Socially stable matchings in the
Hospitals/Residents problem

Matching problems generally involve the assignment of a set (or sets) of agents to one another. Agents may be required to list other agents they find acceptable in order of preference, either explicitly or implicitly through a list of desirable characteristics. Agents may also be subject to capacity constraints, indicating the maximum number of assignments they are allowed to be involved in.

An example of such a matching problem that has received much attention in literature is the Hospitals/Residents problem (HR) [11, 13, 28, 24]. An HR instance consists of a set of residents seeking to be matched to a set of hospitals. Each resident ranks a subset of the hospitals in strict order of preference, and vice versa. Further, each resident forms an acceptable pair with every hospital on his preference list. Finally, each hospital has a capacity, indicating the maximum number of residents that it can be assigned. A matching is a set assignments among acceptable pairs such that no resident is assigned to more than one hospital, and no hospital exceeds its capacity. An acceptable pair forms a blocking pair with respect to a matching, or blocks a matching, if both agents would rather be assigned to each other than remain with their assignees (if any) in the matching. A matching is stable if it admits no blocking pair. HR has a wide range of applications including traditional markets like the assignment of graduating medical students (residents) to hospitals [18, 27] and students to high schools [1, 2], and online markets like oDesk (an online labour market), AirBnB (an online short-term housing rental market), and Match.com/OkCupid/etc. (online dating markets). In applications such as these, it has been convincingly argued that stability is a desirable property of a matching [27].

Although the concept of stability is important in many applications of matching problems, there are classes of matching problems (such as the Stable Roommates problem) for which an instance is not guaranteed to admit a stable matching [11]. Moreover, enforcing the stability requirement tends to reduce the size of matchings discovered [6]. This is an issue particularly in the case of applications where it is desirable to find the largest possible matching. Also, it is generally assumed that a resident-hospital pair that blocks a matching in theory will also block the matching in practice. However this assumption is not always true in some real-life applications, as resident-hospital pairs are more likely to form blocking pairs in practice if social ties exist between them. These factors have motivated studies into alternative, weaker stability definitions that still aim to prevent a given matching from being subverted in practice while increasing the number of agents involved in the matchings.

Arcaute and Vassilvitskii [3] described the Hospitals/Residents problem in the context of assigning job applicants to company positions. They observed that applicants are more likely to be employed by a company if they are recommended by their friends who are already employees of that company. In their model, an applicant-company pair $(a,c)$ may block a matching $M$ if $(a,c)$ blocks it in the traditional sense (as described in the analogous HR context) and $a$ is friends with another applicant $a^{\prime}$ assigned to $c$ in $M$. Thus their problem incorporates both the traditional HR problem and additionally an underlying social network, represented as an undirected graph consisting of applicants as nodes and edges between nodes where the corresponding applicants have some social ties (e.g., are friends). Matchings that admit no blocking pair in this context are called locally stable due to the addition of the informational constraint on blocking pairs. Cheng and McDermid [8] investigated the problem (which they called HR+SN) further and established various algorithmic properties and complexity results. They showed that locally stable matchings can be of different sizes and the problem of finding a maximum locally stable matching is NP-hard. They identified special cases where the problem is polynomially solvable and gave upper and lower bounds on the approximability of the problem.

While the HR+SN model is quite natural in the job market, it makes an assumption that the employed applicant $a^{\prime}$ will always be willing to make a recommendation. This however may not be the case as a recommendation may in practice lead to $a^{\prime}$ being rejected by his assigned company. Ultimately this may lead to a reassignment for $a^{\prime}$ to a worse company or indeed $a^{\prime}$ may end up unmatched. While it is true that a scenario may arise where these social ties between applicants may lead to a blocking pair of a matching, it is arguably equally likely that social ties between an applicant and the company itself will exist. That is, an applicant need not know another applicant who was employed by the company in order to block a matching; it is enough for him to know any employee in the company (for example the Head of Human Resources). Such a model could also be natural in many applications both within and beyond the job market context.

