A Generalization of Birkhoff's Theorem for Distributive Lattices, with Applications to Robust Stable Matchings

04/16/2018 ∙ by Tung Mai, et al. ∙ 0

Birkhoff's theorem, which has also been called the fundamental theorem for finite distributive lattices, states that the elements of any such lattice L are isomorphic to the closed sets of a partial order, say Π. We generalize this theorem to showing that each sublattice of L is isomorphic to a distinct partial order that can be obtained from Π via the operation of compression, defined in this paper. Let A be an instance of stable matching, with L being its lattice of stable matchings, and let B be the instance obtained by permuting the preference list of any one boy or any one girl. Let M_A and M_B be their sets of stable matchings. Our results are the following: - We show that M_A ∩M_B is a sublattice of L and M_A∖M_B is a semi-sublattice of L. - Using our generalization of Birkhoff's Theorem, we give an efficient algorithm for finding the compression of Π that is isomorphic to the lattice of M_A ∩M_B. - Given a polynomial sized domain D of such errors (of permuting one of the preference lists), we give an efficient algorithm that checks if there is a stable matching for A that is stable for each such resulting instance B. We call this a fully robust stable matching. - If yes, the set of all such matchings forms a sublattice of L and our algorithm finds its partial order as well. Using the latter, we can obtain a matching that optimizes (maximizes or minimizes) the weight among all fully robust stable matchings.

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1 Introduction

The two topics, of stable matchings and the design of algorithms that produce robust solutions, have been studied intensively for decades and there are today several books on each of these topics, e.g., see [Knu97, GI89, Man13] and [CE06, BTEGN09]. Yet, at the intersection of these two topics lies a single work, namely [MV18a]. The contribution of that recent paper of ours was two-fold: First, it introduced the problem of finding a stable matching that is robust to errors in the input and gave a polynomial time algorithm for a special case. Second, it initiated work on a new structural question, namely finding relationships between the lattices of solutions of two “nearby” instances of stable matching (in this case, the given and the perturbed instances). In the current paper, we make substantial progress on both fronts.

The domain of errors considered in [MV18a] was too restrictive, it was polynomial sized, and we had asked the question of extending the domain. In the current paper, we extend the domain to a super-exponentially large one, though we need to somewhat weaken the notion of “robust”. Underlying our polynomial time algorithms are new structural properties, of the type described above, of lattices of stable matchings.

Over the last half century, the stable matching problem [GS62]

has been the subject of intense study from numerous different angles in many fields, including computer science, mathematics, operations research, economics and game theory, e.g., see

[Knu97, GI89, Man13]. Over these decades, researchers have unearthed the deep and pristine combinatorial structure of this problem, which in turn has led to efficient algorithms for numerous questions studied about this problem. In 2012, this problem was a subject of the Nobel Prize in Economics, awarded to Roth and Shapley [RS12]. To the best of our knowledge, the issue of robust solutions had not been studied in the context of this problem until [MV18a] (see also Section 1.4), even though the design of algorithms that produce robust solutions is already a very well established field, especially as pertaining to optimization, e.g., see [CE06, BTEGN09].

The setting defined in [MV18a] was the following: Given an instance of stable matching on boys and girls, let be the instance that results after introducing one error from a domain

, chosen via a given discrete probability distribution. We had defined a

robust stable matching as a matching that is stable for and maximizes the probability of being stable for as well. The domain, , of errors was defined via an operation called shift. For a girl , assume her preference list in instance is . Move up the position of so ’s list becomes , and let denote the resulting instance. An analogous operation is defined on a boy ’s list; again some girl on his list is moved up. For each girl and each boy, consider all possible such shifts to get the domain D; clearly, is .

The setting of the current paper is: Let be as above and let denote the set of all possible instances, , obtained by introducing one error of the following type in : For any one girl or any one boy, arbitrarily permute of the preference list of the girl or the boy. Clearly . Let be an arbitrary polynomial sized set. Define a fully robust stable matching to be a matching that is stable for and for each of the instances in . Our main algorithmic result is:

Theorem 1.

For the setting given above, there is a polynomial time algorithm for checking if there is a fully robust stable matching. If the answer is yes, the set of all such matchings form a sublattice of and our algorithm finds a partial order that generates this sublattice.

