18
Weighted Random Popular Matchings* Toshiya Itoh, 1 Osamu Watanabe 2 1 Global Scientific Information and Computing Center, Tokyo Institute of Technology, Meguro-ku, Tokyo 152-8550, Japan; e-mail: [email protected] 2 Department of Mathematical and Computing Sciences, Tokyo Institute of Technology, Meguro-ku, Tokyo 152-8552, Japan; e-mail: [email protected] Received 1 May 2008; accepted 23 August 2009; received in final form 24 August 2009 Published online 1 March 2010 in Wiley Online Library (wileyonlinelibrary.com). DOI 10.1002/rsa.20316 ABSTRACT: For a set A of n applicants and a set I of m items, we consider a problem of computing a matching of applicants to items, i.e., a function M mapping A to I ; here we assume that each applicant x A provides a preference list on items in I . We say that an applicant x A prefers an item p than an item q if p is located at a higher position than q in its preference list, and we say that x prefers a matching M over a matching M if x prefers M(x) than M (x). For a given matching problem A, I , and preference lists, we say that M is more popular than M if the number of applicants preferring M over M is larger than that of applicants preferring M over M, and M is called a popular matching if there is no other matching that is more popular than M. Here we consider the situation that A is partitioned into A 1 , A 2 , ... , A k , and that each A i is assigned a weight w i > 0 such that w 1 > w 2 > ··· > w k > 0. For such a matching problem, we say that M is more popular than M if the total weight of applicants preferring M over M is larger than that of applicants preferring M over M, and we call M an k-weighted popular matching if there is no other matching that is more popular than M. Mahdian (In Proceedings of the 7th ACM Conference on Electronic Commerce, 2006) showed that if m > 1.42n, then a random instance of the (nonweighted) matching problem has a popular matching with high probability. In this article, we analyze the two-weighted matching problem, and we show that (lower bound) if m/n 4/3 = o(1), then a random instance of the two- weighted matching problem with w 1 2w 2 has a two-weighted popular matching with probability o(1); and (upper bound) if n 4/3 /m = o(1), then a random instance of the two-weighted matching problem with w 1 2w 2 has a two-weighted popular matching with probability 1 o(1). © 2010 Wiley Periodicals, Inc. Random Struct. Alg., 37, 477–494, 2010 Keywords: random popular matchings; weighted popular matchings; well-formed matchings Correspondence to: T. Itoh *Supported by JSPS Global COE program “Computationism as a Foundation for the Sciences.” © 2010 Wiley Periodicals, Inc. 477

Weighted random popular matchings

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Weighted Random Popular Matchings*

Toshiya Itoh,1 Osamu Watanabe2

1Global Scientific Information and Computing Center, Tokyo Institute of Technology,Meguro-ku, Tokyo 152-8550, Japan; e-mail: [email protected]

2Department of Mathematical and Computing Sciences, Tokyo Institute ofTechnology, Meguro-ku, Tokyo 152-8552, Japan; e-mail: [email protected]

Received 1 May 2008; accepted 23 August 2009; received in final form 24 August 2009Published online 1 March 2010 in Wiley Online Library (wileyonlinelibrary.com).DOI 10.1002/rsa.20316

ABSTRACT: For a set A of n applicants and a set I of m items, we consider a problem of computinga matching of applicants to items, i.e., a function M mapping A to I; here we assume that eachapplicant x ∈ A provides a preference list on items in I . We say that an applicant x ∈ A prefers anitem p than an item q if p is located at a higher position than q in its preference list, and we say thatx prefers a matching M over a matching M′ if x prefers M(x) than M′(x). For a given matchingproblem A, I , and preference lists, we say that M is more popular than M′ if the number of applicantspreferring M over M′ is larger than that of applicants preferring M′ over M, and M is called apopular matching if there is no other matching that is more popular than M. Here we consider thesituation that A is partitioned into A1, A2, . . . , Ak , and that each Ai is assigned a weight wi > 0 suchthat w1 > w2 > · · · > wk > 0. For such a matching problem, we say that M is more popular than M′