Additionally, many matching markets are cleared by a centralised clearinghouse. While more traditional markets require agents to explicitly list potential matches, many online markets ask agents to list desirable characteristics and then use software to infer the preference lists of the agents. In these markets, communication between agents is facilitated by the centralised clearinghouse. Some agent pairs in the market may have social ties outside the clearinghouse. Often these social ties are due to past interactions within the marketplace and so the clearinghouse is aware of them. These pairs can communicate outside the clearinghouse and might block proposed matchings. Most pairs, however, only become aware of each other when the clearinghouse proposes them as a match. Thus even if they prefer each other to their assigned matches, they will not be able to discover each other and deviate from the matching.

Based on these ideas, we present a variant of HR called the Hospitals / Residents problem under Social Stability (HRSS). In this model, which we describe in the context of assigning graduating medical residents to hospital positions, we assume that a resident-hospital pair will only form a blocking pair in practice if there exists some social relationship between them. Two agents that have such a social relationship are called an acquainted pair, and this is represented by an edge in a social network graph. We call a pair of agents that do not have such a social relationship an unacquainted pair. Such a pair may be part of a matching $M$ (given that $M$ is typically constructed by a trusted third party, i.e., a centralised clearinghouse) but cannot form a blocking pair with respect to $M$. As a consequence, although a resident-hospital pair may form a blocking pair in the classical sense, if they are an unacquainted pair, they will not form a blocking pair in the HRSS context. A matching that admits no blocking pair in this new context is said to be socially stable. We denote the one-to-one restriction of HRSS as the Stable Marriage problem with Incomplete lists under Social Stability (SMISS).

Hoefer and Wagner [15, 16] studied a problem that generalises both HR+SN and HRSS. In their model, the social network graph involves all agents and need not be bipartite. A pair locally blocks a given matching $M$ if (i) it blocks in the classical sense, and (ii) the agents involved are at most $l$ edges apart in the social network graph augmented by $M$. This scenario can be viewed as a generalisation of the HR+SN ($l=2$) and HRSS ($l=1$) models. They studied the convergence time for better-response dynamics that converge to locally stable matchings, and also established a lower bound for the approximabiliy of the problem of finding a maximum locally stable matching (for the case that $l\leq 2$).

Locally stable matchings have also been investigated in the context of the Stable Roommates problem (a non-bipartite generalisation of the Stable Marriage problem) in [7]. Here, the Stable Roommates problem with Free edges (SRF) as introduced was motivated by the observation that, in kidney exchange matching schemes, donors and recipients do not always have full information about others and are more likely to have information only on others in the same transplant centre as them. The problem is defined by the traditional Stable Roommates problem together with a set of free edges. These correspond to pairs of agents in different transplant centres that do not share preference information; such pairs may be involved in stable matchings, but cannot block any matching. It is shown in [7] that the problem of determining whether a stable matching exists, given an SRF instance, is NP-complete.

In this paper, we present some algorithmic results for the HRSS model described above. In Section 2, we present some preliminary definitions and observations. We give a reduction from the HRSS to the HR+SN problem in Section 3. In Section 4 we show that MAX HRSS, the problem of finding a maximum socially stable matching given an HRSS instance $I$, is NP-hard even under certain restrictions on the lengths of the preference lists. This result holds even if $I$ is an instance of SMISS. Then in Section 5, we consider the approximability of MAX HRSS. We give a $3/2$-approximation algorithm for the problem, and also show that it is not approximable within $21/19-\varepsilon$, for any $\varepsilon>0$, unless P=NP, and not approximable within $3/2-\varepsilon$, for any $\varepsilon>0$, assuming the Unique Games Conjecture. in Section 6 we present polynomial-time algorithms for three special cases of MAX HRSS, including the cases where (i) each resident’s list is of length at most 2 and each hospital has capacity 1, (ii) the number of unacquainted pairs is constant, and (iii) the number of acquainted pairs is constant. Finally some open problems are given in Section 7.

