LEGAJO ADMINISTRATIVO

women simultaneously. If pair (mi, wj) is in a stable matching M , we define the

rank of mi in M to be the position of wj on mi’s list and the rank of wj in M as

the position of mi on wj’s list. The weight of M is the sum of the ranks of all the

men and women in M . An egalitarian stable matching has minimum weight over all the possible stable matchings. An efficient algorithm to find an egalitarian stable matching given an sm instance, which relies heavily on the distributive lattice structure of the set of all stable matchings, is described in [57].

2. Minimum regret stable matchings are the stable matchings in which the rank of the worst off person is minimised. An efficient algorithm for finding a minimum regret stable matching, given an instance of sm, is described in [36]

3. Sex equal stable matchings are stable matchings in which the absolute value of the difference between the sum of the ranks of all the men and the sum of the ranks of all the women is minimised. The problem of finding a sex-equal stable matching given an sm instance is NP-hard [71]. This was shown to be true even if the preference lists are of length at most 3 [91].

As stated earlier, although these optimality criteria can be obtained by enumerating the set of all stable matchings, more efficient algorithms to find these optimal stable matchings have been developed where possible [57, 36].

2.3

_{The Hospitals/Residents problem (hr)}

2.3.1

Introduction

The Hospital/Residents problem (hr) [29] is a generalisation of smi in which agents of one set (the hospitals) may be involved in one or more assignments in a given matching. A common application is the matching of residents (i.e., graduating medical students) to hospitals where each resident ranks a set of hospitals and each hospital ranks a set of residents in order of preference.

An instance I of hr involves a set R = {r1, r2, ..., rn1} of residents and H = {h1, h2, ...,

hn2} of hospitals. Each resident ri ∈ R ranks a subset of H in strict order of preference

with each hospital hj ∈ H ranking a subset of R, consisting of those residents who

ranked hj, in strict order of preference. Each hospital hj also has a capacity cj ∈ Z+

indicating the maximum number of residents that can be assigned to it. As in the case of smi, a pair (ri, hj) is called an acceptable pair if hj appears in ri’s preference

2.3. The Hospitals/Residents problem (hr) 16 such that each resident is assigned to at most one hospital and the number of residents assigned to each hospital does not exceed its capacity. We denote the hospital assigned to resident ri in M as M (ri) (if ri is matched in M ) and the set of residents assigned

to hospital hj in M as M (hj). A resident ri is unmatched in M if no acceptable pair

in M contains ri. A hospital hj is undersubscribed in M if |M (hj)|< cj. Otherwise hj

is said to be full in M . An acceptable pair (ri, hj) blocks a matching M , or forms a

blocking pair with respect to M , if ri is either unmatched or prefers hj to M (ri) and

hj is either undersubscribed or prefers ri to at least one member of the set M (hj). A

matching M is said to be stable if there is no blocking pair with respect to M .

This problem was also introduced by Gale and Shapley in [29] although in the context of college admissions. Every instance of hr admits at least one stable matching which can be found in linear time [29]. As described in the context of sm, matchings in the hr context can be hospital-optimal or resident-optimal depending on the orientation of the algorithm used. Given an instance I of hr, a stable matching M is said to be resident-optimal if every resident has their best possible partner in M taken over the set of all stable matchings in I. An analogous definition for hospital-optimal stable matchings exists. An hr instance may admit more than one stable matching, however, all stable matchings in a given hr instance have the same cardinality and contain the same set of residents [30, 101]. Figure 2.4 shows a sample HR instance I consisting of 6 residents and 3 hospitals with each hospital having a capacity of 2. The instance admits a stable matching M = {(r1, h2), (r2, h1), (r3, h1), (r4, h3), (r6, h2)}.

residents’ preferences hospitals’ preferences

r1 : h2 h1 h1 : (2) : r1 r3 r2 r5 r6 r2 : h1 h2 h2 : (2) : r2 r6 r1 r4 r5 r3 : h1 h3 h3 : (2) : r4 r3 r4 : h2 h3 r5 : h2 h1 r6 : h1 h2

Figure 2.4: An instance of the Hospitals/Residents problem

2.3.2

Hospital/Residents with Ties

A generalisation of hr occurs when the preference lists of the residents and hospitals are allowed to contain ties, thus forming the Hospital/Residents Problem with Ties (hrt). A tie in a resident’s or hospital’s preference list is defined analogously to the definition given for ties in the smt context. A blocking pair can be defined with respect

2.3. The Hospitals/Residents problem (hr) 17 to the different stability criteria outlined for smt and smti above and matchings can again be considered in terms of weak stability, strong stability and super stability. In the case of weak stability both the resident and hospital involved in a blocking pair must be better-off before a matching can be undermined. In the case of strong stability either the resident or hospital must be better off with none of them being worse-off. In the case of super stability, a resident-hospital pair can block a matching as long as both are not worse-off. Formal definitions of these stability criteria in the hrt context are as follows. Let (ri, hj) be an acceptable pair in an instance I of hrt and let M be

a matching in I.

