Indifference

A natural generalization of SM and SMI is to allow the agents involved to express some form of

indifference in their preference lists. The most natural way for agents to express indifference is

in the form of ties in the preference list; a tiet on an agent a’s preference list is defined to be a

set of agents all of whom have the same position ona’s list. The notion of a tie is important in

the practical applications of SM and SMI – consider, for example a hospital that must attempt to produce a genuinely strict ranking of hundreds of medical students [105, 106, 107]. We use SMT (SMTI) to stand for the variant of SM (SMI) in which preference lists can contain ties.

Of course, with the inclusion of ties, the definition of a blocking pair must be reconsidered. It stands to reason that a (man,woman) pair should still form a blocking pair if they both improve by becoming matched to each other, but what if, for example,m is indifferent between his current

partner andw?

There are three particularly natural formulations of blocking pair, each with a corresponding notion of stability. These three kinds of stability are defined as follows:

• weak-stability: a (man,woman) pair can block only by both becoming better off

• strong-stability: a (man,woman) pair can block if at least one of them becomes better off, and

2.2 Full preference information 19

• super-stability: a (man,woman) pair can block if neither of them becomes any worse off.

Notice that each form of stability above is increasingly more restrictive than the previous, so super- stability implies strong-stability implies weak-stability. Irving [43] observed that while weakly- stable matchings always exist for an SMT instance, strong- and super-stable matchings need not. He further gave a polynomial-time algorithm for each of the three forms of stability that either returns a stable matching or reports that none exists (in the case of weak-stability, the algorithm always returns a weakly-stable matching). Manlove extended Irving’s results to the SMTI setting [72].

Stability, size, and structure

There is an interesting interplay between the various forms of stability and the cardinality of stable matchings. If a super-stable matching exists for an SMTI instance, then all stable matchings for the instance have equal cardinality, regardless of the definition of stability. Otherwise, if a strongly- stable matching exists, then all strongly-stable matchings have the same size. In general, weakly stable matchings can have different cardinalities, but every strongly-stable matching is at least two- thirds the size of an arbitrary weakly-stable matching [93].

Spieker [95] showed that the set of super-stable matchings for an SMTI instance forms a distribu- tive lattice. Later, Manlove [73] gave an alternative, and perhaps more accessible proof showing that both strong- and super-stable matchings have a distributive lattice structure. The elements of the lattice structure described by Manlove are sets of “equivalent” stable matchings, rather than individual stable matchings. The maximum and minimum elements of the lattice correspond to the sets of man- and woman-optimal stable matchings. Scott [93] extended the notion of a rotation to super-stability, and described polynomial-time algorithms for finding egalitarian and minimum- regret stable matchings, along with algorithms for generating all super-stable matchings and finding all super-stable pairs. Extending such results to the strong-stability case remains an open question, but it seems likely that this can be done in light of the structural results of Manlove [73].

2.2 Full preference information 20

m1: (w2 w1∗) w1: (m2 m∗1) m3

m2: w3 w1 w2: m1 m3

m3: w1 (w3∗ w4) w2 w3: (m∗3 m4) m2

m4: w3 w4: m3

Figure 2.6: An SMTI instance with stable matchings of different sizes.

Weak stability

Irving [43] showed that finding a weakly stable matching in an SMT/SMTI instance is particularly easy: simply arbitrarily break the ties and find any stable matching in the resulting instance. In a sense, this method is the only “easy” thing about weak stability – almost everything else seems to be computationally difficult. In the SMT setting, minimum regret stable matchings and egalitarian stable matchings are both not only NP-hard to find, but are not approximable withinΩ(n) unless

P=NP [74]. It is also NP-hard even to determine if a given (man,woman) pair occurs in a stable matching (i.e. is a stable pair). Identifying any structural relationship involving weakly stable matchings is open, although one can construct SMT/SMTI instances that have neither man- nor woman-optimal stable matchings [88]. Efficiently enumerating all weakly-stable matchings also remains an open question.

We mentioned above, that, in general, the weakly-stable matchings of an SMTI instance can have different cardinality. This fact is illustrated when one uses Irving’s tie-breaking algorithm: the ways in which the ties are broken can have a significant impact on the cardinality of the stable matchings obtained.

