1 has similarities with the algorithm proposed by Kamada and Kojima [37] which finds a weakly stable matching for HER. In contrast to the algorithm of Kamada and Kojima [37], we do not explicitly work with a set of blocking of pairs, but with cutoff scores. Furthermore, a blocking pair (a, p) was satisfied in Kamada and Kojima [37] whenever the new matching \(M\cup \{(a, p)\}\) stayed feasible, leading to a weakly stable solution in the end, whilst we satisfy a blocking pair (a, p) if \((M\cup \{(a, p)\}){\setminus } \{(a,M(a))\}\) stay feasible, leading to a cutoff stable matching. Algorithm 1 can also be viewed as a modified version of the Fleiner–Jankó scoredecreasing algorithm (see subsection 4.3 in [23]). Whereas Fleiner and Jankó [23] do not consider distributional constraints, our goal is to achieve stability properties under general heredity constraints. Finally, in independent and recent work, [39] consider a fixedpoint approach based on cutoffs. However, their heredity constraints apply to individual hospitals/projects and cannot capture SIP or regional constraints.
The idea of Algorithm 1 is simple. We start with maximum cutoffs that induce the empty matching and then we gradually decrease them until we can no longer do so without making the induced matching infeasible. Note that for the classical college admission problem this process is equivalent to the college proposing GaleShapley algorithm [27], that was observed in the Turkish college admission practice [14], and also in the Hungarian college admission scheme [16] in a more general form, as ties have been present in the rankings.
Theorem 3
A cutoff stable matching always exists for a matching problem under any set of distributional constraints that can be represented by a feasibility function, i.e., for the HER model. Algorithm 1 produces one such matching.
Proof
Note that M remains feasible and also fair during the algorithm since it is induced by cutoffs. The final matching is cutoff stable, due to Proposition 3, since the algorithm terminates when the cutoffs are minimal. \(\square \)
We remark here that the argument for Theorem 3 does not require projects to be selected in the order \(P^*\).
The next theorem shows that as long as the feasibility of a matching can be tested in polynomial time, Algorithm 1 runs in polynomial time.
Theorem 4
Suppose checking f(w) takes t time. Then, the running time of Algorithm 1 is \(O(A\cdot P^2t)\).
Proof
We decrease the cutoffs in at most \((A+1)P\) rounds. In every round we potentially need to check the matching induced by decrementing the cutoff of each of P projects. To do this for project \(p_j\), we compute matching \(M^{p_j}\) from the current matching M, by simply comparing whether the newly admitted applicant \(a_i\) with score \(d_M^{p_j}\) from \(p_j\) prefers \(p_j\) to her current match \(M(a_i)\), and if so, whether the new matching \((M\cup \{(a_i,p_j)\}){\setminus } \{(a_i,M(a_i))\}\) is feasible. By assuming that checking the feasibility of a matching takes t time, the overall run time of the algorithm is \(O(AP^2t)\). \(\square \)
We note that Algorithm 1 can be applied to the summer internship problem where the feasibility function is based on the supervisor budgets. Therefore, for the summer internship problem a cutoff stable matching exists. Furthermore, from Theorem 1 we know that the feasibility of a matching can be checked in polynomial time, and thus a cutoff stable matching can be found in polynomial time.
Additionally, using the cutoffdecreasing algorithm we can find a natural relationship between weakly stable and cutoff stable matchings.
Proposition 5
Every weakly stable matching that is not cutoff stable is Paretodominated by a cutoff stable matching for the applicants.
Proof
Let M be a weakly stable matching that is not cutoff stable. Since M is fair, it can be induced by cutoff scores, but these cutoff scores are not minimal, since M is not cutoff stable. Therefore, we can gradually decrease these cutoff scores as described in the cutoff decreasing process until we reach a cutoff stable matching \(M'\). Note that \(M'\) Paretodominates M since decreasing cutoff scores can only make the applicants better off, and every change in the matching means a strict improvement for at least one applicant. \(\square \)
Although Algorithm 1 satisfies cutoff stability, it also has some drawbacks. Next we establish some properties of the algorithm.
Theorem 5
The following properties hold for Algorithm 1.

1.
Algorithm 1 is not strategyproof for the applicants.

2.
Algorithm 1 does not always find a strongly stable matching whenever one exists.

3.
Changing the order of projects ordered after p by \(P^*\) can change p’s allocation.

4.
There exist cutoff stable matchings that cannot be produced as a result of Algorithm 1 by changing the project order \(P^*\).
Proof
We prove each of the statements separately.