1. In the case of weak stability, the pair (ri, hj) blocks M if

(a) ri is unmatched in M or prefers hj to her assigned hospital in M , and

(b) hj is undersubscribed in M or prefers ri to its worst assigned resident in M .

2. In the case of strong stability, the pair (ri, hj) blocks M if either (i)

(a) ri is unmatched in M or prefers hj to her assigned hospital in M , and

(b) hj is undersubscribed in M or prefers ri to its worst assigned resident in M

or is indifferent between them. or (ii)

(a) ri is unmatched in M or prefers hj to her assigned hospital in M or is

indifferent between them, and

(b) hj is undersubscribed in M or prefers ri to its worst assigned resident in M

3. In the case of super stability, the pair (ri, hj) blocks M if

(a) ri is unmatched in M or prefers hj to her assigned hospital in M or is

indifferent between them, and

(b) hj is undersubscribed in M or prefers ri to its worst assigned resident in M

or is indifferent between them.

Every instance of the hrt problem admits at least one weakly stable matching. This can be obtained by breaking the ties in both sets of preference lists in an arbitrary manner, thus giving rise to a hr instance which can then be solved using variants of the Gale-Shapley algorithm for hr [38]. However, in general, the way in which the ties are broken yields stable matchings of varying sizes [86] and the problem of finding a maximum weakly stable matching given an hrt instance (max hrt) is known to be NP-hard [86] as hrt is a many-to-one generalisation of smti.

2.3. The Hospitals/Residents problem (hr) 18 Research in hrt and smti and their variants has been very active since the initial results were published in [84, 64]. In particular, weak stability has attracted considerable attention perhaps due to the fact that under weak stability, hrt and smti instances are guaranteed to admit a stable matching though this need not be the case for strong stability and super stability [51]. Various approximation algorithms for max hrt can be found in the literature [66, 67, 74, 86, 59] with the best having a bound of 3/2 [90, 75, 97]. The parameterized complexity of max hrt has also been studied in the literature. In [88] max hrt was shown to be in FPT with the parameter being the sum of the lengths of the ties in the preference lists.

Concerning inapproximability results, Halldorsson et al. showed that it is NP-hard to approximate max smti to within δ for some δ > 1 [39] (where δ is very close to 1) even if each man’s preference list is of length at most 7 and each woman’s preference list is of length at most 4 and each preference list is derived from two master lists of men and women. Irving et al. presented the same inapproximability result for the restriction where men’s and women’s preference lists are of length at most 3 and 4 respectively [61]. In the case of unbounded length preference lists where each man’s preference list is strictly ordered and each woman’s preference list is either strictly ordered or is a tie of length 2, max smti was shown to be inapproximable to within 21/19 − ε for some ε > 0 unless P=NP [40]. Finally Yanagisawa improved this lower bound to 33/29 for the case where ties are on both sides and each tie is of length 2 [115]. Various heuristics for solving max hrt have also been developed [60, 34, 35].

2.3.3

_{Cloning hr instances}

A technique known as cloning can be used to convert an hr instance I into an instance an smi I0 in polynomial time, such that there is a bijective function between the sets of stable matchings in I and I0 _{[38, 106]. Let I be an instance of hr where R =} {r1, r2, ..., rn1} is the set of residents and H = {h1, h2, ..., hn2} is the set of hospitals.

Let cj be the capacity of hospital hj ∈ H. We can construct an instance I0 of smi as

follows. Each resident in I corresponds to a man in I0. Each hospital hj ∈ H gives rise

to cj women in I0, denoted by hj,1, hj,2, ..., hj,cj, each of whom has the same preference

list as hj in I0 but with a capacity of 1. Each man ri ∈ R starts off with the same

preference list in I0 as he has in I. We then replace each hospital hj on his list by the

cj women hj,1, hj,2, ..., hj,cj listed in strict order (increasing on second subscript). There

is a bijective function between the sets of stable matchings in I and I0. This relation between hr and smi was shown to hold (but not necessarily with a bijective function) when cloning max hrt to max smti instances in [85] thus some of the positive results for max smti can carry over to max hrt. The same technique will be used in Section

2.4. The Stable Roommates problem (sr) 19