2.1 SRTI: Stable roommates problem with ties and incomplete lists

Let $A$ be a finite set of agents. For every agent $x\in A$ , let $A_x \subseteq A \backslash \{x\}$ be a set of agents that are acceptable to $x$ as roommates. For every $y$ in $A_x$ , we assume that $x$ prefers $y$ as a roommate compared to being single.

Let $\prec _{x}$ be a partial ordering of $x$ ’s preferences over $A_x$ where incomparability is transitive. We refer to $\prec _{x}$ as agent $x$ ’s preference list. For two agents $y$ and $z$ in $A_x$ , we denote by $y \prec _{x} z$ that $x$ prefers $y$ to $z$ . In this context, ties correspond to indifference in the preference lists: an agent $x$ is indifferent between the agents $y$ and $z$ , denoted by $y \sim _{x} z$ , if $y \not \prec _{x} z$ and $z \not \prec _{x} y$ . We denote by $\prec$ the collection of all preference lists.

A matching for a given SRTI instance is a function $M: A \mapsto A$ such that, for all $\{x,y\} \subseteq A\times A$ such that $x\in A_y$ and $y\in A_x$ , $M(x)=y$ if and only if $M(y)=x$ . If agent $x$ is mapped to itself, we then say he/she is single.

A matching $M$ is blocked by a pair $\{x, y\} \subseteq A \times A$ ( $x\neq y$ ) if the following hold: both agents $x$ and $y$ are acceptable to each other, $x$ is single with respect to $M$ , or $y \prec _{x} M(x)$ , and $y$ is single with respect to $M$ , or $x \prec _{y} M(y)$ .

A matching for SRTI is called stable if it is not blocked by any pair of agents.

We can declaratively solve SRTI using ASP as described in our earlier work (Erdem et al. Reference Erdem, Fidan, Manlove and Prosser2020). The input $I=(A,\prec )$ of an SRTI instance is formalized by a set $F_I$ of facts using atoms of the forms agent $(x)$ (“ $x$ is an agent in $A$ ”) and prefer2 $(x,y,z)$ (“agent $x$ prefers agent $y$ to agent $z$ , i.e., $y \prec _{x} z$ ”). For every agent $x$ , for every $y\in A_x$ , we also add facts of the form prefer2 $(x,y,x)$ to express that $x$ prefers $y$ as a roommate instead of being single. We recursively define the transitive closure of this preference relation:

\begin{align*} \begin{array}{l} \textit {prefer}(x,y,z) \leftarrow \textit {prefer2}(x,y,z). \\[5pt] \nonumber \textit {prefer}(x,y,z) \leftarrow \textit {prefer2}(x,y,w), \textit {prefer}(x,w,z). \end{array} \end{align*}

and acceptability of agents:

(1) \begin{align} \begin{array}{l} \textit {accept}(x,y) \leftarrow \textit {prefer}(x,y,\_). \\[3pt] \textit {accept}(x,y) \leftarrow \textit {prefer}(x,\_,y). \\[3pt] \textit {accept2}(x,y) \leftarrow \textit {accept}(x,y), \textit {accept}(y,x). \end{array} \end{align}

The output $M: A \mapsto A$ of an SRTI instance is characterized by atoms of the form room $(x,y)$ (“agents $x$ and $y$ are roommates”). For every agent $x$ , exactly one mutual acceptable agent $y$ is nondeterministically chosen as $M(x)$ :

\begin{align*} \begin{array}{l} 1\{\textit {room}(x,y){:} \textit {agent}(y), \textit {accept2}(x,y)\}1 \leftarrow \textit {agent}(x). \\[3pt] \leftarrow \textit {room}(x,y), \textit {not}\ \textit {room}(y,x). \end{array} \end{align*}

and the stability of the generated matching is ensured:

\begin{equation*} \leftarrow \textit {block}(x,y) \qquad (x\neq y) . \end{equation*}

where atoms of the form block $(x,y)$ describe the blocking pairs.

2.2 Personalized-SRTI: SRTI with personalized criteria

Let $B$ be a finite list $\langle b_{1},b_{2},\dots ,b_{k} \rangle$ of criteria. For every $b_{i} \in B$ , let $C_i$ be a finite list $\langle c_{i1},c_{i2},\dots ,c_{im} \rangle$ of its choices for $b_{i}$ , ordered with respect to a “closeness” measure.