Example Figure 2.6 presents an SMTI instance with two different stable matchings of different

cardinality. The example shows two weakly stable matchings, one denoted by underlining, and the other by star. The stable matching denoted by underlining is twice the size of the stable match- ing denoted by star. These matchings can be arrived at by running the (extended) Gale/Shapley algorithm on two of the different ways that the ties of the instance can be broken.

Manlove et al [74] first observed the fact that weakly stable matchings can have different sizes, and further showed that an arbitrary weakly stable matchingM can be as little as one-half the size of

a maximum cardinality stable matching. The obvious question then, is, can we find a maximum cardinality weakly stable matching in polynomial-time? Manlove et al [74] showed that finding a maximum cardinality weakly-stable matching is NP-hard, even in the highly restricted setting in

2.2 Full preference information 21 which the preference lists on one side are strictly ordered, and the preference list of each member of the opposite set is either strictly ordered or is a tie of length two (these conditions holding simul- taneously). Henceforth, we let MAX-SMTI denote the problem of finding a maximum cardinality weakly stable matching of an SMTI instance.

Motivated by the hardness results of Manlove et al [74], researchers have been interested in finding polynomial-time approximation algorithms for MAX-SMTI. As a first step, we may observe that simply computing an arbitrary stable matching is an easy 2-approximation algorithm, because an arbitrary stable matching must be a maximal matching. A number of improvements have since appeared in the recent literature.

For the general case of SMTI, Iwama et al [53] gave a2− clog nn approximation algorithm, wherec

is a positive constant. This algorithm was subsequently improved to yield a performance guarantee of2− √c′_n, wherec′ is a positive constant which is at most1/4√6 [55]. The first approximation

algorithm for general SMTI with a constant performance guarantee better than two was given by Iwama et al [54], with a performance ratio of15/8.

The approximability of several special cases of SMTI have also been studied. Halld´orsson et al [37] gave a(2/(1 + T−2)-approximation algorithm for the restricted case in which ties are only on

one side, and the length of the longest tie isT . This bound can be improved to 13/7 if the ties can

appear in both men’s and women’s preference lists, but are restricted to being size at most two [37]. These same authors later described a randomized algorithm with an expected guarantee of10/7 for

this special case with the additional restriction that ties appear only on one side [39]. Motivated by a restricted case of SMTI arising in practice [44, 107], Irving and Manlove [48] described a

5/3-approximation algorithm for MAX-SMTI instances in which the ties appear only on one side,

say, the women, and each woman may have at most one tie on her preference list, and this tie, if any, appears at the end of her list.

A recent landmark paper of Kir´aly [64] gave two simple algorithms that effectively superseded all previously known approximation algorithms for MAX-SMTI, (save only the randomized algorithm for the very special case studied in [39]). Kir´aly’s first algorithm provides a3/2-approximation for

the restricted case of MAX-SMTI in which ties are allowed to only appear on the women’s side (this is the only restriction). The second algorithm provides a5/3-approximation for the general

MAX-SMTI setting, in which no restrictions are placed on the problem input. In Chapter 3, we describe Kir´aly’s approach in more detail, and give an approximation algorithm with an improved

2.2 Full preference information 22 performance guarantee.

From an inapproximability point of view, it is known that MAX-SMTI is APX-complete [38] and cannot be approximated within21/19 (unless P = NP) [37]. Yanagisawa [103] improved this bound

to33/29, and also showed that MAX-SMTI cannot be approximated within 4/3 under the assump-

tion that the minimum vertex cover problem cannot be approximated within a factor of2− ǫ.

We mention one final result regarding weakly stable matchings. Let(α, β)-SMTI denote an SMTI

instance in which the men’s (women’s) preference lists are of bounded maximum lengthα (β). De-

fine(α, β)-MAX-SMTI similarly. Irving et al [51] showed that (3, 3)-MAX-SMTI is NP-hard, but (2,_{∞)-MAX-SMTI is polynomial-time solvable (the ∞ here denotes preference lists of unbounded}

length). They furthermore showed that there exists a constantδ0such that(4, 3)-MAX-SMTI is not

approximable withinδ0unless P=NP. The inapproximability of(3, 3)-MAX-SMTI remains open.