if the total weight of applicants preferring M over M′ is larger than that of applicants preferring M′

over M, and we call M an k-weighted popular matching if there is no other matching that is morepopular than M. Mahdian (In Proceedings of the 7th ACM Conference on Electronic Commerce,2006) showed that if m > 1.42n, then a random instance of the (nonweighted) matching problemhas a popular matching with high probability. In this article, we analyze the two-weighted matchingproblem, and we show that (lower bound) if m/n4/3 = o(1), then a random instance of the two-weighted matching problem with w1 ≥ 2w2 has a two-weighted popular matching with probabilityo(1); and (upper bound) if n4/3/m = o(1), then a random instance of the two-weighted matchingproblem with w1 ≥ 2w2 has a two-weighted popular matching with probability 1−o(1). © 2010 WileyPeriodicals, Inc. Random Struct. Alg., 37, 477–494, 2010

Keywords: random popular matchings; weighted popular matchings; well-formed matchings

Correspondence to: T. Itoh*Supported by JSPS Global COE program “Computationism as a Foundation for the Sciences.”© 2010 Wiley Periodicals, Inc.

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478 ITOH AND WATANABE

1. INTRODUCTION

For a set A of n applicants and a set I of m items, we consider the problem of computing acertain matching of applicants to items, i.e., a function M mapping A to I . Here, we assumethat each applicant x ∈ A provides its preference list defined on a subset Jx ⊆ I . A preferencelist ��x of each applicant x may contain ties among the items and it ranks subsets Jh

x ’s of Jx;that is, Jx is partitioned into J1

x , J2x , . . . , Jd

x , where Jhx is a set of the hth preferred items. We

say that an applicant x prefers p ∈ Jx than q ∈ Jx if p ∈ Jix and q ∈ Jh

x for i < h. For anymatchings M and M′, we say that an applicant x prefers M over M′ if the applicant xprefers M(x) than M′(x), and we say that M is more popular than M′ if the total numberof applicants preferring M over M′ is larger than that of applicants preferring M′ overM. M is called a popular matching [3] if there is no other matching that is more popularthan M. The popular matching problem is to compute this popular matching for given A, I ,and preference lists. This problem has applications in the real world, e.g., mail-based DVDrental systems such as NetFlix [1].

Here, we consider the (general) situation that the set A of applicants is partitioned intoseveral categories A1, A2, . . . , Ak , and that each category Ai is assigned a weight wi > 0such that w1 > w2 > · · · > wk . This setting can be regarded as a case where the applicantsin A1 are platinum members, the applicants in A2 are gold members, the applicants in A3

are silver members, the applicants in A4 are regular members, etc. In a way similar tothe above, we define the k-weighted popular matching problem [5], where the goal is tocompute a popular matching M in the sense that for any other matching M′, the totalweight of applicants preferring M is larger than that of applicants preferring M′. Noticethat the original popular matching problem, which we will call the single category popularmatching problem, is the one-weighted popular matching problem.

We say that a preference list ��x of an applicant x is complete if Jx = I , that is, x showsits preferences on all items, and a k-weighted popular matching problem (A, I , {��x}x∈A) iscalled complete if ��x is complete for every applicant x ∈ A. We also say that a preferencelist ��x of an applicant x is strict if |Jh

x | = 1 for each h, that is, x prefers each item in Jx

differently, and a k-weighted popular matching problem is called strict if ��x is strict forevery applicant x ∈ A.

1.1. Known Results

For the strict single category popular matching problem, Abraham, Irving, Kavitha, andMehlhorn [2] presented a deterministic O(n + m) time algorithm that outputs a popularmatching if it exists; they also showed, for the single category popular matching problemwith ties, a deterministic O(

√nm) time algorithm. To derive these algorithms, Abraham,

Irving, Kavitha, and Mehlhorn introduced the notions of f -items (the first items) and s-items(the second items), and characterized popular matchings by f -items and s-items. Mestre [5]generalized those results to the k-weighted popular matching problem, and he showed adeterministic O(n + m) time algorithm for the strict case, where it outputs a k-weightedpopular matching if any, and a deterministic O(min(k

√n, n)m) time algorithm for the case

with ties.In general, some instances of the complete and strict single category popular matching

problem do not have a popular matching. Mahdian [4] showed that if m > 1.42n, then arandom instance of the complete and strict single category popular matching problem hasa popular matching with probability 1 − o(1); if m < 1.42n, then a random instance of the

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WEIGHTED RANDOM POPULAR MATCHINGS 479

complete and strict single category popular matching problem has a popular matching withprobability o(1).