Terms used in Theorem 1 are standard but are also defined in the next section. The new structural properties which support this result are described in Section 1.2.

1.1 The lattice of stable matchings

Conway, see [Knu97], proved that the set of stable matchings of an instance forms a lattice, with the meet and join of two stable matchings being the operations of taking the boy-optimal choices and girl-optimal choices, respectively, of the two matchings. Knuth [Knu97] asked if every finite distributive lattice is isomorphic to the lattice arising from an instance of stable matching. A positive answer was provided by Blair [Bla84]; for a much better proof, see [GI89].

It is easy to see that the family of closed sets of a partial order, say , is closed under union and intersection and forms a distributive lattice, with join and meet being these two operations, respectively; let us denote it by . Birkhoff’s theorem [Bir37], which has also been called the fundamental theorem for finite distributive lattices, e.g., see [Sta96], states that corresponding to any finite distributed lattice, , there is a partial order, say , whose lattice of closed sets is isomorphic to , i.e., . We will say that generates .

For the lattice of stable matchings, the partial order defined in Birkhoff’s Theorem, has additional useful structural properties. First, its elements are rotations. A rotation takes matched boy-girl pairs in a fixed order, say , and “cyclically” changes the mates of these agents, see Section 2.3 for details. The number , the pairs, and the order among the pairs are so chosen that when a rotation is applied to a stable matching containing all pairs, the resulting matching is also stable. Moreover, there is no valid rotation on any subset of these pairs, under any ordering. Hence, a rotation can be viewed as a minimal change to the current matching that results in a stable matching. Any boy–girl pair, , belongs to at most one rotation. Consequently, the set of rotations underlying satisfies is , and hence, is a succinct representation of ; the latter can be exponentially large. will be called the rotation poset for .

Second, the rotation poset helps traverse the lattice as follows. For any closed set of , the corresponding stable matching can be obtained as follows: start from the boy-optimal matching in the lattice and apply the rotations in set , in any topological order consistent with . The resulting matching will be . In particular, applying all rotations in , starting from the boy-optimal matching, leads to the girl-optimal matching.

1.2 Overview of structural results

We start by giving a short overview of the structural facts proven in [MV18a]. Let and be two instances of stable matching over boys and girls, with sets of stable matchings and , and lattices and , respectively. Let be the poset on rotations that is isomorphic to . It is easy to see that if is obtained from by changing the lists of only one gender, either boys or girls, but not both, then the matchings in form a sublattice in each of the two lattices (Proposition  2). [MV18a] further show that if is obtained by applying a shift operation, then is also a sublattice of . Using this fact, they show that there is at most one rotation, , that leads from to and at most one rotation, that leads from to ; moreover, these rotations can be efficiently found. Furthermore, for a closed set of , is stable for instance iff .

Our failure at extending the domain of errors in any straightforward manner led to the following abstract question. Suppose is such that and are both sublattices of , i.e., is partitioned into two sublattices. Then is there a polynomial time algorithm for finding a matching in ? This is one of the two abstract questions we will answer in this paper and hence we will call it Case I (See Section 5).

Since we are dealing with arbitrary sublattices, say , of a lattice, say , it would be useful to find a way of obtaining the partial order, , which generates from the partial order, , that generates . A generalization of Birkhoff’s Theorem, proved within category theory [Wika] provides an avenue towards answering this question. In this paper, we provide a purely combinatorial proof in the setting of lattices of stable matching. The advantage being that it gives some crucial notions and insights, such as that the central notion of meta-rotation; just as rotations help traverse a lattice, meta-rotations help traverse a sublattice. An outline of our proof is given in Section 1.3. We give the notion of a compression, which when applied to yields another partial order ; it is formally defined in Definition 1. We prove that there is a one-to-one correspondence between all possible sublattices of and all possible compressions of such that if corresponds to , then .

Via the machinery developed above, we can answer the stated question, i.e., suppose lattice is partitioned into two sublattices over and , then how do you generate a matching in ? We prove that there exists a sequence of rotations such that a closed set of generates a matching in iff it contains but not for some (Proposition 8). Furthermore, this sequence of rotations can be found in polynomial time. However, so far this abstract fact has not yielded a concrete error pattern, beyond shift, that we can add to our domain.