Let $f$ be a function that maps an agent $x\in A$ and a criterion $b_{i} \in B$ to a positive integer $j$ ( $1\leq j \leq |C_{i}|$ ), describing the choice $c_{ij}$ of the agent $x$ . For every agent $x$ , $x$ ’s (preference) profile $P_{x}=\langle f(x,b_{1}),f(x,b_{2}),\dots ,f(x,b_{k})\rangle$ characterizes its choices of for each criterion in $B$ , respectively.

Every criterion in $B$ may have a different importance for each agent. For that, we introduce a weight function $w$ that maps an agent $x\in A$ and a criterion $b_{i} \in B$ to a non-negative integer such that $w(x,b_{i})$ denotes the importance of the criterion $b_{i}$ for $x\in A$ . For every agent $x\in A$ , let us denote by the weight list $W_{x}=\langle w(x,b_{1}),w(x,b_{2}),\dots ,w(x,b_{k})\rangle$ the respective weights of criteria in $B$ for $x$ .

For every agent $x\in A$ , with a profile $P_x$ and a weight list $W_x$ , let us denote the criteria of the same weight $u\gt 0$ and the agent $x$ ’s choices for them, by a nonempty set $E_u$ of tuples: $E_u = \{(f(x,\pi _{i}),\pi _i)\ |\ u=w(x,\pi _i)\gt 0,\ \pi _i \in \{b_{1},b_{2},\dots ,b_{k}\}\}.$ Then, for every agent $x\in A$ , we define a sorted profile $P_{x}^{'}$ for $x$ , with respect to $P_{x}$ and $W_x$ : $P_{x}^{'}=\langle E_{u_1}, E_{u_2}, \dots , E_{u_m} \rangle$ where $m \leq k$ , and, for each $i$ ( $1 {\leq } i {\lt } m$ ), $u_i \gt u_{i+1}$ .

For every agent $y\in A \backslash A_{x}$ (i.e., $y$ is not acceptable to $x$ ), $y$ is choice-acceptable to $x$ if there exists some criterion $b_{i}\in B$ where $w(x,b_i)\gt 0$ such that $f(y,b_{i})=f(x,b_{i})$ . For every agent $x$ with a sorted profile $P_{x}^{'}=\langle E_{u_1}, E_{u_2}, \dots , E_{u_m} \rangle$ ( $m \leq k$ ), for every two agents $y$ and $z$ that are choice-acceptable to $x$ , the agents $y$ and $z$ are choice-equal for $x$ relative to the first $j$ sets $E_{u_1}, E_{u_2}, \dots , E_{u_j}$ in $P_{x}^{'}$ (denoted $y =_{x} z |_j$ ) if $j=0$ , or $j\gt 0$ , $y =_{x} z |_{j-1}$ , and, for every $(f(x,\pi _{i}),\pi _i)\in E_{u_j}$ , $f(x,\pi _{i}) = f(y,\pi _{i}) = f(z,\pi _{i})$ .

We say that $x$ prefers $y$ to $z$ with respect to a sorted profile $P_{x}^{'}$ (denoted $y \prec '_{x} z$ ) if, for some $j\gt 0$ , $y =_{x} z |_{j-1}$ , and $|\{\pi _i |\ (f(x,\pi _{i}),\pi _i){\in } E_{u_j},\ f(x,\pi _{i}) {=} f(y,\pi _{i})\}|\ \gt$ $|\{\pi _i |\ (f(x,\pi _{i}),\pi _i){\in } E_{u_j},\ f(x,\pi _{i}) {=} f(z,\pi _{i})\}|.$

A criteria-based personalized preference list $\prec _{x}^{'}$ is a partial ordering of $x$ ’s preferences over $A'_x$ with respect to a sorted profile $P_{x}^{'}$ , where incomparability is transitive. An extended preference list $\prec _{x}^{''}$ is obtained by concatenating $\prec _{x}$ and $\prec _{x}^{'}$ , depending on the importance given to these two types of lists. Personalized-SRTI is then characterized by $(A,\prec ^{''})$ where $A$ is a finite set of agent, and $\prec ^{''}$ is the collection of the extended preference list of each agent $x\,\in \,A$ .

