Optimizing a Generalized Gini Index in Stable Marriage Problems: NP-Hardness, Approximation and a Polynomial Time Special Case

Optimizing a Generalized Gini Index in Stable Marriage Problems: NP-Hardness, Approximation and a Polynomial Time Special Case

Hugo Gilbert H. Gilbert O. Spanjaard Sorbonne Université, CNRS, Laboratoire d’Informatique de Paris 6, LIP6, F-75005 Paris, France
33email: {hugo.gilbert,olivier.spanjaard}@lip6.fr
   Olivier Spanjaard H. Gilbert O. Spanjaard Sorbonne Université, CNRS, Laboratoire d’Informatique de Paris 6, LIP6, F-75005 Paris, France
33email: {hugo.gilbert,olivier.spanjaard}@lip6.fr
Abstract

This paper deals with fairness in stable marriage problems. The idea studied here is to achieve fairness thanks to a Generalized Gini Index (GGI), a well-known criterion in inequality measurement, that includes both the egalitarian and utilitarian criteria as special cases. We show that determining a stable marriage optimizing a GGI criterion of agents’ disutilities is an NP-hard problem. We then provide a polynomial time 2-approximation algorithm in the general case, as well as an exact algorithm which is polynomial time in the case of a constant number of non-zero weights parametrizing the GGI criterion.

Keywords:
Stable marriage problem Fairness Generalized Gini index Complexity
journal:

1 Introduction

Since the seminal work of Gale and Shapley [1962] on stable marriages, matching problems under preferences have been extensively studied both by economists and computer scientists. These problems involve two sets of agents (also called individuals in the sequel) that should be matched with each other while taking agents’ preferences into account. The results obtained in the field have a tremendous number of applications, among which the National Resident Matching Program in the US (for allocating junior doctors to hospitals), the teacher allocation in France (for allocating newly tenured teachers to schools) or the allocation of lawyers in Germany (for assigning graduating lawyers to legal internship positions). For an overview of the applications of matching models under preferences, the interested reader can refer to a recent book chapter on this topic (Biró, 2017).

The stable marriage problem involves men and women, each of whom ranks the members of the opposite sex in order of preference. The goal is to find a stable matching, i.e., a matching between men and women such that there is no man and woman that prefer each other to their current match. Gale and Shapley [1962] provided an algorithm that computes a stable marriage. However, it is well-known that this algorithm favours one group (men or women, according to the way the algorithm is applied) over the other.

We are interested here in fair stable marriage algorithms, i.e., in procedures favouring stable marriages that fairly share dissatisfactions –also called disutilities– among individuals (irrespective of their sex), the dissatisfaction being defined for each woman (resp. man) as a function of the rank, in order of preferences, of the man (resp. woman) to whom she is paired with. Given the vector of individuals’ dissatisfactions induced by a matching, there are several ways of formalizing the notion of “fairness”. We mean here by fair stable marriage that the vector of individuals’ dissatisfactions should be well-balanced. For example, consider the following instance of the stable marriage problem.

Example 1

The instance consists of 10 men and women with the following preferences, where (resp. ) means that (resp. ) prefers to (resp. to ):

The stable marriages in this instance are:

where a pair means that and are matched.

If one assumes that the dissatisfaction of an individual is equal to the rank of the partner in his/her preference list, then the dissatisfactions induced by the previous stable marriages are:

matching vector of dissatisfactions sum of dissatisfactions max of dissatisfactions
61 10
63 10
63 10
65 10
74 9

where the component of the vector is the dissatisfaction of for , and of for .

In this instance, the matching can be considered as inducing a well-balanced vector of dissatisfactions. The matchings , and indeed favour more some individuals (the men in this case) than others, while matching yields quite high dissatisfactions for numerous agents. The matching is therefore a good compromise between the utilitarian and the egalitarian viewpoints, where the utilitarian viewpoint aims at minimizing the sum of dissatisfactions while the egalitarian viewpoint aims at minimizing the dissatisfaction of the worst off individual. Both the utilitarian and egalitarian approaches have been advocated for promoting fairness in the stable marriage problem (Gusfield, 1987; Gusfield and Irving, 1989). Other approaches aim at treating equally men and women, by minimizing the absolute difference between the total dissatisfactions of the two groups (sex-equal stable marriage problem (Kato, 1993; McDermid and Irving, 2014)) or by minimizing the maximum total dissatisfaction between the two groups (balanced stable marriage problem (Manlove, 2013)). However, note that, in the instance of Example 1, all these criteria favour either (utilitarian) or (egalitarian, sex-equal, balanced). Finally, there exists another type of approach, that is not based on assigning scores to marriages. In a first step, for each man, one lists all his possible matches in a stable marriage, in order of his preferences (this list includes as many elements as there are feasible stable marriages). In a second step, each man is matched with the median woman in the list. This procedure yields a stable marriage, which is called median stable marriage (Teo and Sethuraman, 1998; Cheng, 2010). In the instance of the example, the median stable marriage is . Nevertheless, in this article, we focus on determining a fair stable marriage by using a scoring rule.

