Maximally Satisfying Lower Quotas
in the Hospitals/Residents Problem with Ties^††thanks: This work was partially supported by the joint project of Kyoto University and Toyota Motor Corporation, titled “Advanced Mathematical Science for Mobility Society”.

The Hospitals/Residents model (HR), a many-to-one matching model, has been extensively studied since the seminal work of Gale and Shapley [13]. Its input consists of a set of residents and a set of hospitals. Each resident has a preference over hospitals; similarly, each hospital has a preference over residents. In addition, each hospital is associated with a positive integer called the upper quota, which specifies the maximum number of residents it can accept. In this model, stability is the central solution concept, which requires the nonexistence of a blocking pair, i.e., a resident–hospital pair that has an incentive to deviate jointly from the current matching. In the basic model, each agent (resident or hospital) is assumed to have a strict preference for possible partners. For this model, the resident-oriented Gale–Shapley algorithm (also known as the deferred acceptance mechanism) is known to find a stable matching. This algorithm has advantages from both computational and strategic viewpoints: it runs in linear time and is strategy-proof for residents.

In reality, people typically have indifference among possible partners. Accordingly, a stable matching model that allows ties in preference lists, denoted by HRT in the context of HR, was introduced [21]. For such a model, several definitions of stability are possible. Among them, weak stability provides a natural concept, in which agents have no incentive to move within the ties. It is known that if we break the ties of an instance $I$ arbitrarily, any stable matching of the resultant instance is a weakly stable matching of $I$. Hence, the Gale–Shapley algorithm can still be used to obtain a weakly stable matching. In applications, typically, ties are broken randomly, or participants are forced to report strict preferences even if their true preferences have ties. Hereafter, “stability” in the presence of ties refers to “weak stability,” unless stated otherwise.

It is commonly known that HR plays an important role not only in theory but also in practice; for example, in assigning students to high schools [1, 2] and residents to hospitals [31]. In such applications, “imbalance” is one of the major problems. For example, hospitals in urban areas are generally more popular than those in rural areas; hence it is likely that the former are well-staffed whereas the latter suffer from a shortage of doctors. One possible solution to this problem is to introduce a lower quota of each hospital, which specifies the minimum number of residents required by a hospital, and obtain a stable matching that satisfies both the upper and lower quotas. However, such a matching may not exist in general [17, 29], and determining if such a stable matching exists in HRT is known to be NP-complete (which is an immediate consequence from page 276 of [30]).

In general, it is too pessimistic to assume that a shortage of residents would force hospitals to go out of operation. In some cases, the hospital simply has to reduce its service level according to how much its lower quota is satisfied. In this scenario, a hospital will wish to satisfy the lower quota as much as possible, if not completely. To formulate this situation, we introduce the following optimization problem, which we call HRT to Maximally Satisfy Lower Quotas (HRT-MSLQ). Specifically, let $R$ and $H$ be the sets of residents and hospitals, respectively. All members in $R$ and $H$ have complete preference lists that may contain ties. Each hospital $h$ has an upper quota $u(h)$, the maximum number of residents it can accept. The stability of a matching is defined with respect to these preference lists and upper quotas, as in conventional HRT. In addition, each hospital $h$ is associated with a lower quota $\ell(h)$, which specifies the minimum number of residents required to keep its service level. We assume that $\ell(h)\leq u(h)\leq|R|$ for each $h\in H$. For a stable matching $M$, let $M(h)$ be the set of residents assigned to $h$. The satisfaction ratio, $s_{M}(h)$, of hospital $h\in H$ (with respect to $\ell(h)$) is defined as $s_{M}(h)=\min\left\{1,\frac{|M(h)|}{\ell(h)}\right\}$. Here, we let $s_{M}(h)=1$ if $\ell(h)=0$, because the lower quota is automatically satisfied in this case. The satisfaction ratio reflects a situation in which hospital $h$’s service level increases linearly with respect to the number of residents up to $\ell(h)$ but does not increase after that, even though $h$ is still willing to accept $u(h)-\ell(h)$ more residents. These $u(h)-\ell(h)$ positions may be considered as “marginal seats,” which do not affect the service level but provide hospitals with advantages, such as generous work shifts. Our HRT-MSLQ problem asks us to maximize the total satisfaction ratio over the family $\mathcal{M}$ of all stable matchings in the problem instance, i.e.,