Table 1. A Personalized-SRTI instance, characterized by a set of students $x$ , and their extended preference lists $\prec _{x}^{''}$

3.1 Defining $\boldsymbol{{k}}$ -acceptable pairs with respect to $\boldsymbol{\prec} _{\boldsymbol{{x}}}$

Let $A_x^{+}$ be a set of acceptable agents to $x \in A$ , and $A_x^{-}$ be a set of unwanted agents by $x \in A$ . Note that $A_x^{+} \cup A_x^{-} \subseteq A$ . Let $A^{-}$ be a set of unwanted pairs, that is $A^{-} = \{\{x,y\}: x \in A_y^{-},\,y\in A\}$ . If an agent $x$ states $y$ as acceptable or unwanted, then we say that $y$ is known by $x$ .

Let $K_x$ be a set of agents known by $x$ . We assume that $A_x^{+} \cup A_x^{-} = K_x$ . We denote by $K$ the set of pairs of agents who are known by each other, i.e., $K = \{\{x,y\}: x \in K_y,\,y\in A\}$ . Note that $K$ does not require that agents mutually know each other.

A $k$ -acceptability graph is an undirected graph $G_A=(V,E)$ , where every vertex $x \in V$ uniquely denotes an agent in $A$ , and every edge $\{x,y\}\in E$ denotes pairs of agents that know each other and none of them is unwanted by the other: $\{x,y\} \in K \setminus A^{-}$ . The curious reader may notice that the “known by” relation is not symmetric but the graph $G_A$ is undirected (since it includes an undirected edge to denote one-sided known but non-unwanted pairs). We construct $G_A=(V,E)$ a bit more generously based on the following assumption: if a student $x$ is known by another student $y$ , and neither student finds the other one unwanted, then the pair $\{x,y\} \in E$ regardless of $y$ is known by $x$ . This assumption is reasonable and allows more possibilities for good-enough matchings. Indeed, a student $x$ might state someone $y$ in their preference list without being listed in return because the other student $y$ has not provided any preferences; in such cases, it is reasonable to assume that $x$ is not unwanted by $y$ and it is possible that $y$ might include $x$ in their list if they provide their preferences.

We say that agents $x$ and $y$ form a $k$ -connected pairs if there exists a path of length $k$ that connects $x$ and $y$ in $G_A$ . Let us denote by $P^{k}_{G_A}$ the set of all $k$ -connected pairs in $G_A$ , and, for simplicity, we will refer to it as $P^k$ throughout the remainder of the paper.

3.2 Breaking ties in $\boldsymbol{\prec} _{\boldsymbol{{x}}}^{\boldsymbol{\prime}}$ by considering $\boldsymbol{{k}}$ -connected pairs

Let $I_x$ be a set of agents that are inferred as acceptable to $x$ as roommates, that is $I_x = \{y: y \in \prec '_{x}\}$ . If an agent $x$ is indifferent between $y$ and $z$ in $I_x$ , then this tie is broken by considering $k$ -connected pairs according to the following rule: if there exists a value $k$ such that $\{x,y\} \in P{^k}$ , and there is no $l$ such that $\{x,z\} \in P{^l}$ for $l\leq k$ , then $y \prec '_{x} z$ .

3.3 Defining the $\boldsymbol{{k}}$ -extended preference lists $\boldsymbol{\prec} _{\boldsymbol{{x}}}^{\boldsymbol{{k}}^{\boldsymbol{\prime}\boldsymbol{\prime}}}$

Let $O_{x}$ be a set of agents that may be acceptable to $x$ , that is $O_x = \{y: y \in A \backslash {(K_x} \cup I_x)\}$ . Let $\prec _{x}^{k}$ be a partial ordering of $x$ ’s preferences over $O_{x}$ where incomparability is transitive. For two agents $y$ and $z$ in $O_x$ , $y \prec _{x}^{k} z$ if $\{x,y\} \in P{^{i}}$ and $\{x,z\} \notin P{^{i}}$ , but $\{x,z\} \in P{^{k}}$ for some $0\lt i\lt k$ . Note that for every $y \in O_{x}$ , $y \in \prec _{x}^1$ if $\{x,y\} \in P{^1}$ .

