2.2 Full preference information
2.2.4 Exploiting the structure
The algorithmic consequences of the rotation posetΠ of an SMI instance are numerous. Of im-
mediate interest is the fact that the rotation poset allows for the efficient generation of all stable matchings. Gusfield [34] showed thatM can be enumerated in O(m + n|M|) time – hence there
is only a linear-time delay between the output of each stable matching. He further showed that the set of all stable pairs – the set of (man,woman) pairs that appear in some stable matching – can be computed inO(m) time. Later, Gent et al [33] gave a different approach to enumerating all stable
2.2 Full preference information 15 matchings by using constraint programming. Their method uses arc consistent domains in order to achieve failure-free enumeration of all stable matchings.
An interesting question that arises is whether there are stable matchings that are somehow fair to both the men and women of the instance, as opposed to the extreme unfairness of the stable matchingsM0andMz. We next review several such problems, some of which are polynomial-time
solvable, thanks to the structural results of the rotation poset. Henceforth, for agentsa and b, let pa(b) denote the position of agent b on agent a’s preference list. If a finds b unacceptable, pa(b) is
undefined.
Minimum regret stable matchings
Given a stable matchingM we define the regret of agent a to be pa(M (a)), i.e, the position of a’s
partner inM . The regret of an unmatched agent is undefined. The regret of a matching M is the
maximum regret taken over all agents inM . A minimum regret stable matching is a stable matching
with minimum possible regret.
For the stable marriage setting, Knuth [69] showed that the minimum regret stable matching prob- lem can be solved inO(m2) time, attributing the result to Selkow. Gusfield [34] improved this to
an optimalO(m)-time solution.
Fair stable matchings
Suppose we wish to somehow treat the men and the women of an SMI instance equally. For a stable matchingM , define the egalitarian value e(M ) to be
e(M ) = X
(m,w)∈M
(pm(w) + pw(m)).
An egalitarian stable matching is a stable matchingM that minimizes e(M′) over all M′ ∈ M.
The egalitarian egalitarian stable matching problem (ESM) is to find an egalitarian stable matching. Intuitively, ESM captures the notion of finding a stable matching with the best “social welfare”.
2.2 Full preference information 16 m1: w1 w∗2 . . . w1: m∗n . . . m1 m2: w2 w∗3 . . . w2: m∗1 . . . m2 .. . ... mn−1: wn−1 w∗n . . . wn−1 : m∗n−2 . . . mn−1 mn: wn w∗1 . . . wn: m∗n−1 . . . mn
Figure 2.4: An SM instance with two stable matchings with greatly varying values ofe(·).
Example The example given in Figure 2.4 shows how e(·) can greatly vary for different stable
matchings of a particular SMI instance. The example consists of an SM instance with two stable matchings, one denoted by underlining, and the other by star. The ellipses in the preference lists denote any arbitrary ordering of the remaining agents not explicitly mentioned. The egalitarian valuee(·) for the stable matching denoted by underlining is n2+ n, whereas e(·) for the matching
denoted by star is3n.
Another notion of a fair stable matching arises by attempting to find a stable matching with the property that the men’s and women’s overall happiness is as close as possible. To this end, the
sex-equality measureδ(M ) of a stable matching is defined to be
δ(M ) = X (m,w)∈M pm(w)− X (m,w)∈M pw(m).
A sex-equal stable matching is a stable matchingM that minimizes|δ(M′)| over all M′∈ M. The
sex-equal stable matching problem (SESM) is to find a sex-equal stable matching.
Example The example given in Figure 2.5 shows how|δ(·)| can also greatly vary for different stable
matchings of a particular SMI instance. The example consists of an SM instance with two stable matchings, one denoted by underlining, and the other by star. The ellipses in the preference lists denote any arbitrary ordering of the remaining agents not explicitly mentioned. The absolute value of the sex-equality measureδ(·) for these two stable matchings is n2− n and zero, respectively.
The elimination of a rotation can, in general, result in a stable matching with a different value ofe(·) and/or δ(·). For a rotation ρ = (m0, w0), (m1, w1),. . .,(mr−1, wr−1), we define v(ρ) and
w(ρ) [47, 56] to capture the change in egalitarian value and sex-equality measure, respectively, by
2.2 Full preference information 17 m1 : w1 w∗2 . . . w1 : m2 m∗n . . . m1 m2 : w2 w∗3 . . . w2 : m3 m∗1 . . . m2 .. . ... mn−1 : wn−1 w∗n . . . wn−1: mn m∗n−2 . . . mn−1 mn: wn w∗1 . . . wn: m1 m∗n−1 . . . mn
Figure 2.5: An SM instance with two stable matchings with greatly varying values of|δ(·)|. v(ρ) = r−1 X i=0 (pmi(wi+1)− pmi(wi)) + r−1 X i=0 (pwi(mi−1)− pwi(mi)), w(ρ) = r−1 X i=0 (pmi(wi+1)− pmi(wi))− r−1 X i=0 (pwi(mi−1)− pwi(mi)).
Irving, Leather, and Gusfield [47] showed that ESM can be solved inO(m2) time by assigning the
weight given byv(·) to each rotation in Π, and then finding a maximum weight closed subset of Π.
Later, Feder [28] gave an alternative approach that improves this running time toO(m1.5log n).
At first one would probably suspect that a similar approach would work for the sex-equal stable marriage problem. Indeed, on the surface, these two problems look almost identical: e(M ) is a
sum of the ranks of the mens’ and womens’ partners, whereasδ(M ) is the absolute difference. It
is perhaps surprising then, that the sex-equal stable matching problem is strongly NP-hard [58]. On the positive side, Iwama et al [56] give a polynomial-time approximation algorithm for the so-called
near-sex-equal stable marriage problem. Their algorithms involve assigning the weight w(ρ) to
each rotation inΠ and then attempting to find an appropriate subset of rotations to eliminate. They
further study the problem of finding a minimum regret stable matching amongst the set of all near- sex-equal stable matchings. This latter problem is NP-hard, but there is an approximation algorithm with a performance guarantee better than two [56]. We shall study SESM more extensively in Chapter 4.
As a final word on fair stable matchings, we mention the median stable matching problem. For each (matched) manm in an SMI instance, sort the multiset of women m is matched to inM from m’s most to least preferred. For example, if man m is matched to woman w in exactly ten stable
matchings inM, then w appears ten consecutive times in this sorted list. Let wi(m) denote the ith
2.2 Full preference information 18 towi(m′). Teo and Sethuraman [99] proved the surprising result that Mi is not only a matching,
but is also stable. Theith-median stable matching is defined to be the stable matching obtained by matching every man towi(m). Note that, in general, not every Miso obtained is distinct, and not
every stable matchingM is equal to some Mj for somej.
The definition of a median stable matching does not lend itself to any natural polynomial-time algo- rithm – it appears as though we must explicitly enumerateM to construct a median stable match- ing. Cheng [18] gave a new characterization of the so-called generalized median stable matchings, showing that there is an intimate relationship between median stable matchings and the median el- ements of the lattice ofM. She went on to show that finding a median stable matching is NP-hard,
but is approximable in a formal sense, and even polynomial-time solvable for some special cases. Very recently, Kijima and Nemoto [63] improved upon some of Cheng’s results.