A Generalization of Teo and Sethuraman's Median Stable Marriage Theorem
AA Generalization of Teo and Sethuraman’s Median Stable MarriageTheorem ∗ Vijay K. Garg,The University of Texas at Austin,Department of Electrical and Computer Engineering,Austin, TX 78712, USAJanuary 10, 2020
Abstract
Let L be any finite distributive lattice and B be any boolean predicate defined on L such that theset of elements satisfying B is a sublattice of L . Consider any subset M of L of size k of elements of L that satisfy B . Then, we show that k generalized median elements generated from M also satisfy B . Wecall this result generalized median theorem on finite distributive lattices. When this result is appliedto the stable matching, we get Teo and Sethuraman’s median stable matching theorem. Our proof ismuch simpler than that of Teo and Sethuraman. When the generalized median theorem is applied to theassignment problem, we get an analogous result for market clearing price vectors. ∗ Supported by NSF CNS-1812349, CNS-1563544, and the Cullen Trust for Higher Education Endowed Professorship a r X i v : . [ c s . D M ] J a n Introduction
The Stable Matching Problem (SMP) [4] has wide applications in economics, distributed computing, resourceallocation and many other fields [9, 11] with multiple books and survey articles [8–10, 12]. In the standardSMP, there are n men, m , m , . . . , m n , and n women, w , w , . . . , w n , each with their totally orderedpreference list. The goal is to find a matching between men and women such that there is no blocking pair .A pair of a man and a woman is a blocking pair for a matching if they are not married to each other butprefer each other over their partners in that matching. Teo and Sethuraman [13] have shown the followingresult called generalized median stable matching theorem. Theorem 1. [13] Let M , M , ..., M k be k distinct stable marriage solutions. Each man m i has k possiblemates under these matchings. Assign him the woman whose rank is j among the k (possibly nondistinct)women. Then, this assignment gives rise to another, not necessarily distinct, stable marriage solution. Their proof uses properties of fractional stable marriages. In this paper, we prove a generalized mediantheorem on finite distributive lattices. When this result is applied to the stable matching, we get Teo andSethuraman’s median stable matching theorem. Not only our proof is much simpler than [13], but is alsomore general because it is applicable to any problem for which the set of solutions form a finite distributivelattice. For example, the theorem is applicable to constrained stable matchings [6] such as stable marriagesin which regret of man m i is at most that of m j , or stable marriages with forbidden pairs. When thegeneralized median theorem is applied to the market clearing price vectors [3], we get an analogous resultfor market clearing prices. Let L be any finite distributive lattice. Let B be any boolean predicate defined on L . B is called a regular predicate if the set of elements of L that satisfy B form a sublattice of L [7].In the context of the stable matching problem, we define L as the set of all assignment vectors G of size n such that each component of the vector is a number from 0 to n −
1. The interpretation of G [ i ] = j isthat man m i is assigned his choice j as his spouse. The choice 0 corresponds to the top choice and thechoice n − L is natural. For any twoassignment vectors G and H , G ≤ H ≡ ∀ i : G [ i ] ≤ H [ i ]In other words, an assignment G is less than or equal to H iff every man gets at least as good a choice in G as in H .Note that an assignment may not even be a matching if multiple men are assigned the same woman. Wenow define the predicate B on L . An assignment vector G satisfies B if it is a stable marriage. Formally, G satisfies B if1. G is a matching, i.e., all men are assigned different woman, and2. There is no blocking pair in G . A pair of man and woman ( m, w ) is blocking if they are not matchedand they prefer each other over their assignment in G .It is well-known that the set of elements of L that satisfy B form a sublattice of L , i.e., if G and H arestable marriages, then so are G (cid:116) H and G (cid:117) H where (cid:116) and (cid:117) are the join and the meet operation in thelattice L . This result is stated by Knuth [10] and attributed to Conway.Going back to our general set-up, we have a finite distributive lattice L and a regular predicate B defined on L . Let M be any set of elements in L that satisfy B . We now present the general median2heorem on finite distributive lattices. Before we present the theorem, we need to define the notation G [ i ]for any element G in a general finite distributive lattice L . Let J ( L ) denote the sub-poset of L consistingof all the join-irreducible elements of L . Fig. 1(i) shows a distributive lattice L and Fig. 1(ii) shows thesubposet J ( L ). Consider any chain partition of J ( L ). Suppose that J ( L ) has n chains in its partition. Inour example, J ( L ) is partitioned into two chains. Let H be an order ideal of the poset J ( L ), i.e., H is asubset of J ( L ) such that if y belongs to H and x is less than y , then x is also included in H . It is clearthat H can be equivalently represented as a vector of size n such that H [ i ] equals the number of elementsof chain i of J ( L ) included in H . In our example, the order ideal { a, b, c } is represented as the vector(2 ,
1) because it contains two elements from the first chain and one element from the second chain. FromBirkhoff’s Theorem on finite distributive lattices [1, 2, 5], there is 1-1 correspondence between order idealsof the poset J ( L ) and L . Thus, any element G in L is equivalent to an order ideal H of J ( L ) and G [ i ] issame as H [ i ] which is simply the number of elements of chain i included in H . Fig. 1(iii) shows the vectorrepresentation of all elements in lattice L . With this notation, we are ready to present the generalizedmedian theorem for finite distributive lattices. (i) (ii) (iii) c = (0 , db b abcd = (2 , ca a cdL : J ( L ) : cd = (0 , ac = (1 , a = (1 , acd = (1 , abc = (2 , ∅ = (0 , Figure 1: (i) A finite distributive lattice L (ii) poset J ( L ) (iii) Viewing L as order ideals of J ( L ) Theorem 2.