Considering $k$ -connected pairs, we define $\prec _{x}^{k''} = \prec _{x}^{''} + \prec _{x}^{k}$ as a $k$ -extended preference list. Note that $\prec _{x}^{k''}$ is the same as $\prec _{x}^{''}$ when $k=0$ . Let us denote by $\prec ^{k''}$ the collection of the $k$ -extended preference lists of each agent $x \in A$ .

Table 2. A $k$ -Personalized-SRTI instance, characterized by a set of students $x$ , sets $A_x^{-}$ of unwanted students as roommates, and the $k$ -extended lists $\prec _{x}^{k''}$ of preferred students as roommates for $k=1$ and $k=2$

A $k$ -Personalized-SRTI instance is an SRTI instance characterized by a set $A$ of agents, a set $A^-$ of pairs of agents (denoting which agents are forbidden for which), and a collection $\prec ^{k''}$ of the $k$ -extended preference lists of each agent $x \in A$ . A solution to a $k$ -Personalized-SRTI instance is defined in the following section, as “ $k$ -stable matchings.”

3.4 Defining $\boldsymbol{{k}}$ -matchings with respect to $\boldsymbol{\prec} _{\boldsymbol{{y}}}^{\boldsymbol{{k}}^{\boldsymbol{\prime}\boldsymbol{\prime}}}$

We say that $x$ is $k$ -acceptable to $y$ if $x \in \prec _{y}^{k''}$ . If an agent $x$ is $k$ -acceptable to $y$ and $y$ is $k$ -acceptable to $x$ , then $x$ and $y$ are called $k$ -mutually acceptable.

A $k$ -matching for a given $k$ -Personalized-SRTI instance $(A,A^-,\prec ^{k''})$ , is a function $M: A \mapsto A$ such that, for all $\{x,y\} \subseteq A\times A$ where $x\in A_{y}{^{+}} \cup I_y \cup O_y$ and $y \in A_{x}{^{+}} \cup I_x \cup O_x$ , $M(x){=}y$ if and only if $M(y){=}x$ .

A pair $\{x,y\}\subseteq A\times A\,(x \neq y)$ is a $k$ -blocking pair with respect to $M$ if

1. i) $x$ and $y$ are $k$ -mutually acceptable to each other,

2. ii) $x$ is single in $M$ , or $y \prec _{x}^{k''} M(x)$ , and

3. iii) $y$ is single in $M$ , or $x \prec _{y}^{k''} M(y)$ .

A $k$ -matching $M$ is called $k$ -stable if there is no $k$ -blocking pair with respect to $M$ .

SRTI with $k$ -connection ( $k$ -SRTI) is a special case of $k$ -Personalized-SRTI $(A,\prec ^{k''})$ where, for every agent $x\in A, \prec _{x}^{''} = \prec _{x}$ .

3.5 Observations about $\boldsymbol{{k}}$ -Personalized-SRTI

Table 3. A $k$ -Personalized-SRTI instance, characterized by a set of students $x$ , sets $A_x^{-}$ of unwanted students as roommates, and the $k$ -extended lists $\prec _{x}^{k''}$ of preferred students as roommates for $k=1$ and $k=2$

Observation 2 (A stable matching may not be a $k$ -stable matching.) Consider the example shown in Table 4. Consider the SRTI instance $I=(A,\prec )$ , and Personalized-SRTI instance $I'=(A,\prec ^{''})$ . For both of them, there is a stable matching: $M=\{ab,cd,e\}$ . Then, for the $1$ -SRTI instance $I''=(A,A^{-},\prec ^{1''})$ , $M$ is not a $1$ -stable matching since $be$ blocks the matching $M$ . Hence, $M_1=\{be,cd,a\}$ is a $1$ -stable matching.

Table 4. A $k$ -Personalized-SRTI instance, characterized by a set of students $x$ , sets $A_x^{-}$ of unwanted students as roommates, and the $k$ -extended lists $\prec _{x}^{1''}$ of preferred students as roommates for $k=1$

5.1 Experimental evaluation with objective measures

Benchmark instances. First we randomly generate SRTI instances using Prosser’s software (SRItoolkit 2019) based on Merten’s idea (Reference Mertens2005): (1) we generate a random graph ensemble $G(n,p)$ according to the Erdös-Rényi model (Reference Erdös and Rényi1960), where $n$ is the required number of agents and $p$ is the edge probability (i.e., each pair of vertices is connected independently with probability $p$ ); (2) since the edges characterize the acceptability relations, we generate a random permutation of each agent’s acceptable partners to provide the preference lists.