An instance $I$ of the Hospitals/Residents problem (HR), as defined in [11], contains a set $R=\{r_{1},r_{2},...,r_{n_{1}}\}$ of residents, a set $H=\{h_{1},h_{2},...,h_{n_{2}}\}$ of hospitals. Each resident $r_{i}\in R$ ranks a subset of $H$ in strict order of preference; each hospital $h_{j}\in H$ ranks a subset of $R$, consisting of those residents who ranked $h_{j}$, in strict order of preference. Each hospital $h_{j}$ also has a capacity $c_{j}\in\mathbb{Z}^{+}$ indicating the maximum number of residents that can be assigned to it. A pair $(r_{i},h_{j})$ is called an acceptable pair if $h_{j}$ appears in $r_{i}$’s preference list. We denote by $\mathcal{A}$ the set of all acceptable pairs. A matching $M$ is a set of acceptable pairs such that each resident is assigned to at most one hospital and the number of residents assigned to each hospital does not exceed its capacity. If $r_{i}$ is matched in $M$, we denote the hospital assigned to resident $r_{i}$ in $M$ by $M(r_{i})$. We denote the set of residents assigned to hospital $h_{j}$ in $M$ as $M(h_{j})$. A resident $r_{i}$ is unmatched in $M$ if no pair in $M$ contains $r_{i}$. A hospital $h_{j}$ is undersubscribed in $M$ if $|M(h_{j})|<c_{j}$. A pair $(r_{i},h_{j})$ is said to block a matching $M$, or form a blocking pair with respect to $M$, in the classical sense, if (i) $r_{i}$ is unmatched in $M$ or prefers $h_{j}$ to $M(r_{i})$ and (ii) $h_{j}$ is undersubscribed in $M$ or prefers $r_{i}$ to some resident in $M(h_{j})$. A matching that admits no blocking pair is said to be stable.

We define an instance $(I,G)$ of the Hospitals/Residents Problem under Social Stability (HRSS) as consisting of an HR instance $I$ (as defined above) and a bipartite graph $G=(R\cup H,A)$, where $A\subseteq\mathcal{A}$. A pair $(r_{i},h_{j})$ belongs to $A$ if and only if $r_{i}$ has social ties with $h_{j}$. We call $(r_{i},h_{j})$ an acquainted pair. We also define the set of unacquainted pairs (which cannot block any matching) to be $U=\mathcal{A}\backslash A$. A pair $(r_{i},h_{j})$ socially blocks a matching $M$, or forms a social blocking pair with respect to $M$, if $(r_{i},h_{j})$ blocks $M$ in the classical sense in the underlying HR instance $I$ and $(r_{i},h_{j})\in A$. A matching $M$ is said to be socially stable if there exists no social blocking pair with respect to $M$. If we restrict the hospitals’ capacities to 1, we obtain the Stable Marriage problem with Incomplete lists under Social Stability (SMISS), and refer to the agents as men $\mathcal{U}=\{m_{1},\ldots,m_{n_{1}}\}$ and women $\mathcal{W}=\{w_{1},\ldots,w_{n_{2}}\}$.

Clearly every instance of HRSS admits a socially stable matching. This is because the underlying HR instance is bound to admit a stable matching [11] which is also socially stable. However socially stable matchings could be larger than stable matchings. Consider the SMISS instance $(I,G)$ shown in Figure 1. Matchings $M_{1}=\{(m_{1},w_{1}),(m_{2},w_{2})\}$ and $M_{2}=\{(m_{2},w_{1})\}$ are both socially stable in $(I,G)$ and $M_{2}$ is the unique stable matching. Thus an instance of SMISS (and hence HRSS) can admit a socially stable matching that is twice the size of a stable matching. Clearly the instance shown in Figure 1 can be replicated to give an arbitrarily large SMISS instance with a socially stable matching that is twice the size of a stable matching. This, and applications where we seek to match as many agents as possible, motivates MAX HRSS.