1.2. Main Results

In this article, we consider the complete and strict two-weighted popular matching problem,and investigate when a random instance of the complete and strict two-weighted popularmatching problem has a two-weighted popular matching. Our results are summarized asfollows.

Theorem 4.1. A random instance of the complete and strict two-weighted popularmatching problem with w1 ≥ 2w2 has a two-weighted popular matching with probabilityP2

L(n, m) = O(m3/n4). Thus if m/n4/3 = o(1), then P2L(n, m) = o(1).

Theorem 5.1. A random instance of the complete and strict two-weighted popularmatching problem with w1 ≥ 2w2 has a two-weighted popular matching with probabilityP2

U(n, m) = 1 − O(n4/m3). Thus if n4/3/m = o(1), then P2U(n, m) = 1 − o(1).

These results imply that for an random instance of the complete and strict two-weightedpopular matching problem with w1 ≥ 2w2, there exists a phase transition for the existenceof popular matchings.

2. PRELIMINARIES

In the rest of this article, we consider the complete and strict two-weighted popular matchingproblem. Let A be the set of n applicants and I be the set of m items. We assume that A ispartitioned into A1 and A2, and we refer to A1 (respectively A2) as the first (respectively thesecond) category. For any constant 0 < δ < 1, we also assume that |A1| = δ|A| = δn and|A2| = (1 − δ)|A| = (1 − δ)n. Let w1 > w2 > 0 be weights of the first category A1 and thesecond category A2, respectively.

We define f -items and s-items [2, 5] as follows: For each applicant x ∈ A1, let f1(x) bethe most preferred item in its preference list ��x, and we call it an f1-item of x. We use F1 todenote the set of all f1-items of applicants x ∈ A1. For each applicant x ∈ A1, let s1(x) bethe most preferred item in its preference list ��x that is not in F1, and we use S1 to denotethe set of all s1-items of applicants x ∈ A1. Similarly, for each applicant y ∈ A2, let f2(y)and s2(y) be the most preferred item in its preference list ��y that is not in F1 and not inF1 ∪ F2, respectively, where we use F2 and S2 to denote the set of all f2-items and s2-items,respectively. From this definition, we have that F1 ∩ S1 = ∅, F1 ∩ F2 = ∅, and F2 ∩ S2 = ∅;on the other hand, we may have that S1 ∩ F2 = ∅ or S1 ∩ S2 = ∅.

For characterizing the existence of k-weighted popular matching, Mestre [5] defined thenotion of “well-formed matching,” which generalizes well-formed matching for the singlecategory popular matching problem [2]. We recall this characterization here. Below weconsider any instance (A, I , {��x}x∈A) of the strict (not necessarily complete) two-weightedpopular matching problem.

Definition 2.1. A matching M is well-formed if by M (1) each x ∈ A1 is matched tof1(x) or s1(x); (2) each y ∈ A2 is matched to f2(y) or s2(y); (3) each p ∈ F1 is matched to

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480 ITOH AND WATANABE

some x ∈ A1 such that p = f1(x); and (4) each q ∈ F2 is matched to some y ∈ A2 such thatq = f2(y).

Mestre [5] showed that the existence of a two-weighted popular matching is almostequivalent to that of a well-formed matching. Precisely, he proved the following character-ization.

Proposition 2.1. Let (A, I , {��x}x∈A) be an instance of the strict two-weighted popularmatching problem [5]. Any two-weighted popular matching of (A, I , {��x}x∈A) is a well-formed matching, and if w1 ≥ 2w2, then any well-formed matching of (A, I , {��x}x∈A) is atwo-weighted popular matching.