Then, motivated by the real-world applications, (3) in each SRTI instance, we shorten the given preference lists so as to contain the first 5 choices for each agent. Then, (4) we populate each instance to a set of Personalized-SRTI instances by extending these lists by including the choices inferred from the habitual preferences of agents: we consider 2 and 5 habitual criteria, randomly generate the importance of each criterion, and randomly generate the habitual preferences of 20% and 50% of agents.

For each instance generated with the method above, we define its completeness degree (c.d.) as the probability of an agent occurring in a preference list (i.e., the edge probability). We define the mutual acceptability rate (m.a.p.) of an instance as the ratio of the number of mutually acceptable pairs to the number of all possible pairs. Consider the example shown in Table 2. For a given $2$ -Personalized-SRTI instance, there are $14$ mutually acceptable pairs out of 16 possible pairs. There are two pairs, such as $\{a,c\}$ and $\{b,d\}$ , that are not mutually acceptable. Hence, the mutually acceptability rate is $\frac {14}{16}$ , which simplifies to $0.875$ .

We generate two sets of benchmarks, with respect to their m.a.p.: HMA instances have high m.a.p. of 75% or more, while LMA instances have lower m.a.p..

For HMA instances, first, as described in steps (1) and (2) above, we generate SRTI instances with 40, 60, 80, 100, 150, and 200 agents, with c.d. of 0.075, 0.05, 0.0375, 0.03, 0.02, and 0.015, respectively. Such low c.d. allow us to generate SRTI instances with preference lists with approximate length 3.

In our earlier study (Erdem et al. Reference Erdem, Fidan, Manlove and Prosser2020) that investigates the scalability of our approach to solve SRTI problems, we have already generated benchmark instances using Prosser’s software, as described in steps (1) and (2) above. For LMA instances, we start with those instances with 40, 60, 80, 100, 150, and 200 agents, and with c.d. 0.25.

Then, in each benchmark set, for each number of criteria and the percentage of agents who specify their habitual preferences, we populate each SRTI instance by generating 20 Personalized-SRTI instances, as described in steps (3) and (4) above.

The characteristics of each set of benchmarks are summarized in Tables 5 and 6. Consider, for instance, the first row of Table 5 that describes average values for 20 HMA instances with 40 agents. The SRTI instances (generated after steps (1), (2), and (3)) have an average c.d. of 0.073 and an average m.a.p. of 0.965. After step (4) where we add knowledge about the habitual preferences of 50% of agents by considering 5 criteria, the average c.d. increases to 0.207 while the average m.a.p. decreases to 0.757.

Table 5. HMA instances: average completeness degree (c.d.) and average percentage of mutually acceptability rate (m.a.p.)

Table 6. LMA instances: average completeness degree (c.d.) and average percentage of mutually acceptability rate (m.a.p.)

Experimental results. We have evaluated our method over these two sets of benchmarks to understand the usefulness of adding knowledge about agents’ networks of preferred friends.

Fig 2. Usefulness of $2$ -SRTI over SRTI.

When there is no stable matching to the seed SRTI instances, Figure 2 shows the computational benefit of extending the preference lists by only suitable candidates obtained from networks of preferred friends, considering $2$ -connected pairs (i.e., preferred friend of a preferred friend): while only some HMA instances have good-enough matchings, almost all LMA instances have good-enough matchings. This result is expected because, as the preferences lists are extended, we have a higher number of mutually acceptable pairs, and this allows exploring more number of potential matchings and thus increasing the chances of finding a good-enough solution.

When there is no stable matching to the seed SRTI instances and no good-enough solution after extending the preference lists considering only the agents’ habits and habitual preferences, Figure 3 shows that extending the preference lists by suitable candidates obtained from networks of preferred friends is still useful.

In these experiments, the scalability of SRTI-ASP for $k$ -SRTI is also observed (Figure 4): computing good-enough matchings usually takes a few seconds on a machine with Intel Xeon(R) W-2155 3.30 GHz CPU and 32 GB RAM.