In social choice theory, a scoring rule assigns a score to each alternative by summing the scores given by every individual over the alternative. This summation principle ensures that all individuals contribute equally to the score of an alternative. An alternative is usually a candidate in an election, but it can also be an element of a combinatorial domain. For instance, in proportional representation problems (Procaccia et al., 2008), where one aims at electing a committee, every feasible committee is an alternative. In the setting of stable marriage problems, every stable marriage is an alternative and the utilitarian approach is clearly a scoring rule where each individual evaluates a stable marriage by the rank of his/her match. An interesting extension of the class of scoring rules is the class of rank dependent scoring rules (Goldsmith et al., 2014), where, instead of limiting the aggregation to a summation operation, the scores are aggregated by taking into account their ranks in the ordered list of scores. As emphasized by Goldsmith et al. (2014), rank dependent scoring rules can be used to favour fairness by imposing some conditions on their parameters. A well known class of rank-dependent scoring rules in inequality measurement are the Generalized Gini Indices (GGI) (Weymark, 1981). Furthermore, this class of rank dependent scoring rules circumvents both the utilitarian and egalitarian criteria. Their optimization on combinatorial domains have been studied in several settings (often under the name of Ordered Weighted Averages): assignment problems (Lesca et al., 2018), proportional representation (Elkind and Ismaili, 2015), resource allocation (Heinen et al., 2015). To the best of our knowledge, the problem of determining a GGI optimal stable marriage has not been studied yet. This is precisely the purpose of the present work.

The paper is organized as follows. In Section 2, we introduce notations and we formally define the GGI stable marriage problem studied here. Then, in Section 3, we prove that it is NP-hard to determine an optimal stable marriage according to a GGI criterion applied to agents’ disutilities. In Section 4, we provide a polynomial time 2-approximation algorithm. Finally, in Section 5, we establish a parametrized complexity result with respect to a GGI-specific parameter.

2 The GGI Stable Marriage Problem

Let denote the set of men, and the set of women. As in Example 1, for each (resp. ), a preference relation (resp. ) is defined on (resp. ), where (resp. ) means that (resp. ) prefers to (resp. to ). We denote by the rank of woman in the preference order of man , and similarly for .

A solution of a stable marriage problem is a matching represented by a binary matrix , where means that is matched with . A matching induces a matching function defined by and if . In a perfect matching (called indifferently matching or marriage from now on), every man (resp. woman) is matched with a different woman (resp. man). More formally, a matching is defined by:

(1)
(2)

A matching is said to be stable if there exists no man and woman who prefer each other to their current partner. More formally, a perfect matching is stable if the following constraints hold (Vande Vate, 1989):

(3)

The set of stable marriages, i.e. binary matrices such that constraints 1, 2 and 3 hold, is denoted by . In their seminal paper, Gale and Shapley (1962) states that there always exists at least one stable marriage, which can be computed in .
The Gale-Shapley algorithm is based on a sequence of proposals from men to women. Each man proposes to the women following his preference order, pausing when a women agrees to be matched with him but continuing if his proposal is rejected. When a woman receives a proposal, she rejects it if she already has a better proposal according to her preferences. Otherwise, she agrees to hold it for consideration and rejects any former proposal that she might had. Such a sequence of proposals always leads to a stable marriage called man-optimal stable marriage and denoted by (if the role of men and women is reversed, we obtain the woman-optimal stable marriage denoted by ). In the man-optimal stable marriage, each man has the best partner, and each woman has the worst partner, that is possible in any stable marriage. Contrarily, in the woman-optimal stable marriage, each woman has the best partner, and each man has the worst partner, that is possible in any stable marriage.

Two important properties of the Gale-Shapley algorithm are that:
– if proposes to , then there is no stable marriage in which has a better match than .
– if proposes to , then there is no stable marriage in which has a worse match than .
These properties justify the notion of preference shortlists obtained through the Gale-Shapley algorithm by removing any man from a woman ’s preference list and vice-versa, when receives a proposal from a man she prefers to . Note that the shortlists that are obtained at the end of the algorithm do not depend on the order in which the proposals are made.