For an instance of the single category popular matching problem, it suffices to consideronly a set F of f -items and a set S of s-items [4]. For an instance of the two-weighted popularmatching problem, however, we need to separately consider f1-items, s1-items, f2-items, ands2-items. Some careful analysis is necessary, in particular, because in general, we may havethe situation S1 ∩ F2 = ∅, which makes our probabilistic analysis much harder than (andquite different from) the single category case.

Consider an instance (A, I , {��x}x∈A) of the strict (not necessarily complete) two-weightedpopular matching problem with weights w1 ≥ 2w2. As shown above, the existence of atwo-weighted popular matching is characterized by that of a well-formed matching, whichis determined by the structure of f1-, f2-, s1-, and s2-items. Here, we introduce a graphG = (V , E) for investigating this structure, and in the following discussion, we will mainlyuse this graph. The graph G = (V , E) is defined by a set V = F1 ∪ S1 ∪ F2 ∪ S2 of vertices,and the following set E of edges.

E = {(f1(x), s1(x)) : x ∈ A1} ∪ {(f2(y), s2(y)) : y ∈ A2}.We use E1 and E2 to denote the sets of edges defined for applicants in A1 and A2, respectively,i.e., the former and the latter sets of the above. In the following, the graph G = (V , E)

defined above is called an fs-relation graph for (A, I , {��x}x∈A). Note that this fs-relationgraph G = (V , E) consists of M = |V | ≤ m vertices and n = |A| edges. If e1 ∈ E1 ande2 ∈ E2 are incident with the same vertex p ∈ V , then we have either p ∈ S1 ∩ F2 orp ∈ S1 ∩ S2.

We now characterize the existence of a well-formed matching as follows.

Lemma 2.1. An instance (A, I , {��x}x∈A) of the strict two-weighted popular matchingproblem has a well-formed matching iff its fs-relation graph G = (V , E) has an orientationO on edges such that (a) each p ∈ V has at most one incoming edge in E1 ∪ E2; (b) eachp ∈ F1 has one incoming edge in E1; and (c) each q ∈ F2 has one incoming edge in E2.

Proof. Consider any instance (A, I , {��x}x∈A) of the strict two-weighted popular matchingproblem, where A = A1 ∪ A2, and let G = (V , E) be its fs-relation graph.

First assume that this instance has a well-formed matching M. Define an orientationO on edges of the graph G = (V , E) as follows: For each applicant a ∈ Ai, orient anedge ea = (fi(a), si(a)) ∈ Ei toward M(a). Since M is a matching between A and I , wehave that each p ∈ V has at most one incoming edge. From the condition (3) of Definition2.1, it follows that each p ∈ F1 has one incoming edge in E1, and from the condition (4) ofDefinition 2.1, it follows that each q ∈ F2 has one incoming edge in E2. Thus the orientationO on edges of G = (V , E) satisfies the conditions (a), (b), and (c).

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Assume that the graph G = (V , E) has an orientation O on edges satisfying the conditions(a), (b), and (c). Then we define a matching M as follows: For each x ∈ A1, its f1-item f1(x)(respectively s1-item s1(x)) is matched to x if O orients the edge ex = (f1(x), s1(x)) ∈ E1

by f1(x) ← s1(x) (respectively f1(x) → s1(x)), and for each y ∈ A2, its f2-item f2(y)(respectively s2-item s2(y)) is matched to y if O orients the edge ey = (f2(y), s2(y)) ∈ E2

by f2(y) ← s2(y) (resp. f2(y) → s2(y)). From the condition (a) of the orientation O, it isimmediate to see that M is a matching for (A, I , {��x}x∈A). From the definition of the graphG = (V , E), we have that M satisfies the conditions (1) and (2) of Definition 2.1. Thecondition (b) of the orientation O implies that each p ∈ F1 is matched to x ∈ A1 by M,where f1(x) = p, and the condition (c) of the orientation O guarantees that each q ∈ F2 ismatched to y ∈ A2 by M, where f2(y) = q. Thus the matching M for (A, I , {��x}x∈A) satisfiesthe conditions (1), (2), (3), and (4) of Definition 2.1.