Fig 3. Usefulness of $2$ -SRTI over Personalized-SRTI: (top) 2 criteria and (bottom) 5 criteria.

Fig 4. Scalability of SRTI-ASP in computation time, for $2$ -SRTI over Personalized-SRTI: (left) 2 criteria and (right) 5 criteria. Legend as in Figure 3.

5.2 Experimental evaluation with subjective measures

We evaluate the usefulness of the additional knowledge from the perspective of the users. We design and perform poll survey and interviews with students to obtain their feedback and use subjective measures to evaluate the results.

Online poll survey with students. The objective is to find out whether it makes sense to match roommates considering the networks of preferred friends. It includes one question to learn whether the students prefer to roommate with someone from the preference lists of their preferred friends, when the dormitory system cannot match them with any of their friends on their preference lists. The survey, shown in Figure 1, is conducted online with 101 students at Sabanci University. In the end, 76 students prefer option of extending their preference lists with their preferred friends’ preferences (i.e., option (a) in the poll).

Interviews with students. The objective is to get more detailed information about their poll answers and learn the underlying reasons of students’ answers. Face-to-face interviews are conducted with 20 volunteers (8 female and 12 male) with diverse academic backgrounds. Each interview session is done one-to-one, taking an average of $12-15$ minutes. Before the interview, each volunteer is informed about the objective and the process of the interview, and their permission is requested for recording the interviews. In the interview, the following questions are asked in an appropriate order based on their answers: “Why did you choose this option at the poll?”, “Can you say a little more about …?”, “Did you have such an experience?”, “What if …”, “Why do you think …?”. To provide concrete examples to illustrate the kinds of questions asked and the kinds of responses gathered, two interview transcripts are included in the supplementary material accompanying the paper at the TPLP archive.

Among the volunteers, 4 students have experienced to be roommates with a friend of their friend, positively. At the beginning of the interview, 14 volunteers have chosen option (a) of the poll, that is to be roommates with someone from the preference lists of their preferred friends. During the interview, 4 of the volunteers who chose option (b) of the poll, that is to be roommates with anyone, have changed their decision. According to their final decisions, 18 volunteers choose option (a).

Table 7 shows the volunteers’ reasons for choosing option (a) of the poll, that is to be roommates with someone from the preference lists of their preferred friends, and how many of them give the same reason. For instance, most of the volunteers (16 out of 18) indicate that they trust the person their friend prefers to be roommate with, instead of someone they do not know anything about.

Table 7. Out of 20 students, 14 students have chosen option (a), i.e., they prefer as roommates someone from the preference lists of their preferred friends, compared to anyone. The reasons they have provided for this decision, the number of students who have given the same reason for their decisions before the interview and changed their minds during/after the interview are also provided

Table 8 displays the volunteers’ reasons for choosing option (b) of the poll, that is to be roommates with anyone, and how many of them give this reason. There are 6 such students, and 4 of them have changed their decision during the interview. For instance, one volunteer have said “I may have some friends, who have friends I do not like, in my preference list.” As a follow-up question, we have asked “What if the dormitory system allows you to specify also the students you do not want to be matched with as roommates?”, and the volunteer changed their preference from option (b) to option (a).

Table 8. Out of 20 students, 6 students have chosen option (b), i.e., they prefer as roommates anyone compared to someone from the preference lists of their preferred friends. The reasons they have provided for this decision, the number of students who have given the same reason for their decisions before the interview and changed their minds during/after the interview are also provided

Experimental results: quantitative comparisons. Most of the students (75.2%) who participated in the online poll survey have chosen option (a) at the poll survey. This result illustrates that it makes sense to consider the preferred friends of preferred friends while assigning roommates and extending the preference lists with suitable candidates accordingly. On the other hand, only 10% of the volunteers in the interviews have chosen option (b).

Experimental results: qualitative comparisons. The reasons for choosing option (a), obtained from the students significantly coincide. For instance, getting assigned to a roommate from the preference lists of their preferred friends is preferable (compared to random assignment) because they would be more comfortable and get along well with each other. These results also indicate the usefulness of our knowledge-based approach from the users’ point of view.