The following are some remarks on our problem: (1) To our best knowledge, almost all previous works on lower quotas have investigated cases with no ties and have assumed lower quotas to be hard constraints. Refer to the discussion at the end of this section. (2) Our assumption that all preference lists are complete is theoretically a fundamental scenario used to study the satisfaction ratio for lower quotas. Moreover, there exist several cases in which this assumption is valid [5, 15]. For example, according to Goto et al. [15], a complete list assumption is common in student–laboratory assignment in engineering departments of Japanese universities because it is mandatory that every student be assigned. (3) If preference lists contain no ties, the satisfaction ratio $s_{M}(h)$ is identical for any stable matching $M$ because of the rural hospitals theorem [14, 31, 32]. Hence, there is no chance for algorithms to come into play if the stability is not relaxed. In our setting (i.e., with ties), the rural hospitals theorem implies that our task is essentially to find an optimal tie-breaking. However, it is unclear how to find such a tie-breaking. (4) Alternative objective functions may be considered to reflect our objective of satisfying the lower quotas. In Appendix E, we introduce three such natural objective functions and briefly discuss their behaviors.

Subsequently, we consider our problem algorithmically. Note that the aforementioned maximum gap corresponds to the worst-case approximation factor of the arbitrarily tie-breaking Gale–Shapley algorithm, which is frequently used in practice; this algorithm first breaks ties in the preference lists of agents arbitrarily and then applies the Gale–Shapley algorithm on the resulting preference lists. This correspondence easily follows from the rural hospitals theorem, as explained in Proposition 20 in Appendix C.

In this paper, we show that there are two types of difficulties inherent in our problem HRT-MSLQ for all scenarios except (iv). Even for scenarios (i)–(iii), we show that (1) the problem is NP-hard and that (2) there is no algorithm that is strategy-proof for residents and always returns an optimal solution; see Section 6 and Appendix A.1.

We then consider strategy-proof approximation algorithms. We propose a strategy-proof algorithm Double Proposal, which is applicable in all above possible scenarios, whose approximation factor is substantially better than that of the arbitrary tie-breaking method. The approximation factors are listed in the second row of Table 1, where $\phi$ is a function defined by $\phi(1)=1$, $\phi(2)=\frac{3}{2}$, and $\phi(n)=n(1+\lfloor\frac{n}{2}\rfloor)/(n+\lfloor\frac{n}{2}\rfloor)$ for any $n\geq 3$. Note that $\frac{\theta^{2}+\theta-1}{2\theta-1}<\theta$ holds whenever $\theta>1$.

The main difficulty in our problem originates from the complicated form of our objective function $s(M)=\sum_{h\in H}\min\{1,\frac{|M(h)|}{\ell(h)}\}$. In particular, non-linearity of $s(M)$ makes the analysis of the approximation factor of Double Proposal considerably hard. We therefore introduce some new ideas and techniques to analyze the maximum gap $\Lambda$ and approximation factor of our algorithm, which is one of the main novelties of this paper.

To estimate the approximation factor of the algorithm, we need to compare objective values of a stable matching $M$ output by the algorithm and an (unknown) optimal stable matching $N$. A typical technique used to compare two matchings is to consider a graph of their union. In the marriage model, the connected components of the union are paths and cycles, both of which are easy to analyze; however, this is not the case in a general many-to-one matching model. For some problems, this approach still works via “cloning,” which transforms an instance of HR into that of the marriage model by replacing each hospital $h$ with an upper quota of $u(h)$ by $u(h)$ hospitals with an upper quota of $1$. Unfortunately, however, in HRT-MSLQ there seems to be no simple way to transform the general model into the marriage model because of the non-linearity of the objective function.