Next, we address the case that is not a sublattice of . We start by proving that if is obtained by permuting the preference list of any one boy, then must be a join semi-sublattice of (Lemma 24). Hence our second abstract case, which we will call Case II, is that lattice is partitioned into a sublattice and a join semi-sublattice (see Section 6). For this case, we obtain a more elaborate characterization of compressions that generate the sublattice of (Theorem 9), and a more elaborate condition on rotations which is satisfied by a closed set of iff the corresponding matching is in the sublattice (Proposition 10). Furthermore, we show how to efficiently find these rotations (Theorem 11), hence leading to an efficient algorithm for finding a matching in .

Finally, consider the setting given in the Introduction, with being the super-exponential set of all possible erroneous instances obtained by permuting the preference list of one boy or one girl, and a polynomial sized set of instances which the algorithm needs to consider. We show that the set of all such matchings that are stable for and for each of the instances in forms a sublattice of and we obtain the compression of that generates this sublattice (Section 8.2). Each matching in this sublattice is a fully robust stable matching. Moreover, since we have obtained the poset generating it, we can go further: given a weight function on all boy-girl pairs, we can obtain, using the algorithm of [MV18b], a matching that optimizes (maximizes or minimizes) the weight among all fully robust stable matchings.

1.3 Our proof of the generalization of Birkhoff’s Theorem

As stated above, we will prove the generalization (Theorem 4) in the context of stable matching lattices; as remarked earlier, such lattices are as general as arbitrary finite distributive lattices. Let be a stable matching lattice which is generated by poset . We first give one definition of compression, in Definition 1, which when applied to yields another partial order . Our proof involves showing that each compression of generates a sublattice of (Section 3.1), and corresponding to each sublattice of , there is a compression of that generates (Section 3.2).

The second part is quite non-trivial. It involves first identifying the correct partition of the set of rotations of by considering pairs of matchings, in such that is a direct successor of the , and obtaining the set of rotations that takes us from to . This set will be a meta-rotation for . Consider one such meta-rotation . To obtain all predecessors of in , consider all paths that go from the boy-optimal matching in to the girl-optimal matching by going through the lattice . Find all meta-rotations that always occur before does on all such paths. Then each of these meta-rotations precedes . These are the precedence relations between meta-rotations in .

A second definition of compression: We next present a different, equivalent, definition of compression (Section 4). This definition is in terms of a set of directed edges, , that needs to be added to to yield, after some prescribed operations, the desired partial order . Let be the sublattice generated by . Then we will say that edges define .

The advantage of this definition is that it is much easier to work with for the applications presented later. Its drawback is that several different sets of edges may yield the same compression. Therefore, there is no one-to-one correspondence between sublattices of and the sets of edges that can be added to to yield compressions. Hence this definition is not suitable for proving the generalization of Birkhoff’s Theorem.

1.4 A matter of nomenclature

Assigning correct nomenclature to a new issue under investigation is clearly critical for ease of comprehension. In this context we wish to mention that very recently, Genc et. al. [GSOS17] defined the notion of an -supermatch as follows: this is a stable matching in which if any pairs break up, then it is possible to match them all off by changing the partners of at most other pairs, so the resulting matching is also stable. They showed that it is NP-hard to decide if there is an -supermatch. They also gave a polynomial time algorithm for a very restricted version of this problem, namely given a stable matching and a number , decide if it is a -supermatch. Observe that since the given instance may have exponentially many stable matchings, this does not yield a polynomial time algorithm even for deciding if there is a stable matching which is a -supermatch for a given .

Genc. et. al. [GSSO17] also went on to defining the notion of the most robust stable matching, namely a -supermatch where is minimum. We would like to point out that “robust” is a misnomer in this situation and that the name “fault-tolerant” is more appropriate. In the literature, the latter is used to describe a system which continues to operate even in the event of failures and the former is used to describe a system which is able to cope with erroneous inputs, e.g., see the following pages from Wikipedia [Wikc, Wikb].

2 Preliminaries

2.1 The stable matching problem

The stable matching problem takes as input a set of boys and a set of girls ; each person has a complete preference ranking over the set of opposite sex. The notation indicates that girl strictly prefers to in her preference list. Similarly, indicates that the boy strictly prefers to in his list.

A matching is a one-to-one correspondence between and . For each pair , is called the partner of in (or -partner) and vice versa. For a matching , a pair is said to be blocking if they prefer each other to their partners. A matching is stable if there is no blocking pair for .