Example 2

For instance, with the preferences of Example 1, the Gale-Shapley algorithm leads to the following shortlists:

These shortlists makes it possible to identify some transformations that can be applied from the man-optimal stable marriage to obtain other stable marriages (more favourable to women). These transformations are called rotations (Irving and Leather, 1986). A rotation is a sequence of man-woman pairs such that, for each (), (1) is first in ’s shortlist and (2) ( taken modulo ) is second in ’s shortlist. Such a rotation is said to be exposed in the shortlists.

Example 3

Continuing Example 1, there are two rotations exposed in the shortlists, and .

Given a rotation, if each exchanges his current partner for , then the matching remains stable. Eliminating a rotation amounts to removing all successors of in ’s shortlist together with the corresponding appearances of in the shortlists of men . The obtained stable marriage can then be read from the modified shortlists by matching each man with the first woman in his shortlist. In this new stable marriage, each woman (resp. man) is better off (resp. worse off) than before eliminating the rotation.

Once an exposed rotation has been identified and eliminated, then one or more rotations may be exposed in the resulting (further reduced) shortlists. This process may be repeated, and once all rotations have been eliminated, we obtain the woman optimal stable marriage. A rotation is said to be a predecessor of a rotation , denoted by , if cannot be exposed in the men shortlists before is eliminated. This notion of predecessors makes it possible to define what is called the rotation poset where is the set of all rotations and is the precedence relation that we have just mentioned. A closed set in a poset is a subset of such that .

The following theorem is crucial to understand the importance of the rotation poset.

Theorem 1

(Irving and Leather, 1986) The stable marriages of a given stable marriage instance are in one-to-one correspondence with the closed subsets of the rotation poset.

In this correspondence, each closed subset represents the stable marriage obtained by eliminating the rotations in starting from .

The rotation poset can be represented as a directed acyclic graph, with the rotations as nodes and an arc from to iff is an immediate predecessor of (i.e., and there is no rotation such that ). Note that this graph has at most nodes, i.e., there are at most rotations (Irving et al., 1987). Indeed, there are at most pairs that can be involved in rotations (the pairs of cannot be involved in a rotation). Each pair belong to at most one rotation and there are at least two pairs in each rotation. We will take advantage of the rotation poset in multiple places in the paper. Importantly, note that the rotation poset (actually a subgraph whose transitive closure is the rotation poset) can be generated in (Gusfield and Irving, 1989).

Example 4

For instance, with the preferences of Example 1, the rotations and their immediate predecessors are given in the following table.

Rotation New pairs Immediate predecessors

Figure 1: Rotation poset in Example 1.

This rotation poset shows that there are (potentially many) other stable marriages than the man-optimal or woman-optimal stable marriages. These other stable marriages are likely to be fairer than and as they are both extreme cases. In order to compute a fair stable marriage, the optimization of several aggregation functions has been investigated.
– Utilitarian approach: , which can be minimized in (Feder, 1994)).
– Egalitarian approach: , which can also be minimized in (Gusfield, 1987).
– Sex-equal stable marriage: , the minimization of which is NP-hard (Kato, 1993).
– Balanced stable marriage: , the minimization of which is NP-hard (Manlove, 2013).

Our contribution differs with previous works on the fair stable marriage problem. Indeed, we optimize a generalized Gini index on disutility values.

Given a matching , the disutility (also called dissatisfaction) of a man is defined by , where , is a strictly increasing function called disutility function. The disutility values are defined similarly for women. Every stable marriage induces therefore a disutility vector:

with components. Note that the use of disutility values (often called weights) is a common way to extend the traditional framework where the aggregation function is applied on rank values (see e.g., Teo and Sethuraman (1998); Gusfield and Irving (1989)). Using a unique disutility function for all agents guarantees that they all have the same importance in the aggregation operation. Indeed, the disutility values assigned to the ranks do not depend on the agent’s identity. Note that both the egalitarian and the utilitarian variants of the stable marriage problem remain polynomially solvable if one uses disutility values.

Example 5

We come back to Example 1. Let be the disutility function defined by , then the disutility values are given by the matrices and below where (resp. ) is the disutility of (resp. ) if he (resp. she) is matched with (resp. ).