1.
Consider the following instance.
If we set \(P^*=(p_1,p_2)\) then the algorithm outputs \(M=\{(a_1,p_1)\}\). However, if \(a_2\) were to modify their preferences to \(\succ _{a_2}:p_2,p_1\), then the algorithm will output \(M=\{(a_2,p_2)\}\), which is preferred by \(a_2\). Thus, the algorithm is not strategyproof for applicants.

2.
For the example above, since \((a_2,p_2)\) is the only strongly stable matching, the algorithm does not find the strongly stable matching when one exists.

3.
Consider Example 3. We can obtain each of the two cutoff stable matchings by changing the order the two projects are processed.

4.
This is a consequence of the projectproposing nature of the algorithm. Consider the following instance.
Both possible project orders produce \(M=\{(a_2,p_1),(a_1,p_2)\}\), but the matching \(M=\{(a_1,p_1),(a_2,p_2)\}\) is also cutoff stable. Thus, there exist weakly stable matchings that cannot be produced by the algorithm.
This completes the proof. \(\square \)
Since the examples above contain exactly one supervisor, Theorem 5 also applies to REG and any setting that contains REG as a special case.
5.2 Finding a maximum size cutoff stable matching
First, we observe that this problem is NPhard for the REG model by part i) of the proof of Theorem 2. Then we provide a mixed integer linear programming formulation.
Theorem 6
Finding a maximum size cutoff/weakly stable matching is NPhard for REG, even if every hospital has one seat and each region has at most two hospitals.
Proof
Part i) of the proof of Theorem 2 established a onetoone correspondence in between the weakly stable matchings of an SMTI instance I and the strongly/cutoff/weakly stable matchings of a SIP instance \(I'\), where the sizes of the corresponding matchings are the same. Since finding a maximum size weakly stable matching for SMTI is NPhard, so is the problem of finding a maximum size cutoff/weakly stable matching for SIP. Note that in this construction each project has capacity one and every supervisor has at most two distinct projects, which corresponds to the case of REG, where each hospital has capacity one and every region has at most two hospitals. \(\square \)
Furthermore, since MAXSMTI is not approximable within a factor of 21/19 unless \(P=NP\) [32], due to the onetoone relation of the sizes of the matchings in our reduction, the same inapproximability result applies for our problems as well.
5.3 MILPformulation
The mixed integer linear program will have three main parts. The first set of constraints describes the feasibility of the funding allocation. The matching in between applicants and projects are described with (0–1) binary variables defined for every mutually acceptable applicantproject pair, as follows, let \(y_{a, p}=1\iff (a, p)\in M\). Let \(x_{s,p}\) be a nonnegative continuous variable denoting the funding for project p provided by supervisor s. The following three sets of conditions ensure the feasibility of the solution.
The second set of constraints will describe the fairness of the matching by means of cutoff scores. Let \(z_{a, p}\) be the score of applicant a at project p, a given integer constant in the interval [0, A]. For every project p, let d(p) be an integer variable in the range of \([0,A+1]\) denoting the cutoff of project p. We can link the cutoff scores with the induced matching M by the following set of constraints, as also used in Ágoston [4] and Delorme et al. [21].
The above constraint enforces that if an applicant is assigned to a project then she reached the cutoff there.
This constraint implies that if the applicant is rejected from project p, so she is not admitted there and nor to any better project of her preference (in which case the sum term is zero on the right hand side), then the cutoff must be higher at p than her score there. These two sets of conditions together imply that every student is admitted to her most preferred project where she achieved the cutoff, which means that we get the matching induced by the cutoffs.
For ensuring cutoff stability we have to provide cutoff nonwastefulness by enforcing that the cutoffs are minimal. This can be achieved by simply minimizing the sum of the cutoff scores in the objective function of the MILP. To see this we just need to observe that if the matching would not be cutoff minimal, so we could decrease some of the cutoffs by keeping the solution feasible, but then the solution was not optimal with respect to this objective function.
Finally, if we want to maximize the number of applicants matched in the cutoff stable solution then we can use the following combined objective function, where W is a large enough constant, namely, \(W > (A+1)\cdot P\).
This completes the MILP for finding a maximum size cutoff stable matching.
6 Egalitarian budget allocations
Given a feasible matching M, there can exist multiple ways to allocate supervisor budgets among projects to fund all applicants matched to them. Our goal is to find a method for the fairest such allocation. We have chosen to deal with fairness postmatch, in order to find a solution that does not constrain the set of feasible matchings.