2.2 The lattice of stable matchings

Let and be two stable matchings. We say that dominates , denoted by , if every boy weakly prefers his partner in to . It is well known that the dominance partial order over the set of stable matchings forms a distributive lattice [GI89], with meet (greatest lower bound) and join (least upper bound) defined as follows. The meet of and , , is defined to be the matching that results when each boy chooses his more preferred partner (or equivalently, each girl chooses her less preferred partner) from and ; it is easy to show that this matching is also stable. The join of and , , is defined to be the matching that results when each boy chooses his less preferred partner (or equivalently, each girl chooses her more preferred partner) from and ; this matching is also stable. These operations distribute, i.e., given three stable matchings ,

It is easy to see that the lattice must contain a matching, , that dominates all others and a matching that is dominated by all others. is called the boy-optimal matching, since in it, each boy is matched to his most favorite girl among all stable matchings. This is also the girl-pessimal matching. Similarly, is the boy-pessimal or girl-optimal matching.

2.3 Rotations help traverse the lattice

A crucial ingredient needed to understand the structure of stable matchings is the notion of a rotation, which was defined by Irving [Irv85] and studied in detail in [IL86]. A rotation takes matched pairs in a fixed order, say and “cyclically” changes the mates of these agents, as defined below, to arrive at another stable matching. Furthermore, it represents a minimal set of pairings with this property, i.e, if a cyclic change is applied on any subset of these pairs, with any ordering, then the resulting matching has a blocking pair and is not stable. After rotation, the boys’ mates weakly worsen and the girls’ mates weakly improve. Thus one can traverse from to by applying a suitable sequence of rotations (specified by the rotation poset defined below). Indeed, this is precisely the purpose of rotations.

Let be a stable matching. For a boy let denote the first girl on ’s list such that strictly prefers to her -partner. Let denote the partner in of girl . A rotation exposed in is an ordered list of pairs such that for each , , is , where is taken modulo . In this paper, we assume that the subscript is taken modulo whenever we mention a rotation. Notice that a rotation is cyclic and the sequence of pairs can be rotated. is defined to be a matching in which each boy not in a pair of stays matched to the same girl and each boy in is matched to . It can be proven that is also a stable matching. The transformation from to is called the elimination of from .

Lemma 1 ([Gi89], Theorem 2.5.4).

Every rotation appears exactly once in any sequence of elimination from to .

Let be a rotation. For , we say that moves from to , and moves from to . If is either or is strictly between and in ’s list, then we say that moves below . Similarly, moves above if is or between and in ’s list.

2.4 The rotation poset

A rotation is said to precede another rotation , denoted by , if is eliminated in every sequence of eliminations from to a stable matching in which is exposed. If precedes , we also say that succeeds . If neither nor , we say that and are incomparable Thus, the set of rotations forms a partial order via this precedence relationship. The partial order on rotations is called rotation poset and denoted by .

Lemma 2 ([Gi89], Lemma 3.2.1).

For any boy and girl , there is at most one rotation that moves to , below , or above . Moreover, if moves to and moves from then .

Lemma 3 ([Gi89], Lemma 3.3.2).

contains at most rotations and can be computed in polynomial time.

A closed set of a poset is a set of elements of the poset such that if an element is in then all of its predecessors are also in . There is a one-to-one relationship between the stable matchings and the closed subsets of . Given a closed set , the correponding matching is found by eliminating the rotations starting from according to the topological ordering of the elements in the set . We say that generates and that generates the lattice of all stable matchings of this instance.

Let be a subset of the elements of a poset, and let be an element in . We say that is a minimal element in if there is no predecessors of in . Similarly, is a maximal element in if it has no successors in .

The Hasse diagram of a poset is a directed graph with a vertex for each element in poset, and an edge from to if and there is no such that . In other words, all precedences implied by transitivity are suppressed.

2.5 Sublattice and Semi-sublattice

A sublattice of a distributive lattice is subset of such that for any two elements , and whenever .

A join semi-sublattice of a distributive lattice is subset of such that for any two elements , whenever .

Similarly, meet semi-sublattice of a distributive lattice is subset of such that for any two elements , whenever .

Note that is a sublattice of iff is both join and meet semi-sublattice of .

Proposition 2.

