CANCELACIONES DE REGISTROS

in aj’s preference list and vice versa. A matching M is a set of acceptable pairs where

each agent appears in at most one pair. The definition of a blocking pair also changes slightly. An acceptable pair {ai, aj} forms a blocking pair with respect to M if ai is

unmatched or prefers aj to M (ai), and aj is unmatched or prefers ai to M (aj). The

algorithm for finding a stable matching given an sr instance (should one exist) can be extended to the sri case [38, Section 4.5.2]. Figure 2.6 shows a sample sri instance I with stable matching M = {{a1, a2}, {a3, a4}}.

a1 : a2 a4 a3

a2 : a1 a3

a3 : a2 a1 a4

a4 : a3 a1

Figure 2.6: An instance of the Stable Roommates problem with Incomplete Lists

2.5

Locally Stable Matchings

2.5.1

Introduction

Although the concept of stability is important in many applications of matching prob- lems, there are classes of matching problems (such as the Stable Roommates problem) for which an instance is not guaranteed to admit a stable matching [29]. Moreover, enforcing the stability requirement tends to reduce the size of the matchings discovered [15]. This is an issue particularly in the case of applications where it is desired to find a largest possible matching. It may be argued that in some applications where the size of the matching produced is just as important (if not more important) than the stability of the matching, relaxing the stability definition to allow for larger matchings is justified.

One approach taken was to weaken the stability criterion by tolerating a number of blocking pairs. The smaller the number of blocking pairs, the more stable a matching is said to be. The problem then becomes to find a maximum size matching with as few blocking pairs as possible. Such a model of “almost” stable matchings was considered in [15]. The authors identified a number of practical applications in which such an approach would be feasible. They defined the problem of finding a maximum matching with the minimum number of blocking pairs given an smi instance as max size min bp smi. They showed that the problem is NP-hard to approximate to within n1−ε for any ε > 0 where n is the number of men in a given instance. They also showed the

2.5. Locally Stable Matchings 22

problem to be NP-hard to approximate to within δ for some δ > 1 even if all preference lists are of length at most 3. They then provideed a polynomial-time algorithm for the special case in which the preference lists of one sex are of length at most 2. Hamada et al. strengthened this inapproximability bound to within n1−ε even if all preference lists are of length at most 3 [42].

Another approach is to leverage any information about the social relationship between agents in the matching market. Instead of trying to minimise the number of blocking pairs, we may redefine the stability criterion to allow for certain blocking pairs that are unlikely to lead to a matching being undermined. It is generally assumed that, in the hr context, a resident-hospital pair that blocks a matching in theory will also block the matching in practice. However this assumption is not always true in some real-world applications, as resident-hospital pairs are more likely to form blocking pairs in practice if social ties exist between them. This observation, coupled with the need to find the largest possible matchings, have motivated studies into alternative, weaker stability definitions that still aim to prevent a given matching from being subverted while increasing the number of agents involved in the matchings. If social ties do not exist between a pair of agents, it is assumed that they are unlikely to subvert a matching in practice as they may not become aware that they could form a blocking pair with one another in reality. They are therefore not considered as blocking pairs. Redefining stability on the bases of agents who are most likely to form blocking pairs in practice can yield larger matchings whilst still providing a degree of robustness against a matching being undermined. This section discusses this concept, which we call social stability (also referred to as local stability in the literature), in the context of sm and hr as well as similar ideas described in the literature.

2.5.2

Locally Stable Matchings in the Job Market Context

Arcaute and Vassilvitskii [11] studied the Hospitals/Residents problem in the context of assigning job applicants to company positions. They observed that applicants are more likely to be employed by a company if they are recommended by their friends who are already employees of that company. Given the large amount of applications that may be submitted for vacancies, companies are increasingly reliant on personal in-house recommendations for making employment decisions. In their model applicants have a limited knowledge of the matching market and can only form blocking pairs with com- panies in which their friends are employees. In the context of stable matching theory, an applicant-company pair (a, c) blocks a matching M if (a, c) blocks it in the tradi- tional sense (as described in the analogous hr context) and a is friends with another applicant a0 _{assigned to c in M . Thus their problem (which was later called hr+sn}

2.5. Locally Stable Matchings 23

by Cheng and McDermid [22]) incorporates both the traditional hr problem and ad- ditionally an underlying social network, represented as an undirected graph consisting of applicants as nodes and edges between nodes where the corresponding applicants have some social ties (e.g., are friends). They call matchings that admit no blocking pair in this context locally stable due to the addition of the informational constraint on blocking pairs. Concerning the static properties of locally stable matchings, Arcaute and Vassilvitskii [11] showed that the set of locally stable matchings does not form a distributive lattice. They also considered a dynamic version of their model (where ap- plicants and/or companies arrive and are matched over time), providing a decentralised version of the Gale-Shapley algorithm for finding a locally stable matching.

Cheng and McDermid [22] investigated the problem further and presented some algo- rithmic results. They showed that locally stable matchings can be of different sizes and max hr+sn the problem of finding a maximum locally stable matching is NP-hard. They identified special cases where the problem is polynomially solvable and gave upper and lower bounds on the approximability of the problem. They showed that when the social network is a complete graph, local stability and classic stability become the same thus finding a maximum locally stable matching is polynomially solvable. If the social network is an empty graph, then every maximum cardinality matching is a locally sta- ble matching thus finding a maximum locally stable matching is also computationally easy. Concerning the approximability of max hr+sn, they observed that every stable matching in the classical sense is also a locally stable matching and is at least 1/2 the size of a maximum locally stable matching. They also proved that it is NP-hard to approximate max hr+sn to within 21/19 − δ for any δ > 0.

The problem of finding an approximation algorithm for max hr+sn with a perfor- mance guarantee better than 2 still remains open. Given the close relationship between max hr+sn and max hrt and the fact that max hrt can be approximated to within 3/2, Cheng and McDermid conjectured that a 3/2 approximation algorithm for max hr+sn also exists. They proposed a strategy similar to those used for max hrt in [75, 90]. They concluded the paper by describing a special case where social ties exist only between applicants who are matched in every stable matching and applicants who are not. In this case every stable matching is a 3/2 approximation of a maximum locally stable matching. Askalidis et al. [13] suggested that the elusiveness of a 3/2 approximation algorithm for max hr+sn is due to the fact that the model is dynamic in nature – blocking pairs are not determined from the problem instance alone but change depending on the matching being considered.

2.6. The Student/Project Allocation problem 24