As defined in [3, 8], an instance $(I,G^{\prime})$ of the HR+SN problem involves a Hospitals / Residents instance $I$, defined in [11], containing a set $R=\{r_{1},r_{2},...,r_{n_{1}}\}$ of residents, a set $H=\{h_{1},h_{2},...,h_{n_{2}}\}$ of hospitals, and a graph $G$ describing the social network (SN) of the residents. In the graph $G^{\prime}=(V,E)$, $V=R$ and an edge $\{r_{i},r_{k}\}$ belongs to $E$ if and only if $r_{i}$ and $r_{k}$ have social ties. A pair $(r_{i},h_{j})$ is a local blocking pair with respect to a matching $M$, or locally blocks $M$, if $(r_{i},h_{j})$ blocks $M$ in the classical sense and there is some resident $r_{k}$ such that $\{r_{i},r_{k}\}\in E$ and $r_{k}\in M(h_{j})$. A matching $M$ is said to be locally stable if there exists no local blocking pair with respect to $M$. In the HR+SN (respectively HRSS) context we refer to a resident-complete locally (respectively socially) stable matching as one in which all the residents are matched.

In this section we show the close relationship between the HRSS and HR+SN problems. Consider an instance $(I,G)$ of HRSS where $I$ is the underlying HR instance and $G$ is the social network graph. $I$ involves of a set of residents $R_{0}=\{r_{1},r_{2},...,r_{n_{1}}\}$ and a set of hospitals $H_{0}=\{h_{1},h_{2},...,h_{n_{2}}\}$. We construct an instance $(I^{\prime},G^{\prime})$ of HR+SN from $(I,G)$ as follows: let $I^{\prime}$ consist of a set of residents $R=R_{0}\cup R_{1}$ where $R_{1}=\{r_{n_{1}+1},r_{n_{1}+2},...,r_{n_{1}+n_{2}}\}$. Every resident $r_{n_{1}+j}\in R_{1}$ has a single entry $h_{j}$ in his preference list. Every resident $r_{i}\in R_{0}$ has an identical preference list in $I^{\prime}$ as in $I$. Let $I^{\prime}$ also involve a set of hospitals $H$, where $H=H_{0}$ such that every hospital $h_{j}\in H$ has resident $r_{n_{1}+j}$ as the first entry in its preference list and has capacity $c_{j}^{\prime}=c_{j}+1$. $h_{j}$’s preference list in $I$ is then appended to $r_{n_{i}+j}$ to yield $h_{j}$’s preference list in $I^{\prime}$. To construct $G^{\prime}$, let the vertices in $G^{\prime}$ correspond to the residents in $R$ and add edge $\{r_{i},r_{n_{1}+j}\}$ to $G^{\prime}$ if and only if $(r_{i},h_{j})\in A$, where $A=E(G)$ is the set of acquainted pairs in $(I,G)$.

• Proof

  Suppose $M$ is socially stable in $(I,G)$. Then no (classical) blocking pair with respect to $M$ in $I$ is contained in $G$. Let $M^{\prime}=M\cup\{(r_{n_{1}+j},h_{j}):r_{n_{1}+j}\in R_{1}\}$. If some pair ($r,h$) locally blocks $M^{\prime}$ in $(I^{\prime},G^{\prime})$ then (i) $(r,h)$ must be a blocking pair with respect to $M^{\prime}$ in $I^{\prime}$, and (ii) $\{r,r^{\prime}\}\in E$ for some $r^{\prime}\in M^{\prime}(h)$. By construction, for every edge in $E$, one resident is in $R_{0}$ and the other in $R_{1}$. If $r\in R_{1}$ then $r$ cannot form any blocking pair with respect to $M^{\prime}$ as he is matched to his only choice. If $r\in R_{0}$ then $r^{\prime}=r_{n_{1}+j}\in R_{1}$ for some $j$ ($1\leq j\leq n_{2}$), and $h=M^{\prime}(r_{n_{1}+j})=h_{j}$. Thus $(r,h)\in A$. By the construction of the preference lists in $(I^{\prime},G^{\prime})$, as $(r,h)$ is a (classical) blocking pair of $M^{\prime}$ in $I^{\prime}$, $(r,h)$ is also a (classical) blocking pair of $M$ in $I$. Hence $(r,h)$ socially blocks $M$ in $(I,G)$, a contradiction.

  Conversely suppose $M^{\prime}$ is a resident-complete locally stable matching in $(I^{\prime},G^{\prime})$. Then there is no blocking pair $(r,h)$ of $M^{\prime}$ in $I^{\prime}$ such that $\{r,r^{\prime}\}\in E$ for some $r^{\prime}\in R$ where $r^{\prime}\in M^{\prime}(h)$. Let $M=M^{\prime}\backslash\{(r_{n_{1}+j},h_{j}):r_{n_{1}+j}\in R_{1}\}$. Clearly $M$ is a resident-complete matching in $I$. If some pair $(r,h)$ socially blocks $M$ in $(I,G)$, (i) $(r,h)$ must block $M$ in $I$ (and thus $M^{\prime}$ in $I^{\prime}$) and (ii) $(r,h)\in A$. By construction, if $(r,h)\in A$, then $\{r,r^{\prime}\}\in E$ where $r\in R_{0}$, $r^{\prime}=r_{n_{1}+j}\in R_{1}$ and $h=h_{j}$. But as $M^{\prime}$ is resident-complete, $(r_{n_{1}+j},h_{j})\in M^{\prime}$ for each $j~(1\leq j\leq n_{2})$. Thus $r^{\prime}=r_{n_{1}+j}\in M^{\prime}(h)$, a contradiction to the initial assumption that $M^{\prime}$ is locally stable in $(I^{\prime},G^{\prime})$. $\Box$

Although the converse statement in Theorem 3.1 places a severe restriction on $M^{\prime}$ (it must be a resident-complete locally stable matching in the HR+SN instance $(I^{\prime},G^{\prime})$), it can be relaxed slightly to the case that $M^{\prime}$ is any locally stable matching in which all the residents $r_{n_{1}+j}\in R_{1}$ are matched. It remains to be shown that a reduction exists from HRSS to HR+SN that does not place such a restriction on $M^{\prime}$.

We now show that MAX SMISS, the problem of finding a maximum socially stable matching given an SMISS instance is NP-hard. Indeed we prove NP-completeness for COM SMISS, the problem of deciding whether there exists a complete socially stable matching (i.e., a socially stable matching in which all men and women are matched) in an instance of SMISS. It is obvious that, given a matching $M$ of size $k$, it can be verified in polynomial time whether $M$ is socially stable, thus the problem is in NP. Next we define the Stable Marriage problem with Ties and Incomplete Lists (SMTI) as a variant of the classical Stable Marriage problem [11] where men and women are allowed to find each other unacceptable (thus causing incomplete preference lists) and preference lists may include ties, representing indifference. For example two women may be tied in a given man’s preference list. It was shown in [25] that COM SMTI, the problem of deciding whether a complete stable matching exists in an instance of SMTI, is NP-complete even if the ties occur in the men’s lists only and each tie occurs at the tail of some list. An SMTI instance $I$ satisfying these restrictions can be reduced to an SMISS instance $(I^{\prime},G)$ in polynomial time such that a matching $M$ is a complete stable matching in $I$ if and only if $M$ is a complete socially stable matching in $(I^{\prime},G)$. These observations lead to the following result.

• Proof

  Consider an instance $I$ of SMTI where the ties occur only on the mens’ preference lists and each man has one tie which occurs at the end of the list (a tie may be of length $1$ for this purpose). We define $t(m_{i})$ as the set of women contained in the tie in man $m_{i}$’s preference list. We can construct an instance $(I^{\prime},G)$ of SMISS such that $I^{\prime}$ is the Stable Marriage instance with incomplete lists formed by breaking the ties in $I$ in an arbitrary manner. Let $G=(\mathcal{U}\cup\mathcal{W},A)$, where $\mathcal{U}$ and $\mathcal{W}$ are the sets of all men and women in $I$ respectively and $A=\bigcup_{m_{i}\in\mathcal{U}}\bigcup_{w_{j}\notin t(m_{i})}(m_{i},w_{j})$. We claim that a matching $M$ is a complete stable matching in $I$ if and only if $M$ is a complete socially stable matching in $(I^{\prime},G)$.