Let be an instance of stable matching and let be another instance obtained from by changing the lists of only one gender, either boys or girls, but not both. Then the matchings in form a sublattice in each of the two lattices.

Proof.

It suffices to show that is a sublattice of . Assume and let and be two different matchings in . Let and be the join operations under and respectively. Likewise, let and be the meet operations under and .

By definition of join operation in Section 2.2, is the matching obtained by assigning each boy to his less preferred partner (or equivalently, each girl to her more preferred partner) from and according to instance . Without loss of generality, assume that is an instance obtained from by changing the lists of only girls. Since the list of each boy is identical in and , his less preferred partner from and is also the same in and . Therefore, . A similar argument can be applied to show that .

Hence, and are both in as desired. ∎

Corollary 1.

Let be an instance of stable matching and let be other instances obtained from each by changing the lists of only one gender, either boys or girls, but not both. Then the matchings in form a sublattice in .

Proof.

Assume and let and be two different matchings in . Therefore, and are in for each . By Proposition 2, is a sublattice of . Hence, and are in for each . The claim then follows. ∎

In Section 8.1, we show that for any instance obtained by permuting the preference list of one boy or one girl, forms a semi-sublattice of (Lemma 24). In particular, if the list of a boy is permuted, forms a join semi-sublattice of , and if the list of a girl is permuted, forms a meet semi-sublattice of . In both cases, is a sublattice of according to Proposition 2, and the set of matchings in can be characterized in the same manner.

2.6 Robust Stable Matching

Let be a stable matching instance, and let be a discrete probability distribution over stable matching instances. A robust stable matching is a stable matching maximizing the probability that , where . We denote if prefers to with respect to instance . When the probability is 1, is said to be a fully robust stable matching. In other words, for all in the domain of .

3 A Generalization of Birkhoff’s Theorem

Let be a finite poset. For simplicity of notation, in this paper we will assume that must have two dummy elements and ; the remaining elements will be called proper elements and the term element will refer to proper as well as dummy elements. Further, precedes all other elements and succeds all other elements in . A proper closed set of is any closed set that contains and does not contain . It is easy to see that the set of all proper closed sets of form a distributive lattice under the operations of set intersection and union. We will denoted this lattice by . Birkhoff’s Theorem states that every finite distributive lattice is isomorphic to the proper closed sets of some poset.

Theorem 3.

(Birkhoff [Bir37]) Every finite distributive lattice is isomorphic to , for some finite poset .

Our proof of the generalization of Birkhoff’s Theorem deals with the sublattices of a finite distributive lattice. First, in Definition 1 we state the critical operation of compression of a poset.

Definition 1.

Given a finite poset , first partition its elements; each subset will be called a meta-element. Define the following precedence relations among the meta-elements: if are elements of such that is in meta-element , is in meta-element and precedes , then precedes . Assume that these precedence relations yield a partial order, say , on the meta-elements (if not, this particular partition is not useful for our purpose). Let be any partial order on the meta-elements such that the precedence relations of are a subset of the precedence relations of . Then will be called a compression of . Let and denote the meta-elements of containing and , respectively.

* 0.4 * 0.41324 * 0.4 * 0.4
Figure 1: Two examples of compressions. Lattice . and are compressions of , and they generate the sublattices in , of red and blue elements, respectively.

For examples of compressions see Figure 1. Clearly, precedes all other meta-elements in and succeeds all other meta-elements in . Once again, by a proper closed set of we mean a closed set of that contains and does not contain . Then the lattice formed by the set of all proper closed sets of will be denoted by .

Theorem 4.

[Wika] There is a one-to-one correspondence between the compressions of and the sublattices of . Furthermore, if a sublattice of corresponds to compression , then is isomorphic to .

We will prove Theorem 4 in the context of stable matching lattices; this is w.l.o.g. since stable matching lattices are as general as finite distributive lattices. In this context, the proper elements of partial order will be rotations, and meta-elements are called meta-rotations. Let be the corresponding stable matching lattice.

Clearly it suffices to show that:

  • Given a compression , is isomorphic to a sublattice of .

  • Any sublattice is isomorphic to for some compression .

These two proofs are given in Sections 3.1 and 3.2, respectively.