In our analysis of the uniform model, the union graph of $M$ and $N$ may have a complex structure. We categorize hospitals using a procedure like breadth-first search starting from the set of hospitals $h$ with the satisfaction ratio $s_{N}(h)$ larger than $s_{M}(h)$, which allows us to provide a tight bound on the approximation factor. For the general model, instead of using the union graph, we define two vectors that distribute the values $s(M)$ and $s(N)$ to the residents. By making use of the local optimality of $M$ proven in Section 3, we compare such two vectors and give a tight bound on the approximation factor.

We finally remark that the improvement of Double Proposal over the maximum gap shows that our problem exhibits a different phenomenon from that of MAX-SMTI because the approximation factor of MAX-SMTI cannot be improved from a naive tie-breaking method if strategy-proofness is imposed [18].

Let $R=\{r_{1},r_{2},\ldots,r_{n}\}$ be a set of residents and $H=\{h_{1},h_{2},\ldots,h_{m}\}$ be a set of hospitals. Each hospital $h$ has a lower quota $\ell(h)$ and an upper quota $u(h)$ such that $\ell(h)\leq u(h)\leq n$. We sometimes denote a hospital $h$’s quota pair as $[\ell(h),u(h)]$ for simplicity. Each resident has a preference list over hospitals, which is complete and may contain ties. If a resident $r$ prefers a hospital $h_{i}$ to $h_{j}$, we write $h_{i}\succ_{r}h_{j}$. If $r$ is indifferent between $h_{i}$ and $h_{j}$ (including the case that $h_{i}=h_{j}$), we write $h_{i}=_{r}h_{j}$. We use the notation $h_{i}\succeq_{r}h_{j}$ to signify that $h_{i}\succ_{r}h_{j}$ or $h_{i}=_{r}h_{j}$ holds. Similarly, each hospital has a preference list over residents and the same notations as above are used. In this paper, a preference list is denoted by one row, from left to right according to the preference order. When two or more agents are of equal preference, they are enclosed in parentheses. For example, “$r_{1}$:  $h_{3}$   (  $h_{2}$  $h_{4}$  )  $h_{1}$” is a preference list of resident $r_{1}$ such that $h_{3}$ is the top choice, $h_{2}$ and $h_{4}$ are the second choice with equal preference, and $h_{1}$ is the last choice.

An assignment is a subset of $R\times H$. For an assignment $M$ and a resident $r$, let $M(r)$ be the set of hospitals $h$ such that $(r,h)\in M$. Similarly, for a hospital $h$, let $M(h)$ be the set of residents $r$ such that $(r,h)\in M$. An assignment $M$ is called a matching if $|M(r)|\leq 1$ for each resident $r$ and $|M(h)|\leq u(h)$ for each hospital $h$. For a matching $M$, a resident $r$ is called matched if $|M(r)|=1$ and unmatched otherwise. If $(r,h)\in M$, we say that $r$ is assigned to $h$ and $h$ is assigned $r$. We sometimes abuse notation $M(r)$ to denote the unique hospital where $r$ is assigned. A hospital $h$ is called deficient or sufficient if $|M(h)|<\ell(h)$ or $\ell(h)\leq|M(h)|\leq u(h)$, respectively. Additionally, a hospital $h$ is called full if $|M(h)|=u(h)$ and undersubscribed otherwise.

A resident–hospital pair $(r,h)$ is called a blocking pair for a matching $M$ (or we say that $(r,h)$ blocks $M$) if (i) $r$ is either unmatched in $M$ or prefers $h$ to $M(r)$ and (ii) $h$ is either undersubscribed in $M$ or prefers $r$ to at least one resident in $M(h)$. A matching is called stable if it admits no blocking pair. Recall that the satisfaction ratio of a hospital $h$ (which is also called the score of $h$) in a matching $M$ is defined by $s_{M}(h)=\min\{1,\frac{|M(h)|}{\ell(h)}\}$, where we define $s_{M}(h)=1$ if $\ell(h)=0$. The total satisfaction ratio (also called the score) of a matching $M$, is the sum of the scores of all hospitals, that is, $s(M)=\sum_{h\in H}s_{M}(h)$. The Hospitals/Residents problem with Ties to Maximally Satisfy Lower Quotas, denoted by HRT-MSLQ, is to find a stable matching $M$ that maximizes the score $s(M)$. The optimal score of an instance $I$ is denoted by ${\rm OPT}(I)$.