Let denote a disutility vector. The generalized Gini index (Weymark, 1981) is defined as follows:

Definition 1

Let be a vector of weights such that . The aggregation function induced by is defined by:

where denotes the vector ordered by nonincreasing values, i.e., .

The weights of the GGI aggregation function may be defined in a variety of manner. For instance, the weights initially proposed for the Gini social-evaluation function are:

(4)
Example 6

Coming back to Example 1, if the weights are defined by Equation 4 and the disutility function is defined by , the GGI values of the different stable marriages are (the lower the better):

matching ordered vectors
4.4525
4.4725
4.4725
4.3925
4.74

We thus observe that using a GGI aggregation function makes it possible to obtain as an optimal stable marriage.

The GGI is also known in multicriteria decision making under the name of ordered weighted average (Yager, 1988). This aggregation function, to minimize, is well-known to satisfy the Pigou-Dalton transfer principle if :

Definition 2

An aggregation function satisfies the transfer principle if for any and where :

This condition states that the overall welfare should be improved by any transfer of disutility from a “less happy” agent to a happier agent given that this transfer reduces the gap between the disutilities of agent and . We can now define the GGI Stable Marriage problem.
GGI Stable Marriage (GGISM) INSTANCE: Two disjoint sets of size , the men and the women; for each person, a preference list containing all the members of the opposite sex; a vector of weight parameters and a disutility function . SOLUTION: A stable marriage . MEASURE: (to minimize).

3 Complexity of the GGISM Problem

The GGISM problem extends both the egalitarian and the utilitarian approaches to the stable marriage problem. Indeed, if the weights of the GGI operator are , one obtains the sum operation. If the weights are , one obtains the max operation. While both variants are polynomially solvable problems, the following result states that the GGISM problem is NP-hard:

Theorem 2

The GGISM problem is NP-hard.

Proof

We make a reduction from Minimum 2-Satisfiability, which is strongly NP-hard (Kohli et al., 1994).

Minimum 2-Satisfiability (Min 2-SAT): INSTANCE: A set of variables, a collection of disjunctive clauses of at most 2 literals, where a literal is a variable or a negated variable in . SOLUTION: A truth assignment for . MEASURE: Number of clauses satisfied by the truth assignment (to minimize).

To illustrate the reduction, we will use the following 2-SAT instance:

(5)
(6)

As a preliminary step, note that we can get rid of variables that are present in only one clause. Such a variable is set to true if it is present as a negative literal in the clause and to false otherwise. It can then be removed from the instance. Furthermore, we can make sure that there are exactly two literals in each clause (by duplicating literals). For example, the instance described by Equations 5 and 6 can be modified to:

(7)
(8)

In the following we will denote by the number of variables and by the number of clauses. In the previous example and . Furthermore, we will denote by the clause in .

We are now going to create an instance of the GGISM problem such that:

  • There is a one-to-one correspondence between the stable marriages and the truth assignments for .

  • A stable marriage minimizing the GGI of the agent’s disutilities corresponds to a truth assignment of minimizing the number of clauses that are satisfied.

In order to create a one-to-one correspondence between the stable marriages and the truth assignments for , we are going to create a rotation for each variable . Each of these rotations will be exposed in the shortlists from the man-optimal stable marriage for the instance under construction. Additionally, we will ensure that these rotations will be the only ones of the stable marriage instance. In other words, the rotation poset will have one vertex per variable and no edge, as illustrated in Figure 2.

Figure 2: Rotation poset of the stable marriage instance generated by the reduction.

We now give the “meaning” of these rotations. Let’s recall that in a stable marriage there is a one-to-one correspondence between the closed subsets of nodes of the rotation poset and the stable marriages. Now let be a stable marriage corresponding to a closed subset of rotations, then the corresponding truth assignment over consists in setting if and otherwise. Thus in the generated stable marriage instance, the man-optimal stable marriage (i.e., ) corresponds to a truth assignment where all variables in are set to 0 while the woman-optimal stable marriage (i.e., ) corresponds to a truth assignment where all variables in are set to 1.

We now describe more precisely the fashion in which rotations are generated. For each variable , we create a man-woman pair for each clause that involves either as a positive or negative literal. If variable is present two times in a clause , then two man-woman pairs and are created. This induces the creation of men and women in the instance. The rotation then involves all the men and women induced by variable . For example, in the instance described by Equations 7 and 8, involves men and women as variable is present in and . Let denote the number of times variable appears in . The rotation is then induced by the following patterns in the shortlists of men and women :

For instance, rotation is induced by the following pattern in the shortlists: