Robust and Approximately Stable Marriages under Partial Information

04/24/2018 ∙ by Vijay Menon, et al. ∙ University of Waterloo 0

We study the stable marriage problem in the partial information setting where the agents, although they have an underlying true strict linear order, are allowed to specify partial orders. Specifically, we focus on the case where the agents are allowed to submit strict weak orders and we try to address the following questions from the perspective of a market-designer: i) How can a designer generate matchings that are robust? ii) What is the trade-off between the amount of missing information and the "quality" of solution one can get? With the goal of resolving these questions through a simple and prior-free approach, we suggest looking at matchings that minimize the maximum number of blocking pairs with respect to all the possible underlying true orders as a measure of "quality", and subsequently provide results on finding such matchings. In particular, we first restrict our attention to matchings that have to be stable with respect to at least one of the completions and show that in this case arbitrarily filling-in the missing information and computing the resulting stable matching can give a non-trivial approximation factor for our problem in certain cases. We complement this result by showing that, even under severe restrictions on the preferences of the agents, the factor obtained is asymptotically tight in many cases. We then investigate a special case, where only agents on one side provide strict weak orders and all the missing information is at the bottom of their preference orders, and show that here the negative result mentioned above can be circumvented in order to get a much better approximation factor; this result, too, is tight in many cases. Finally, we move away from the restriction mentioned above and show a general hardness of approximation result and also discuss one possible approach that can lead us to a near-tight approximation bound.

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1 Introduction

Two-sided matching markets have numerous applications, e.g., in matching students to dormitories (i.e., Stable Roommates problem (SR) irv85), residents to hospitals (i.e., Hospital-Resident problem (HR) man08) etc., and hence are ubiquitous in practice. Perhaps unsurprisingly, then, this line of research has received much attention, with plenty of work done on investigating numerous problems like SR and HR, and their many variations (we refer the reader to the excellent books by Gusfield and Irving [gus89] and Manlove [man13] for a survey on two-sided matching problems). The focus of this paper, too, is on one such problem—one that is perhaps the most widely-studied, but yet the simplest—called the Stable Marriage problem (SM), first introduced by Gale and Shapley [gale62]. In SM we are given two disjoint sets (colloquially referred to as the set of men and women) and each agent in one set specifies a strict linear order over the agents in the other set, and the aim is to find a stable matching, i.e., a matching where there is no man-woman pair such that each of them prefers the other over their partner in the matching. (Such a pair, if it exists, is called a blocking pair.)

While the assumption that the agents will be able to specify strict linear orders is not unreasonable in small markets, in general, as the markets get larger, it may not be feasible for an agent to determine a complete ordering over all the alternatives. Furthermore, there may arise situations where agents are simply unwilling to provide strict total orders due to, say, privacy concerns. Thus, it is natural for a designer to allow agents the flexibility to specify partial orders, and so in this paper we assume that the agents submit strict weak orders222All our negative results naturally hold for the case when the agents are allowed to specify strict partial orders. As for our positive results, most of them can be extended for general partial orders, although the resulting bounds will be worse. (i.e, strict partial orders where incomparability is transitive) that are consistent with their underlying true strict linear orders. Although the issue of partially specified preferences has received attention previously, we argue that certain aspects have not been addressed sufficiently. In particular, the common approach to the question of what constitutes a “good” matching in such a setting has been to either work with stable matchings that arise as a result of an arbitrary linear extension of the submitted partial orders (these are known as weakly-stable matchings) or to look at something known as super-stable matchings, which are matchings that are stable with respect to all the possible linear extensions of the submitted partial orders irv94,ras14. In the case of the former, one key issue is that we often do not really know how “good” a particular weakly-stable matching is with the respect to the underlying true orders of the agents, and in the case of the latter they often do not exist. Furthermore, we believe that it is in the interest of the market-designer to understand how robust or “good” a matching is with respect to the underlying true orders of the agents, for, if otherwise, issues relating to instability and market unravelling can arise since the matching that is output by a mechanism can be arbitrarily bad with respect to these true orders. Hence, in this paper we propose to move away from the extremes of working with either arbitrary weakly-stable matchings or super-stable matchings, and to find a middle-ground when it comes to working with partial preference information. To this end, we aim to answer two questions from the perspective of a market-designer: i) How should one handle partial information so as to be able to provide some guarantees with respect to the underlying true preference orders? ii) What is the trade-off between the amount of missing information and the quality of a matching that one can achieve? We discuss our proposal in more detail in the following sections.

1.1 How does one work with partial information?

When agents do not submit full preference orderings, there are several possible ways to cope with the missing information. For instance, one approach that immediately comes to mind is to assume that there exists some underlying distribution from which the agents’ true preferences are drawn, and then use this information to find a “good” matching—which is, say, the one with the least number of blocking pairs in expectation. However, the success of such an approach crucially depends on having access to information about the underlying preference distributions which may not always be available. Therefore, in this paper we make no assumptions on the underlying preference distributions and instead adopt a prior-free and absolute-worst-case approach where we assume that any of the linear extensions of the given strict partial orders can be the underlying true order, and we aim to provide solutions that perform well with respect to all of them. We note that similar worst-case approaches have been looked at previously, for instance, by Chiesa et al. [chi12] in the context of auctions.

The objective we concern ourselves with here is that of minimizing the number of blocking pairs, which is well-defined and has been considered previously in the context of matching problems (for instance, see abr05,biro10). In particular, for a given instance our aim is to return a matching that has the best worst case—i.e., a matching that has the minimum maximum ‘regret’ after one realises the true underlying preference orders. (We refer to as the minimax optimal solution.) More precisely, let denote an instance, where , , and are the set of men and women respectively, and is the strict partial order submitted by agent . Additionally, let denote the set of linear extensions of , be the Cartesian product of the s, i.e., , denote the set of blocking pairs that are associated with the matching according to some linear extension , and denote the set of all possible matchings. Then the matching that we are interested in is defined as .

While we are aware of just one work by Drummond and Boutilier [dru13] who consider the minimax regret approach in the context of stable matchings (they consider it mainly in the context of preference elicitation; see Section 1.4 for more details), the approach, in general, is perhaps reminiscent, for instance, of the works of Hyafil and Boutilier [hya04] and Lu and Boutilier [lu11] who looked at the minimax regret solution criterion in the context of mechanism design for games with type uncertainty and preference elicitation in voting protocols, respectively.

Remark: In the usual definition of a minimax regret solution, there is a second term which measures the ‘regret’ as a result of choosing a particular solution. That is, in the definition above, it would usually be , where is the optimal matching (with respect to the objective function ) for the linear extension . We do not include this in the definition above because as every instance of the marriage problem with linear orders has a stable solution (which by definition has zero blocking pairs). Additionally, the literature on stable matchings uses the term “regret” to denote the maximum cost associated with a stable matching, where the cost of a matching for an agent is the rank of its partner in the matching and the maximum is taken over all the agents (for instance, see man02). However, here the term regret is used in the context of the minimax regret solution criterion.

1.2 How does one measure the amount of missing information?

For the purposes of understanding the trade-off between the amount of missing information and the “quality” of solution one can achieve, we need a way to measure the amount of missing information in a given instance. There are many possible ways to do this, however in this paper we adopt the following. For a given instance , the amount of missing information, , is the fraction of pairwise comparisons one cannot infer from the given strict partial orders. That is, we know that if every agent submits a strict linear order over alternatives, then we can infer comparisons from it. Now, instead, if an agent submits a strict partial order , then we denote by the fraction of these comparisons one cannot infer from (this is the “missing information” in ). Our here is equal to . Although, given a strict partial order , it is straightforward to calculate , we will nevertheless assume throughout that is part of the input. Hence, our definition of an instance will be modified the following way to include the parameter for missing information: .

Remark: denotes the case when all the preferences are strict linear orders. Also, for an instance with agents on each side, the least value of when the amount of missing information is non-zero is (this happens in the case where there is only one agent with just one pairwise comparison missing). However, despite this, in the interest of readability, we sometimes just write statements of the form “for all ”. Such statements need to be understood as being true for only realizable or valid values of that are greater than zero.

1.3 Our Contributions

The focus of our work is on computing the minimax optimal matching, i.e., a matching that, when given an instance , minimizes the maximum number of blocking pairs with respect to all the possible linear extensions (see Section 2.1 for a formal definition of the problem). Towards this end, we make the following contributions:

  • We formally define the problem and show that, interestingly, the problem under consideration is equivalent to the problem of finding a matching that has the minimum number of super-blocking pairs (i.e., man-woman pairs where each of them weakly-prefers the other over their current partners).

  • While an optimal answer to our question might involve matchings that have man-woman pairs such that each of them strictly prefers the other over their partners, we start by focusing our investigation on matchings that do not have such pairs. Given the fact that any matching with no such pairs are weakly-stable, through this setting we address the question “given an instance, can we find a weakly-stable matching that performs well, in terms of minimizing the number of blocking pairs, with respect to all the linear extensions of the given strict partial orders?” We show that by arbitrarily filling-in the missing information and computing the resulting stable matching, one can obtain a non-trivial approximation factor (i.e., one that is ) for our problem for many values of . We complement this result by showing that, even under severe restrictions on the preferences of the agents, the factor obtained is asymptotically tight in many cases.

  • By assuming a special structure on the agents’ preferences—one where strict weak orders are specified by just agents on one side and all the missing information is at the bottom of their preference orders—we show that one can obtain a -approximation algorithm for our problem. The proof of the same is via finding a 2-approximation for another problem (see Problem 3) that might be of independent interest.

  • In Section 4 we remove the restriction to weakly-stable matchings and show a general hardness of approximation result for our problem. Following this, we discuss one possible approach that can lead to a near-tight approximation guarantee for the same.

1.4 Related Work

There has recently been a number of papers that have looked at problems relating to missing preference information or uncertainty in preferences in the context of matching.

Drummond and Boutilier [dru13]

used the minimax regret solution criterion in order to drive preference elicitation strategies for matching problems. While they discussed computing robust matchings subject to a minimax regret solution criteria, their focus was on providing an NP-completeness result and heuristic preference elicitation strategies for refining the missing information. In contrast, in addition to focusing on understanding the exact trade-offs between the amount of missing information and the solution “quality”, we concern ourselves with arriving at approximation algorithms for computing such robust matchings.

Rastegari et al. [ras14] studied a partial information setting in labour markets. However, again, the focus of this paper was different than ours. They looked at pervasive-employer-optimal matchings, which are matchings that are employer-optimal (see ras14 for the definitions) with respect to all the underlying linear extensions. In addition, they also discussed how to identify, in polynomial time, if a matching is employer-optimal with respect to some linear extension.

Recent work by Aziz et al. [aziz16]

looked at the stable matching problem in settings where there is uncertainty about the preferences of the agents. They considered three different models of uncertainty and primarily studied the complexity of computing the stability probability of a given matching and the question of finding a matching that will have the highest probability of being stable. In contrast to their work, in this paper we do not make any underlying distributional assumptions about the preferences of the agents and instead take an absolute worst-case approach, which in turn implies that our results hold irrespective of the underlying distribution on the completions.

Finally, we also briefly mention another line of research which deals with partial information settings and goes by the name of interview minimization (see, for instance, ras13,dru14). One of the main goals in this line of work is to come with a matching that is stable (and possibly satisfying some other desirable property) by conducting as few ‘interviews’ (which in turn helps the agents in refining their preferences) as possible. We view this work as an interesting, orthogonal, direction from the one we pursue in this paper.

2 Preliminaries

Let and be two disjoint sets. The sets and are colloquially referred to as the set of men and women, respectively, and . We assume that each agent in and has a true strict linear order (i.e., a ranking without ties) over the agents in the other set, but this strict linear order may be unknown to the agents or they may be unwilling to completely disclose the same. Hence, each agent in and specifies a strict partial order over the agents in the other set (which we refer to as their preference order) that is consistent with their underlying true orders, and and , respectively, denote the collective preference orders of all the men and women. For a strict partial order associated with agent , we denote the set of linear extensions associated with by and denote by the Cartesian product of the s, i.e., . We refer to the set as “the set of all completions” where the term completion refers to an element in . Also, throughout, we denote strict preferences by and use to denote the relation ‘weakly-prefers’. So, for instance, we say that an agent strictly prefers to and denote this by and use to denote that either strictly prefers to or finds them incomparable. As mentioned in the introduction, we restrict our attention to the case when the strict partial orders submitted by the agents are strict weak orders over the set of agents in the other set.

Remark: Strict weak orders are defined to be strict partial orders where incomparability is transitive. Hence, although the term tie is used to mean indifference, it is convenient to think of strict weak orders as rankings with ties. Therefore, throughout this paper, whenever we say that agent finds and to be tied, we mean that finds and to be incomparable. Additionally, we will use the terms ties and incomparabilities interchangeably.

An instance of the stable marriage problem (SM) is defined as , where denotes the amount of missing information in that instance and this in turn, as defined in Section 1.2, is the average number of pairwise comparisons that are missing from the instance, and and are as defined above. Given an instance , the aim is usually to come up with a matching —which in turn is a set of disjoint pairs , where and —that is stable. There are different notions of stability that have been proposed and below we define two of them that are relevant to our paper: i) weak-stability and ii) super-stability. However, before we look at their definitions we introduce the following terminology that will be used throughout this paper. (Note that in the definitions below we implicitly assume that in any matching all the agents are matched. This is so because of the standard assumption that is made in the literature on SM (i.e., the stable marriage problem where every agent has a strict linear order over all the agents in the other set) that an agent always prefers to be matched to some agent than to remain unmatched.)

Definition 1 (blocking pair/obvious blocking pair).

Given an instance and a matching associated with , is said to be a blocking pair associated with if and . The term blocking pair is usually used in situations where the preferences of the agents are strict linear orders, so in cases where the preferences of the agents have missing information, we refer to such a pair as an obvious blocking pair.

Definition 2 (super-blocking pair).

Given an instance where the agents submit partial preference orders and a matching associated with , we say that is a super-blocking pair associated with if and .

Given the definitions above we can now define weak-stability and super-stability.

Definition 3 (weakly-stable matching).

Given an instance and matching associated with , is so said to be weakly-stable with respect to if it does not have any obvious blocking pairs. When the preferences of the agents are strict linear orders, such a matching is just referred to as a stable matching.

Definition 4 (super-stable matching).

Given an instance and matching associated with , is so said to be super-stable with respect to if it does not have any super-blocking pairs.

2.1 What problems do we consider?

As mentioned in the introduction, we are interested in finding the minimax optimal matching where the objective is to minimize the number of blocking pairs, i.e., to find, from the set of all possible matchings, a matching that has the minimum maximum number of blocking pairs with respect to all the completions. This is formally defined below.

Problem 1 (-minimax-matching).

Given a and an instance , where is the amount of missing information and are the preferences submitted by men and women respectively, compute where .

Although the problem defined above is our main focus, for the rest of this paper we will be talking in terms of the following problem which concerns itself with finding an approximately super-stable matching (i.e., a super-stable matching with the minimum number of super-blocking pairs). As we will see below, the reason we do the same is because both the problems are equivalent.

Problem 2 (-min-bp-super-stable-matching).

Given a and an instance , where is the amount of missing information and are the preferences submitted by men and women respectively, compute where and is the set of super-blocking pairs associated with for the instance .

Below we show that both the problems described above are equivalent. However, before that we prove the following lemma.

Lemma 1.

Let be a matching associated with some instance , denote the maximum number of blocking pairs associated with for any completion of , and denote the number of super-blocking pairs associated with for the instance . Then, .

Proof.

First, it is easy to see that if there are blocking pairs associated with for a completion, then there are at least as many super-blocking pairs associated with . Therefore, .

Next, we will show that if is the number of super-blocking pairs associated with , then has at least one completion such that it has number of blocking pairs associated with . To see this, for each and for each such that is a super-blocking pair, do the following:

  • if finds incomparable to , then construct a new partial order for such that it is the same as except for the fact that in we have that strictly prefers over .

  • if finds incomparable to , then construct a new partial order for such that it is the same as except for the fact that in we have that strictly prefers over .

Once the above steps are done, if there still exists any agent whose preference order is partial, then complete it arbitrarily. Now, consider this instance that is obtained. Then, again, it is easy to see that every which was a super-blocking pair associated with in forms a blocking pair in with respect to . Therefore, this completion has blocking pairs, and since the maximum number of blocking pairs in any completion is , we have that . Combining this with the case above, we have that . ∎

Given the lemma above, we can now prove our theorem.

Theorem 2.

For any , the -minimax-matching and -min-bp-super-stable-matching problems are equivalent.

Proof.

To see this, let be some instance, where , and be some matching associated with . We show that is an optimal solution for the -minimax-matching problem if and only if it is an optimal solution for the -min-bp-super-stable-matching problem.

Let us suppose that is not an optimal solution for the -min-bp-super-stable-matching problem. This implies that there exists some other such that . However, from Lemma 1 we know that the maximum number of blocking pairs associated with for any completion with respect to is equal to , which in turn contradicts the fact that was optimal for the -minimax-matching problem.

We can prove this analogously. ∎

For the rest of this paper, we assume that we are always dealing with instances which do not have a super-stable matching as this can be checked in polynomial-time [Theorem 3.4]irv94. So, now, in the context of the -min-bp-super-stable-matching problem, it is easy to show that if the number of super-blocking pairs in the optimal solution is a constant, then we can solve it in polynomial-time. We state this in the theorem below. Later, in Section 4, we will see that the problem is NP-hard, even to approximate.

Theorem 3.

An exact solution to the -min-bp-super-stable-matching problem can be computed in time, where is the number of super-blocking pairs in the optimal solution.

Proof.

We will describe the algorithm below whose main idea is based on the following observation.

For an instance , consider its optimal solution and let the super-blocking pairs associated with be . Next, for each such pair , put () at the end of ’s (’s) preference list (i.e., make every other man (woman), except those involved in another blocking pair with (), rank better than ()). If either of them are involved in multiple blocking pairs in , then make those partners as incomparable at the end of the preference list. Let us call the new instance . Notice that is super-stable with respect to as the pairs in are no longer blocking and no new blocking pairs are created because of our manipulations to the preference list.

Given the above observation, we can now describe the exponential algorithm.

  • Initially . Given a , try out every possible set of pairs of size to see if they are the right blocking pairs.

  • For each set generated in the previous step, modify the original instance to as described above and see if has a super-stable matching (this can be done in polynomial time). If yes, then return the super-stable matching as that is the solution. Otherwise, if none of the sets of size result in a “yes”, then go back to step 1 and try again with the next value of .