  Suppose $M$ is a complete stable matching in $I$. Suppose also that $M$ is not socially stable in $(I^{\prime},G)$. Then there exists some pair $(m_{i},w_{j})\in A$ that socially blocks $M$ in $(I^{\prime},G)$. Since $(m_{i},w_{j})\in A$, $w_{j}\notin t(m_{i})$. Thus $m_{i}$ prefers $w_{j}$ to $M(m_{i})$ in $I$. Also $w_{j}$ prefers $m_{i}$ to $M(w_{j})$ since there are no ties in $w_{j}$’s preference list. Thus $(m_{i},w_{j})$ blocks $M$ in $I$, a contradiction to our initial assumption.

  Conversely, suppose $M$ is a complete socially stable matching in $(I^{\prime},G)$. Suppose also that $M$ is not stable in $I$. Then there exists some pair $(m_{i},w_{j})$ that blocks $M$ in $I$. If $M(m_{i})\in t(m_{i})$ then $w_{j}\notin t(m_{i})$ so $m_{i}$ prefers $w_{j}$ to $M(m_{i})$ in $(I^{\prime},G)$. If $M(m_{i})\notin t(m_{i})$ then $m_{i}$ prefers $w_{j}$ to $M(m_{i})$ in $(I^{\prime},G)$. $w_{j}$ has the same preference list in $I$ and $(I^{\prime},G)$. So $(m_{i},w_{j})\in A$ and thus the pair socially blocks $M$ in $(I^{\prime},G)$, a contradiction to our initial assumption. $\Box$

As discussed in [20], some centralised matching schemes usually require the agents in one or more sets to have preference lists bounded in length by some small integer. For example, until recently, in the Scottish Foundation Allocation Scheme (the centralised clearinghouse for matching medical residents in Scotland) [18], medical graduates were required to rank only 6 hospitals in their preference lists. We denote by $(p,q)$-MAX HRSS the problem of finding a maximum socially stable matching in an HRSS instance where each resident is allowed to rank at most $p$ hospitals and each hospital at most $q$ residents. We set $p=\infty$ and $q=\infty$ to represent instances where the residents and hospitals respectively are allowed to have unbounded-length preference lists. Analogously we may obtain the definition of $(p,q)$-MAX SMISS and $(p,q)$-COM SMISS from MAX SMISS and COM SMISS respectively. It turns out that $(p,q)$-COM SMISS is NP-complete even for small values of $p$ and $q$.

• Proof

  We prove this by inspecting the hardness result described for the $(3,3)$-COM SMTI problem by Irving et al. [20]. They showed that $(3,3)$-COM SMTI, the problem of deciding whether a complete stable matching exists in an instance of SMTI where each preference list is of length at most $3$, is NP-complete using a reduction from a variant of the SAT problem.

  By inspecting the instance $I$ of SMTI, constructed in the proof, we observe that all the ties appear on the women’s side of the instance and appear at the ends of the preference list. We have shown in Theorem 4.1 that an instance $I$ of SMTI in this form can be reduced in polynomial time to an instance $(I^{\prime},G)$ of SMISS such that a matching $M$ is a complete stable matching in $I$ if and only if $M$ is a complete socially stable matching in $(I^{\prime},G)$. We conclude that $(3,3)$-COM SMISS is also NP-complete. $\Box$

As shown in Section 4, MAX HRSS is NP-hard. In order to deal with this hardness, polynomial-time approximation algorithms can be developed for MAX HRSS. In this section we present a $3/2$-approximation algorithm for MAX HRSS. We show this is tight assuming the Unique Games Conjecture (UCG), and also show a $21/19-\varepsilon$ lower bound assuming $P\not=NP$. The lower bounds hold even for MAX SMISS.