Let L be any finite distributive lattice. Let B be any boolean predicate defined on L such that B is regular. Let M = { M , M , . . . , M k } be any subset of L of size k such that all elements of M satisfy B .Then, for each index r , we get a multiset { M i [ r ] | ≤ i ≤ k } . For any j , we construct the general j -medianstate, G j as follows. For any r , G j [ r ] is given by the j th element in the sorted multiset { M i [ r ] | ≤ i ≤ k } .Then, G j also satisfies B . Before, we give the proof, we illustrate the theorem on an example. Consider the lattice L in Fig. 1.Suppose that we are given M = { (1 , , (0 , , (0 , } such that all three elements in M satisfy B . When wesort the first index, we get the multiset { , , } , and when we sort the second index, we get the multiset { , , } . By using respective components, the median elements generated from M are { (0 , , (0 , , (1 , } .Theorem claims that the median elements also satisfy B . It is easy to verify that the claim is true for thisexample. The element (0 ,
0) satisfies B because both (1 ,
0) and (0 ,
1) satisfy B which is closed under meets.The element (1 ,
2) satisfies B because both (1 ,
0) and (0 ,
2) satisfy B which is also closed under joins.Observe that k = 1, then Theorem 2 is trivially true for any predicate B . When k = 2, it is true iff B isregular because the generalized medians correspond to the meet and the join of two elements. Theorem 2works for any k .We now give the proof. 3 roof. Without loss of generality, we can assume that M k is a maximal element in the set M . We useinduction on k . When k equals 1, the theorem holds trivially because G and M are identical. Supposethat theorem holds for all values of k less than t . Consider the set M = { M i | ≤ i ≤ t − } ∪ { M t } . Let G , G , . . . , G t be median elements of M . We show that they satisfy B . We have two cases. Case 1 : First suppose that M t ≥ M i for all i < t .In this case, the assertion holds by induction since G , G , G t − are the median elements for { M i | ≤ i ≤ t − } and G t is identical to M t . Case 2 : M t is not greater than or equal to M i for some i < t .Let P = { P i | ≤ i < t } be the median elements for { M i | ≤ i < t } . From the induction hypothesis, wecan assume that all elements in P i satisfy B . From the definition of the median elements, we know that P t − ≥ P i for all 1 ≤ i ≤ t −
1. We construct P (cid:48) , a set of size t , from P as follows: P (cid:48) := { P , P , . . . , P t − , P t − (cid:117) M t , P t − (cid:116) M t } . Since t >
1, we are guaranteed that P t − exists and therefore the construction is valid.We first claim that the set of median elements for M and P (cid:48) are identical. Let Q be defined as P ∪ { M t } .It is clear that the median elements of M and Q are identical because P is the set of median elementsof { M i | ≤ i ≤ t − } . We show that the median elements of P (cid:48) and Q are identical. Each of P i for1 ≤ i ≤ t − P (cid:48) [ j ] and the multiset Q [ j ] for any j . Since the lasttwo elements in P (cid:48) are P t − (cid:117) M t , P t − (cid:116) M t , we get that P t − and M t also contribute exactly once for P (cid:48) [ j ]and Q [ j ]. Hence, median elements of P (cid:48) and Q are identical which implies that the median elements of P (cid:48) and M are identical.We next claim that the median elements of P (cid:48) satisfy B . Since, P t − (cid:116) M t is greater than or equal toall elements in P (cid:48) , case 1 is applicable and we get that all the median elements of P (cid:48) satisfy B .By using L as the set of all assignments to men and B as the predicate that the assignment is a stablemarriage, we get the following Teo and Sethurmanan’s result on generalized median stable matching [13]. Corollary 1.
Let M = { M , M , . . . , M k } be any set of stable marriages for any instance of the stablemarriage problem. For any j , if every man is assigned the j th element in the sorted multiset { M i [ m ] | ≤ i ≤ k } , then the resulting assignment is a stable marriage. Moreover, we get the following result on market clearing prices.
Corollary 2.
Let M = { M , M , . . . , M k } be any set of market clearing prices for any instance of themarket clearing prices problem. For any j , if every item is assigned the j th price in the sorted multiset { M i [ m ] | ≤ i ≤ k } , then the resulting price assignment is market clearing. I thank Changyong Hu and Xiong Zheng for some discussions on the topic.
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