Now, it is easy to see that we end up with the optimal solution this way since we try all possible sets of blocking pairs. As for the time, we know that for each we have at most choices of sets and for each set we need at most time to do the necessary manipulations to the instance and to check for super-stability. Hence, the total time required is . ∎

3 Investigating weakly-stable matchings

In this section we focus on situations where obvious blocking pairs are not permitted in the final matching. In particular, we explore the space of weakly-stable matchings and ask whether it is possible to find weakly-stable matchings that also provide good approximations to the -min-bp-super-stable-matching problem (and thus the -minimax-matching problem).

3.1 Approximating -min-bp-super-stable-matching with weakly-stable matchings

It has previously been established that a matching is weakly-stable if and only if it is stable with respect to at least one completion [Section 1.2]man02. Therefore, given this result, one immediate question that arises in the context of approximating the -min-bp-super-stable-matching problem is “what if we just fill in the missing information arbitrarily and then compute a stable matching associated with such a completion?” This is the question we consider here, and we show that weakly-stable matchings do give a non-trivial (i.e., one that is , as any matching has only super-blocking pairs) approximation bound for our problem for certain values of . The proof of the following theorem is through a simple application of the Cauchy-Schwarz inequality.

Theorem 4.

For any and an instance where , any weakly-stable matching with respect to gives an -approximation for the -min-bp-super-stable-matching problem.

Proof.

Let be a weakly-stable matching associated with . By the definition of weakly-stable matchings we know that does not have any obvious blocking pairs. This implies that for every super-blocking pair associated with , either finds incomparable to his partner or finds incomparable to her partner . If it is the former then we refer to the super-blocking pair as one that is associated with and if not we say that it is associated with . Next, let us suppose that there are agents who have a blocking pair associated with them and let denote the number of super-blocking pairs associated with agent . So, now, the number of super-blocking pairs, , associated with can be written as

(1)

where refers to the length of largest tie associated with agent and the inequality follows from the definition of an association of a super-blocking pair with an agent.

Additionally, for each , we know that at least pairwise comparisons are missing with respect to (since has a tie of length ). Therefore, using the Cauchy-Schwarz inequality, we have that

(2)

Also, since the total amount of missing information in the instance is less than or equal to and since each (as it is a weak order and a tie, if it exists, is of length at least 2) we have that

(3)

Now, using Equation 3 and the fact that is also upper-bounded by (as there are only agents in the instance), we have that . Therefore, using Equation 2 and again using the fact that is the maximum amount of missing information, we have,

(4)

This in turn implies that if we solve for , we have,

So, now, we can use the fact that to see that

Finally, this along with Equation 1 gives our result since the number of super-blocking pairs in the optimal solution is at least 1 (since, as mentioned in Section 2.1, we are only considering instances that do not have a super-stable matching). ∎

3.2 Can we do better when restricted to weakly-stable matchings?

While Theorem 4 established an approximation factor for the -min-bp-super-stable-matching problem when considering only weakly-stable matchings, it was simply based on arbitrarily filling-in the missing information. Therefore, there remains the question as to whether one can be clever about handling the missing information and as a result obtain improved approximation bounds. In this section we consider this question and show that for many values of the approximation factor obtained in Theorem 4 is asymptotically the best one can achieve when restricted to weakly-stable matchings.

Theorem 5.

For any , if there exists an -approximation algorithm for -min-bp-super-stable-matching that always returns a matching that is weakly-stable, then . Moreover, this result is true even if we allow only one side to specify ties and also insist that all the ties need to be at the top of the preference order.

Proof.

At a very high-level, the key idea in this proof is to create an instance such that if we insist on there being no obvious blocking pairs, then this results in some kind of a “cascading effect”, thus in turn causing a very sharp blow-up in the number of super-blocking pairs. With this intuition, we first construct an instance as shown in Figure 1, where ties appear only on the women’s side. Furthermore, we define the following:

  • , (for simplicity we assume that and are integers; we can appropriately modify the proof if that is not the case)

  • ,

  • for some set , place all the women (men) with index in in the increasing order of their indices

  • for some set , place all the women (men) with index in as tied

  • place all the remaining alternatives in some strict order.


Men

  Women

 

Figure 1: The instance that is used in the proof of Theorem 5

Next, we will show that all the weakly-stable matchings associated with have super-blocking pairs, whereas the optimal solution has exactly one super-blocking pair. To do this, first note that the optimal solution associated with the instance is , where is the only super-blocking pair (and it is an obvious blocking pair). Also, it can be verified that the total amount of missing information in is at most . So, next, we prove the following claim.

Claim 1.

If is a weakly-stable matching associated with the instance , then .

Proof.

First, note that in any weakly-stable matching will always be matched to as otherwise it will result in an obvious blocking pair. Next, let us suppose that there exists an such that . Now, we will consider the following two cases and show that in both the cases this is impossible.

Case 1. is matched to a woman in : In this case one can see that forms an obvious blocking pair.