3.1 is isomorphic to a sublattice of

Let be a closed subset of ; clearly is a set of meta-rotations. Define to be the union of all meta-rotations in , i.e.,

We will define the process of elimination of a meta-rotation of to be the elimination of the rotations in in an order consistent with partial order . Furthermore, elimination of meta-rotations in will mean starting from stable matching in lattice and eliminating all meta-rotations in in an order consistent with . Observe that this is equivalent to starting from stable matching in and eliminating all rotations in in an order consistent with partial order . This follows from Definition 1, since if there exist rotations in such that is in meta-rotation , is in meta-rotation and precedes , then must also precede . Hence, if the elimination of all rotations in gives matching , then elimination of all meta-rotations in will also give the same matching.

Finally, to prove the statement in the title of this section, it suffices to observe that if and are two proper closed sets of the partial order then

It follows that the set of matchings obtained by elimination of meta-rotations in a proper closed set of are closed under the operations of meet and join and hence form a sublattice of .

3.2 Any sublattice of is isomorphic to , for a compression of

We will obtain compression of in stages. First, we show how to partition the set of rotations of to obtain the meta-rotations of . We then find precedence relations among these meta-rotations to obtain . Finally, we show .

Notice that can be represented by its Hasse diagram . Each edge of contains exactly one (not necessarily unique) rotation of . Then, by Lemma 1, for any two stable matchings such that , all paths from to in contain the same set of rotations.

Definition 2.

For , is said to be an -direct successor of iff and there is no such that . Let be a sequence of matchings in such that is an -direct successor of for all . Then any path in from to containing , for all , is called an -path.

Let and denote the boy-optimal and girl-optimal matchings, respectively, in . For with , let denote the set of rotations contained on any -path from to . Further, let and denote the set of rotations contained on any path from to and to , respectively in . Define the following set whose elements are sets of rotations.

Lemma 4.

is a partition of .

Proof.

First, we show that any rotation must be in an element of . Consider a path from to in the such that goes from to via an -path. Since is a path from to , all rotations of are contained on by Lemma 1. Hence, they all appear in the sets in .

Next assume that there are two pairs of -direct successors such that and . The set of rotations eliminated from to is

Similarly,

Therefore,

Let , we have

Hence,

Since and , . Therefore,

and hence is not a -direct successor of , leading to a contradiction. ∎

We will denote and by and , respectively. The elements of will be the meta-rotations of . Next, we need to define precedence relations among these meta-rotations to complete the construction of . For a meta-rotation , , define the following subset of :

Lemma 5.

For each meta-rotation , , forms a sublattice of .

Proof.

Take two matchings such that and are supersets of . Then and are also supersets of . ∎

Let be the boy-optimal matching in the lattice . Let be any -path from to and let be the set of meta-rotations appearing before on .

Lemma 6.

The set does not depend on . Furthermore, on any -path from containing , each meta-rotation in appears before .

Proof.

Since all paths from to give the same set of rotations, all -paths from to give the same set of meta-rotations. Moreover, must appear last in the any -path from to ; otherwise, there exists a matching in preceding , giving a contradiction. It follows that does not depend on .

Let be an -path from that contains matchings , where is an -direct successor of . Let denote the meta-rotation that is contained on edge . Suppose there is a meta-rotation such that does not appear before on . Then contains but not . Therefore is a matching in preceding , giving is a contradiction. Hence all matchings in must appear before on all such paths . ∎

Finally, add precedence relations from all meta-rotations in to , for each meta-rotation in . Also, add precedence relations from all meta-rotations in to . This completes the construction of . Below we show that is indeed a compression of , but first we need to establish that this construction does yield a valid poset.

Lemma 7.

satisfies transitivity and anti-symmetry.

Proof.

First we prove that satifies transitivity. Let be meta-rotations such that and . We may assume that . Then and . Since , is a superset of . By Lemma 5, . Similarly, . Therefore , and hence .

Next we prove that satisfies anti-symmetry. Assume that there exist meta-rotations such that and . Clearly . Since , . Therefore, is a superset of . It follows that . Applying a similar argument we get . Now, we get a contradiction, since and are different meta-rotations. ∎

Lemma 8.

is a compression of .

Proof.

Let be rotations in such that . Let be the meta-rotation containing and be the meta-rotation containing . It suffices to show that . Let be an -path from to . Since , must appear before in . Hence, also appears before in . By Lemma 6, as desired. ∎

Finally, the next two lemmas prove that .