For each supervisor \(s\in S\), and each project they supervise \(p\in P_s\) set \(0< t_{s,p}\le 1\) such that, for each \(p\in P\), \(\sum _{s\in S_p} t_{s,p}=1\). Call this the normative ‘target’  how much we would want s to contribute to the funding of any applicant matched to p. For instance, if we would ideally want the supervisors of any given project to contribute equally to its funding, we would set: \(t_{s,p}=\frac{M(p)}{S_p} \ \ \forall p\in P, s\in S_p.\) However, given a feasible matching M, some supervisors may lack sufficient funding to reach these targets. Therefore, we seek to find a funding allocation that is closest to the target allocations. In order to define closest, we consider specific lexicographic comparisons. Let \(X=\{x_{s,p}\}_{s\in S,p\in P_s}\) be a feasible funding allocation for matching M. Denote by \(\phi _X\) the vector corresponding to the weakly decreasing ordering of the set: \(\Big \{\frac{x_{s,p}}{t_{s,p}}\Big \}_{s\in S,p\in P_s}.\) Denote by \(\varPhi _M\) the set of such vectors corresponding to all feasible funding allocations for matching M.
The egalitarian feasible funding allocation for matching M is the funding allocation corresponding to \(\phi ^* \in \varPhi _M\) such that for all \(\phi \in \varPhi _M\), \(\phi ^*\prec _{lex} \phi \), where \(\prec _{lex}\) refers to the wellknown lexicographic order. This is exactly equivalent to finding the leximin optimum of the following set: \(\Big \{\frac{x_{s,p}}{t_{s,p}}\Big \}_{s\in S,p\in P_s}.\) The egalitarian feasible funding allocation can be achieved via Algorithm 2, which runs a series of linear programs.
Theorem 7
Given a feasible matching, the egalitarian funding allocation can be computed in time \(O(S^2P^2)LP(O(S\cdot P))\), where LP refers to the running time of the linear programming algorithm used.
Proof
The algorithm solves a series of linear programs with \(O(S\cdot P)\) constraints. The while loop iterates at most \(M\le SP\) times, as the algorithm adds at least one element to the set \(M_{tight}\) in every iteration. The for loop also iterates at most M times. Thus, the overall complexity is \(O(S^2P^2)LP(O(S\cdot P))\), where LP refers to the running time of the linear programming algorithm used. Finding a feasible funding allocation for matching M at the beginning of the algorithm does not increase its complexity. Since linear programs can be solved in polynomial time, the egalitarian funding allocation can also be computed in polynomial time. It is possible that this could be improved to strongly polynomial time by reducing to the fair integral flow problem [26]. We leave the existence of such a reduction as an open question. \(\square \)
7 Conclusion
We presented a novel matching model that captures many realworld scenarios. For the model, we presented a compelling solution that is polynomialtime and satisfies stability and fairness properties. Our central algorithm computes a cutoff stable matching even for the general matching model with heredity constraints. Hence, it applies to many applications such as refugee matching that involve heredity constraints.
Several directions and problems arise as a result of our study. Our approach to finding a fair budget allocation was to first compute a cutoff stable matching and then find an egalitarian budget allocation. It will be interesting to explore a fair outcome that is fairest in some global sense across all weakly stable matchings. It had been open whether there exists an algorithm that is strategyproof and satisfies weak stability or cutoff stability. Just recently, Cho et al. [19] proved that strategyproofness and weak stability are incompatible in general. Understanding the conditions under which strategyproofness and stability concepts are compatible remains an interesting direction.
Notes
For an overview of reallife matching markets, please see http://www.matchinginpractice.eu, and a recent survey [15].
See, for example, the discussion by Kamada and Kojima [39] who point out that even intraproject heredity constraints capture problems including college admissions with students with disabilities; refugee match, and daycare allocation.
For an overview of network flows, see Ahuja et al. [6].
In the school choice literature this property is also called as justified envyfreeness, see, e.g., Abdulkadiroğlu and Sönmez [2].
Most of the computational hardness results for distributional constraints concern overlapping regions (see, e.g. Goto et al. [29]).
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Acknowledgements
Aziz gratefully acknowledges support from Defence Science and Technology (DST). Baychkov’s work was supported by the CSIRO undergraduate vacation scholarship program, and he extends many thanks to Haris Aziz and Gavin Walker for their invaluable mentorship. Biró is supported by the Hungarian Academy of Sciences, Momentum Grant No. LP20212, and the Hungarian Scientific Research Fund, OTKA, Grant No. K128611.
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This paper is the extension of our conference paper [11]. The main addition is Sect. 5 on our new solution concept of cutoff stability and an algorithm to achieve it.
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Aziz, H., Baychkov, A. & Biró, P. Cutoff stability under distributional constraints with an application to summer internship matching. Math. Program. (2022). https://doi.org/10.1007/s10107022019171
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DOI: https://doi.org/10.1007/s10107022019171
Keywords
 Stable matching
 Distributional constraints
 Cutoff scores
 NPhardness
 Integer programming
Mathematics Subject Classification
 05C70
 91B68
 90C11
 91A12
 05C85