Case 2. is matched to a woman in : Note that if this is the case, then there is at least one such that is matched with a woman . Now, notice that forms an obvious blocking pair. ∎

Given the claim above, consider a man whose index is in some block and let his index value be . From the way the preferences are defined, it is easy to see that in any weakly-stable matching, will be matched to a woman whose index lies in the same block (because otherwise it will result in an obvious blocking pair). At the same time, from the claim above we know that this ’s index is not . Now, let us consider the woman who is the woman with the highest index value in and let denote the man who is matched to in a weakly-stable matching. From the observation above we know that has an index value in . Additionally, given the way the preferences are defined for and using the fact that any woman such that finds all the men in to be incomparable, one can see that forms super-blocking pairs (with all the women in except and the one with the same index value as ). Also, by using the same argument again, but with respect to , we can show that partner of in the matching forms at least super-blocking pairs (with all women except , , and the one with the same index). Continuing this way we see that each block contributes super-blocking pairs. And so, since there are blocks and for all , we have that there are super-blocking pairs in any weakly-stable matching. ∎

3.3 The case of one-sided top-truncated preferences: An approximation algorithm for -min-bp-super-stable-matching

Although Theorem 5 is an inherently negative result, in this section we consider an interesting restriction on the preferences of the agents and show how this negative result can be circumvented. In particular, we consider the case where only agents on one side are allowed to specify ties and all the ties need to be at the bottom. Such a restriction has been looked at previously in the context of matching problems and as noted by Irving and Manlove [irv08] is one that appears in practise in the Scottish Foundation Allocation Scheme (SFAS). Additionally, restricting ties to only at the bottom models a very well-studied class of preferences known as top-truncated preferences, which has received considerable attention in the context of voting (see, for instance, bau12).

Top-truncated preferences model scenarios where an agent is certain about their most preferred choices, but is indifferent among the remaining ones or is unsure about them. More precisely, in our setting, the preference order submitted by, say, a woman is said to be a top-truncated order if it is a linear order over a subset of and the remaining men are all considered to be incomparable by . In this section we consider one-sided top-truncated preferences, i.e., where only men or women are allowed to specify top-truncated orders, and show an -approximation algorithm for -min-bp-super-stable-matching under this setting. (Without loss of generality we assume throughout that only the women submit strict weak orders.) Although arbitrarily filling-in the missing information and computing the resulting weakly-stable matching can lead to an -approximate matching even for this restricted case (see Appendix A for an example), we will see that not all weakly-stable matchings are “bad” and that in fact the -approximate matching we obtain is weakly-stable.

However, in order to arrive at this result, we first introduce the following problem which might be of independent interest. (To the best of our knowledge, this has not been previously considered in the literature.) Informally, in this problem we are given an instance and are asked if we can delete some of the agents to ensure that the instance, when restricted to the remaining agents, will have a perfect super-stable matching.

Problem 3 (min-delete-super-stable-matching).

Given an instance , where is the amount of missing information and are the preferences submitted by men and women respectively, compute the set of minimum cardinality such that the instance , where , has a perfect super-stable matching (i.e., every agent in is matched in a super-stable matching).

Below we first show a 2-approximation for the min-delete-super-stable-matching problem when restricted to the case of one-sided top-truncated preferences. Subsequently, we use this result in order to get an -approximation for our problem. However, before that, we introduce the following terminology which will be used throughout in this section.

  • An instance of the min-delete-super-stable-matching problem can also be thought of as the set of agents along with their preference lists. Initially for every agent this list has all the agents in the other set listed in some order. Now, during the course of our algorithm sometimes we use the operation “delete” which removes agent from ’s list and from ’s. After such a deletion (or after a series of such deletions) our instance now refers to the set of agents along with their updated lists.

  • We say that a matching is internally super-stable with respect to an instance if is super-stable with respect to the instance that is obtained by only considering the matched agents in (i.e., consider and remove all the agents who are not matched in from ).

  • We say that an instance with no ties has an exposed rotation if, in , is the first agent in ’s list and is the second agent in ’s list (here is done modulo ).

1:proposeWith 2:assign each agent to be free 3:while some is free do 4:       first agent on ’s list 5:      if  is already engaged to agent && finds and incomparable then 6:            delete 7:      else 8:            if  is already engaged to agent  then 9:                 assign to be free 10:            end if 11:            assign and to be engaged 12:            for each agent in ’s list such that  do 13:                 delete 14:            end for 15:      end if 16:end while 17:for each man  do 18:       first woman on ’s list 19:      if there exists a man such that finds and incomparable then 20:            delete 21:      end if 22:end for deletions in this loop only happen once and results in the removal of all the remaining ties 23:return this returns the updated lists 24: 25:a one-sided top-truncated instance 26: 27: 28:while there exists some exposed rotation in  do 29:      delete for all 30:       31:       32:end while 33: for all men , match with the only woman in his list 34:construct where , if is a super-blocking pair in w.r.t.  35: minimum vertex cover of 36:for each  do 37:       38:end for 39:return

Algorithm 1: For the case of one-sided top-truncated preferences, the set returned by the algorithm is a 2-approximation for the min-delete-super-stable-matching problem and the matching returned is an -approximation for -min-bp-super-stable-matching
Proposition 6.