Lemma 9.

Any matching in must be in .

Proof.

For any proper closed subset in , let be the matching generated by eliminating meta-rotations in . Let be another proper closed subset in such that , where is a maximal meta-rotation in . Then is a matching in by induction. Since contains , . Therefore, . It follows that . ∎

Lemma 10.

Any matching in must be in .

Proof.

Suppose there exists a matching in such that . Then it must be the case that cannot be partitioned into meta-rotations which form a closed subset of . Now there are two cases.

First, suppose that can be partitioned into meta-rotations, but they do not form a closed subset of . Let be a meta-rotation such that , and there exists such that . By Lemma 5, and hence is a superset of all meta-rotations in , giving is a contradiction.

Next, suppose that cannot be partitioned into meta-rotations in . Since the set of meta-rotations partitions , there exists a meta-rotation such that is a non-empty subset of . Let be the set of meta-rotations preceding in .

is the matching generated by meta-rotations in . Obviously, is a closed subset in . Therefore, . By Lemma 9, . Since , as well. The set of rotations contained on a path from to in is exactly . Therefore, can not be a subset of any meta-rotation, contradicting the fact that is a non-empty subset of . ∎

4 An Alternative View of Compression

In this section we give an alternative definition of compression of a poset; this will be used in the rest of the paper. We are given a poset for a stable matching instance; let be the lattice it generates. Let denote the Hasse diagram of . Consider the following operations to derive a new poset : Choose a set of directed edges to add to and let be the resulting graph. Let be the graph obtained by shrinking the strongly connected components of ; each strongly connected component will be a meta-rotation of . The edges which are not shrunk will define a DAG, , on the strongly connected components. These edges give precedence relations among meta-rotation for poset .

Let be the sublattice of generated by . We will say that the set of edges defines . It can be seen that each set uniquely defines a sublattice ; however, there may be multiple sets that define the same sublattice. Observe that given a compression of , a set of edges defining can easily be obtained. See Figure 2 for examples of sets of edges which define sublattices.

Proposition 5.

The two definitions of compression of a poset are equivalent.

Proof.

Let be a compression of obtained using the first definition. Clearly, for each meta-rotation in , we can add edges to so the strongly connected component created is precisely this meta-rotation. Any additional precedence relations introduced among incomparable meta-rotations can also be introduced by adding appropriate edges.

The other direction is even simpler, since each strongly connected component can be defined to be a meta-rotation and extra edges added can also be simulated by introducing new precedence constraints. ∎

* 0.41324 * 0.41324
Figure 2: (red edges) and (blue edges) define the sublattices in Figure 1, of red and blue elements, respectively.

For a (directed) edge , is called the tail and is called the head of . Let be a closed set of . Then we say that:

  • separates an edge if and .

  • crosses an edge if and .

If does not separate or cross any edge , is called a splitting set w.r.t. .

Lemma 11.

Let be a sublattice of and be a set of edges defining . A matching is in iff the closed subset generating does not separate any edge .

Proof.

Let be a compression corresponding to . By Theorem 4, the matchings in are generated by eliminating rotations in closed subsets of .

First, assume separates . Moreover, assume for the sake of contradiction, and let be the closed subset of corresponding to . Let and be the meta-rotations containing and respectively. Notice that the sets of rotations in and are identical. Therefore, and . Since , there is an edge from to in . Hence, is not a closed subset of .

Next, assume that does not separate any . We show that the rotations in can be partitioned into meta-rotations in a closed subset of . If cannot be partitioned into meta-rotations, there must exist a meta-rotation such that is a non-empty proper subset of . Since consists of rotations in a strongly connected component of , there must be an edge from to in . Hence, separates . Since is a closed subset, can not be an edge in . Therefore, , which is a contradiction. It remains to show that the set of meta-rotations partitioning is a closed subset of . Assume otherwise, there exist meta-rotation and such that there exists an edge from to in . Therefore, there exists , and , which is a contradiction. ∎

Remark 6.

We may assume w.l.o.g. that the set defining is minimal in the following sense: There is no edge such that is not separated by any closed set of . Observe that if there is such an edge, then defines the same sublattice . Similarly, there is no edge such that each closed set separating also separates another edge in .

Definition 3.

W.r.t. an element in a poset , we define four useful subsets of :