Note that if $|R|\geq\sum_{h\in H}u(h)$, then all hospitals are full in any stable matching (recall that preference lists are complete). Hence, all stable matchings have the same score $|H|$, and the problem is trivial. Therefore, throughout this paper, we assume $|R|<\sum_{h\in H}u(h)$. In this setting, all residents are matched in any stable matching as an unmatched resident forms a blocking pair with an undersubscribed hospital.

In this section, we present our algorithm Double Proposal for HRT-MSLQ along with a few of its basic properties. Its strategy-proofness and approximation factors for several models are presented in the following sections.

Our proposed algorithm Double Proposal is based on the resident-oriented Gale–Shapley algorithm but allows each resident $r$ to make proposals twice to each hospital. Here, we explain the ideas underlying this modification.

Let us apply the ordinary resident-oriented Gale–Shapley algorithm to HRT-MSLQ, which starts with an empty matching $M\coloneqq\emptyset$ and repeatedly updates $M$ by a proposal-acceptance/rejection process. In each iteration, the algorithm takes a currently unassigned resident $r$ and lets her propose to the hospital at the top of her current list. If the preference list of resident $r$ contains ties, the proposal order of $r$ depends on how to break the ties in her list. Hence, we need to define a priority rule for hospitals that are in a tie. Recall that our objective function is given by $s(M)=\sum_{h\in H}\min\{1,\frac{|M(h)|}{\ell(h)}\}$. This value immediately increases by $\frac{1}{\ell(h)}$ if $r$ proposes to a deficient hospital $h$, whereas it does not increase if $r$ proposes to a sufficient hospital $h^{\prime}$, although the latter may cause a rejection of some resident if $h^{\prime}$ is full. Therefore, a naive greedy approach is to let $r$ first prioritize deficient hospitals over sufficient hospitals and then prioritize those with small lower quotas among deficient hospitals. This approach is useful for attaining a larger objective value for some instances; however, it is not enough to improve the approximation factor in the sense of worst case analysis, as a deficient hospital $h$ in some iteration might become sufficient later and it might be better if $r$ had made a proposal to a hospital other than $h$ in the tie. Furthermore, this naive approach sacrifices strategy-proofness as demonstrated in Appendix A.2. This failure of strategy-proofness follows from the adaptivity of this tie-breaking rule, in the sense that the proposal order of each resident is affected by the other residents’ behaviors.

In our algorithm Double Proposal, each resident can propose twice to each hospital. If the head of $r$’s preference list is a tie when $r$ makes a proposal, then the hospitals to which $r$ has not yet proposed are prioritized. This idea was inspired by an algorithm of [18]. Recall that each hospital $h$ has an upper quota $u(h)$ and a lower quota $\ell(h)$. In our algorithm, we use $\ell(h)$ as a dummy upper quota. Whenever $|M(h)|<\ell(h)$, a hospital $h$ accepts any proposal. If $h$ receives a new proposal from $r$ when $|M(h)|\geq\ell(h)$, then $h$ checks whether there is a resident in $M(h)\cup\{r\}$ who has not been rejected by $h$ so far. If such a resident exists, $h$ rejects that resident regardless of the preference of $h$. Otherwise, we apply the usual acceptance/rejection operation, i.e., $h$ accepts $r$ if $|M(h)|<u(h)$ and otherwise replaces $r$ with the worst resident $r^{\prime}$ in $M(h)$. Roughly speaking, the first proposals are used to implement priority on deficient hospitals, and the second proposals are used to guarantee stability.