Algorithm 1 is a polynomial-time 2-approximation algorithm for the min-delete-super-stable-matching problem when restricted to the case of one-sided top-truncated preferences.

Proof.

The main idea for Algorithm 1 is inspired by the work of tan90 who looked at the problem of finding the maximum internally stable matching for the stable roommates problem (which is equivalent to the problem of finding the minimum number of agents to delete so that the rest of the agents will have a stable matching when the instance is just restricted to themselves). Informally, at a very high level, the key idea in tan90’s algorithm was to show that some of the entries in each agent’s list can be deleted by running the proposal-rejection sequence like in Gale-Shapley algorithm and through rotation eliminations, while at the same time maintaining at least one solution of the maximum size. As we will see below, this is essentially what we do here as well, adapting this idea as necessary for our case when there are ties on one side but only at the bottom.

Before we go on to the main lemmas, let us suppose that is an arbitrary instance of the min-delete-super-stable-matching problem when restricted to the case of one-sided top-truncated preferences, where is the amount of missing information, are the preference orders submitted by the men and women, respectively, and . Also, let be the optimal solution for this instance. This in turn implies that we can form a perfect and internally super-stable matching of size , and that in fact is the maximum size of any such matching (as otherwise cannot be optimal). Next, for now, let us assume the correctness of the following lemmas (note that all the instances we talk about in this section are restricted to the case of one-sided top-truncated preferences). We will prove them later, in Sections 3.3.1, 3.3.2, and 3.3.3.

Lemma 7.

Let denote some instance and denote the instance returned by the procedure proposeWith), where the set represents the proposing side. If there exists a matching of size in that is internally super-stable with respect to , then there exists a matching of size in that is internally super-stable with respect to .

Lemma 8.

Let denote some instance that does not contain any ties, be a rotation that is exposed in , and be the instance that is obtained by deleting the entries for all from . If there exists a matching of size in that is internally super-stable with respect to , then there exists a matching of size in that is internally super-stable with respect to .

Lemma 9.

If is an instance that does not contain any exposed rotation, then the list of every man in has only one woman and vice versa.

Given the above lemmas and given the fact that the instance has an internally super-stable matching of size , we can start with the instance and repeatedly apply Lemma 7 and see that the instance that we obtain after line 27 of Algorithm 1 has a matching of size that is internally super-stable matching with respect to . Next, starting with the instance , we can again repeatedly apply Lemma 8 and see that the instance that remains after line 32 of Algorithm 1 also has a matching of size that is internally super-stable matching with respect to . Additionally, from Lemma 9 we know that in this instance each man has only one woman in his list and vice versa. So, now, consider the matching that can be obtained by matching each man with the only woman in his list. Next, consider the bipartite graph where and if is a super-blocking pair with respect to the original instance , and consider a minimum vertex cover of . We show below that .

Suppose this is false and that has an internally super-stable matching of size greater than . Now, from the discussion above, we know that this implies also has a matching of size greater than such that it is internally super-stable with respect to . This in turn implies that we can remove less than agents and have a matching that is internally super-stable with respect to . That is, we can remove less than agents and also at the same time ensure that for every , this matching has only one of or matched in it, for if otherwise it will not be internally super-stable with respect to . However, this implies that is not the size of the minimum vertex cover of , and hence we have a contradiction.

Now, to get our approximation bound, consider the set that is returned by the algorithm. We know that , where the first inequality arises because of lines 36-38 in Algorithm 1 and the second inequality uses the observation above that . Finally, it is easy to see that all the steps can be done in polynomial-time (it is well-known through the Kőnig’s Theorem that one can find a minimum vertex cover of a bipartite graph in polynomial-time). ∎

Given Proposition 6, we can now prove the following theorem.

Theorem 10.

For any , Algorithm 1 is a polynomial-time -approximation algorithm for the -min-bp-super-stable-matching problem when restricted to the case of one-sided top-truncated preferences. Moreover, the -approximate matching that is returned is also weakly-stable.

Proof.