Formally, our algorithm Double Proposal is described in Algorithm 1. For convenience, in the preference list, a hospital $h$ that is not included in any tie is regarded as a tie consisting of $h$ only. We say that a resident is rejected by a hospital $h$ if she is chosen as $r^{\prime}$ in Lines 12 or 17. To argue strategy-proofness, we need to make the algorithm deterministic. To this end, we remove arbitrariness using indices of agents as follows. If there are multiple hospitals (resp., residents) satisfying the condition to be chosen at Lines 5 or 7 (resp., at Lines 12 or 17), take the one with the smallest index (resp., with the largest index). Furthermore, when there are multiple unmatched residents at Line 3, take the one with the smallest index. In this paper, Double Proposal always refers to this deterministic version.

In addition to stability, the output of Double Proposal satisfies the following property, which plays a key role in the analysis of the approximation factors in Section 5.

Lemma 2 states some local optimality of Double Proposal. Suppose that we reassign $r$ from $h$ to $h^{\prime}$. Then, $h$ may lose and $h^{\prime}$ may gain score, but Lemma 2 says that the objective value does not increase. To see this, note that if the objective value were to increase, $h^{\prime}$ must gain score and $h$ would either not lose score or lose less score than $h^{\prime}$ would gain. The former and the latter are the “if” parts of (ii) and (i), respectively, and in either case the conclusion $|M(h^{\prime})|\geq\ell(h^{\prime})$ implies that $h^{\prime}$ cannot gain score by accepting one more resident.

An algorithm is called strategy-proof for residents if it gives residents no incentive to misrepresent their preferences. The precise definition follows. An algorithm that always outputs a matching deterministically can be regarded as a mapping from instances of HRT-MSLQ into matchings. Let $A$ be an algorithm. We denote by $A(I)$ the matching returned by $A$ for an instance $I$. For any instance $I$, let $r\in R$ be any resident, who has a preference $\succeq_{r}$. Additionally, let $I^{\prime}$ be an instance of HRT-MSLQ which is obtained from $I$ by replacing $\succeq_{r}$ with some other $\succeq^{\prime}_{r}$. Furthermore, let $M\coloneqq A(I)$ and $M^{\prime}\coloneqq A(I^{\prime})$. Then, $A$ is strategy-proof if $M(r)\succeq_{r}M^{\prime}(r)$ holds regardless of the choices of $I$, $r$, and $\succeq^{\prime}_{r}$.

In the setting without ties, it is known that the resident-oriented Gale–Shapley algorithm is strategy-proof for residents (even if preference lists are incomplete) [9, 16, 33]. Furthermore, it has been proved that no algorithm can be strategy-proof for both residents and hospitals [33]. As in many existing papers on two-sided matching, we use the term “strategy-proofness” to refer to strategy-proofness for residents.

Before proving the strategy-proofness of Double Proposal, we remark that the exact optimization and strategy-proofness are incompatible even if a computational issue is set aside. The following fact is demonstrated in Appendix A.1.

This proposition implies that, if we require strategy-proofness for an algorithm, then we should consider approximation even in the absence of computational constraints. Now, we show the strategy-proofness of our approximation algorithm.

In this section, we analyze the approximation factors of our algorithm, together with the maximum gaps $\Lambda$ for the four models mentioned in Section 1. All results in this section are summarized in the first and second rows of Table 1 in Section 1. Most of the proofs are deferred to Appendix C, which gives the full version of this section.

For an instance $I$ of HRT-MSLQ, let ${\rm OPT}(I)$ and ${\rm WST}(I)$ respectively denote the maximum and minimum scores over all stable matchings of $I$, and let ${\rm ALG}(I)$ be the score of the output of our algorithm Double Proposal. Proposition 20 in Appendix C shows that ${\rm WST}(I)$ can be the score of the output of the algorithm that first breaks ties arbitrarily and then applies the Gale–Shapley algorithm for the resultant instance . Therefore, the maximum gap is equivalent to the approximation factor of such arbitrary tie-breaking GS algorithm.

For a model $\cal I$ (i.e., subfamily of problem instances of HRT-MSLQ), let

In subsequent subsections, we provide exact values of $\Lambda({\cal I})$ and ${\rm APPROX}({\cal I})$ for the four fundamental models. Recall our assumptions that preference lists are complete, $|R|<\sum_{h\in H}u(h)$, and $\ell(h)\leq u(h)\leq n$ for each $h\in H$.