Consider an arbitrary instance of the -min-bp-super-stable-matching problem when restricted to the case of one-sided top-truncated preferences. Let be the optimal solution associated with . Next, consider the same instance for the min-delete-super-stable-matching problem and let us consider the matching that is returned by Algorithm 1 for this instance. Also, let be the optimal solution of the min-delete-super-stable-matching problem for the instance and be the set that is returned by Algorithm 1. (We can assume throughout that , for if otherwise this implies that it has a super-stable matching, and as mentioned in Section 2.1 we do not consider such instances.) First, it is easy to see that this matching is weakly-stable (because in every instance that results after the first proposal-rejection sequence (which is in line 23), a matching that is formed by matching each man with the first woman on his list will be weakly-stable). Second, note that we can rewrite as , where and . Now, if () denotes the number of super-blocking pairs associated with men in (), then

(5)

where the second step is using the fact that the men in can form at most super-blocking pairs with women outside of and the men in can in the worst case form a blocking pair with all the women, and the last step is using the fact that and that , which we know is true by Proposition 6.

Also, if is the optimal solution for the -min-bp-super-stable-matching problem, then we know that

(6)

as otherwise one can delete all the men who are involved in super-blocking pairs in and their corresponding partners and get a super-stable matching on the remaining agents.

Finally, using Equations 3.3 and 6 we have our theorem. ∎

3.3.1 Proof of Lemma 7

Let us first prove the following claims. However, before that we introduce the following terminology. When the procedure proposeWith is executed, for every run of the while loop in line 3 with respect to an agent , we can track the instance that is currently being used. That is, initially we have the instance and this is referred to as the instance that is “currently being used” by the agent where is the first agent with respect whom the while loop is executed. Now, after the first run of the while loop (w.r.t. ) we have an updated instance (because of some delete operations that happened in lines 4-15), say, . Therefore, the next time the while loop is run with respect to some agent , this is the instance that is “currently being used” with respect to . We use this terminology in the following claim.

Claim 2.

Let be an agent who is assigned to be ‘free’ in the procedure proposeWith, denote the instance that is currently being used with respect to , and be the instance obtained by running lines 4-15 with respect to agent . If there exists a matching of size in that is internally super-stable with respect to , then there exists a matching of size in that is internally super-stable with respect to .

Proof.

First, note that every time the procedure proposeWith() is called with the set , the agents in this set do not have any ties. This is because, when proposeWith() is called with the set of women for the first time in line 27, all ties are already broken due to the first call of proposeWith() with the set of men in line 26. Therefore, in all the arguments below we do not need to concern ourselves with issues that can arise as a result of the agents in having ties.

Now, to prove this claim we consider the following two cases separately.

  1. when , the first agent in ’s list, is either engaged to , but prefers to , or is not engaged currently

  2. when is engaged to and finds and incomparable

Case 1: In this case the only change that happens to the instance are due to deletions of the form where . Let the resulting instance be . So, in order to prove that has a matching of size that is internally super-stable with respect to , we just need to prove that there exists a matching of size in that is internally super-stable matching with respect to and does not have any matches of the form . Now, let us suppose that’s not the case and that in every matching of size in that is internally super-stable with respect to there exists such a pair. This in turn implies that is unmatched in , for if otherwise will form an obvious-blocking pair. Hence, we can form the matching . Note that is internally-stable with respect to as this does not introduce any new super-blocking pair (since is the first agent on ’s list and does not block any other agent in as he prefers the new partner over his old partner ) and has the same size as . This in turn is a contradiction and hence we have that has a matching of size that is internally super-stable with respect to .

Case 2: In this case, the only change that happens to the instance is the deletion of . Let the resulting instance be . So, in order to prove that has a matching of size that is internally super-stable with respect to , we just need to prove that there exists a matching of size in that is internally super-stable matching with respect to and does not have any matches of the form . Suppose that’s not the case and that every matching of size in that is internally super-stable matching with respect to has . This in turn implies that is unmatched in , for if otherwise will form a super-blocking pair (since is the first agent on ’s list and finds and incomparable). Hence, we can now form the matching , which is internally super-stable with respect to as this does not introduce any new super-blocking pair (since i) is the first agent on ’s list and ii) does not block any other agent in as it finds and incomparable and so if he blocks someone now he also blocked them when was present, thus contradicting the fact that was internally super-stable) and has the same size as . This in turn is a contradiction and hence, again, we have that has a matching of size that is internally super-stable with respect to . ∎

Next, we prove the following claim whose proof is omitted since it can be proved in a way that is similar to Case 2 in the proof of Claim 2.

Claim 3.

Let be a man, be the first woman on ’s list, be some instance obtained after line 16, and be the instance obtained after deleting each in line 20. If there exists a matching of size in that is internally super-stable with respect to , then there exists a matching of size in that is internally super-stable with respect to .

Given the two lemmas above, we are now ready to prove Lemma 7. First, starting with and by repeatedly using Claim 2 with respect to each free agent , we can see that the instance that we obtain at the end of the while loop (i.e., line 16) has a matching of size that is internally super-stable matching with respect to the initial instance . Second, starting with