3. CHARACTERIZATION FOR THE TWO-WEIGHTED POPULARMATCHING PROBLEM

In this section, we present necessary and sufficient conditions for an instance of the stricttwo-weighted popular matching problem to have a two-weighted popular matching. For aninstance (A, I , {��x}x∈A) of the strict two-weighted popular matching problem, we discuss thecondition in terms of the structure of its fs-relation graph G = (V , E).

Theorem 3.1. An instance (A, I , {��x}x∈A) of the strict two-weighted popular matchingproblem has a well-formed matching iff its fs-relation graph G = (V , E) contains, as itssubgraph, none of the graphs G1

1, G21, G3

1, G2, nor G3 of Fig. 1.

Proof. Assume that the graph G = (V , E) contains one of the graphs G11, G2

1, G31, G2, and

G3 of Fig. 1. We first consider the case where G contains G11 (in a way similar to this case,

we can handle the cases where G contains G21 or G3

1). If the edge (vi2 , vi3) ∈ E1 of G11 is

oriented by vi2 ← vi3 , then the edge (vi1 , vi2) ∈ E2 is oriented by vi1 ← vi2 to meet thecondition (a) of Lemma 2.1. However, this does not meet the condition (c) of Lemma 2.1,since the vertex vi2 ∈ S1 ∩ F2 ⊆ F2 has no incoming edges in E2. So the edge (vi2 , vi3) ∈ E1

must be oriented by vi2 → vi3 . It is also the case for the edge (vik−2 , vik−1) ∈ E1, that is,(vik−2 , vik−1) ∈ E1 must be oriented by vik−1 → vik−2 . These facts imply that there existsa vertex vij , 2 < j < k − 1, that has at least two incoming edges under the orientation,which violates the condition (a) of Lemma 2.1. Thus if G contains G1

1, then the instancedoes not have a well-formed matching. Similarly we can show that if G contains G2, thenthe instance does not have a well-formed matching. For the case where G contains G3, thecycle C1 of G3 must be oriented in one of the clockwise and counterclockwise orientationsto satisfy the condition (a) of Lemma 2.1. This implies that some vertex vi1 on C1 has oneincoming edge. It is also the case for C2 and we may assume a similar vertex vik on C2.Let P = vi1 , vi2 , . . . , vik−1 , vik be a path connecting vi1 and vik . To meet the condition (a) ofLemma 2.1, the edges (vi1 , vi2) and (vik−1 , vik ) must be oriented by vi1 → vi2 and vik → vik−1 ,respectively. This implies that there exists some vertex vij , 2 < j < k, on P that has at leasttwo incoming edge under the orientation, which violates the condition (a) of Lemma 2.1.Thus if G contains G3, then the instance does not have a well-formed matching.

Now we prove the if-direction, and for this, assume that the graph G = (V , E) does notcontain any of the graphs G1

1, G21, G3

1, G2, and G3. We argue by considering cycles and the

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482 ITOH AND WATANABE

Fig. 1. (a) a path P = vi1 , vi2 , . . . , vik that has vertices vi2 , vik−1 ∈ S1 ∩ F2 such that (vi2 , vi3) ∈ E1 and(vik−2 , vik−1) ∈ E1; (b) vi1 = vik in the subgraph G1

1; (c) vi2 = vik−1 in the subgraph G11; (d) a cycle C

and a path P = vi1 , vi2 , . . . , vik joining C at vik that has a vertex vi2 ∈ S1 ∩ F2 such that (vi2 , vi3) ∈ E1;(e) a connected component including cycles C1 and C2.

other part separately. Let C be the set of cycles in G. Delete all cycle edges in G = (V , E)

and obtain a forest F . Let FJ and FNJ be the partition of F such that

FJ = {T ∈ F : T is incident with some cycle in G = (V , E)};FNJ = {T ∈ F : T is not incident with any cycle in G = (V , E)}.

We first orient edges in each cycle C of C. Since G = (V , E) does not contain any of G11, G2

1,and G3

1, we can define an orientation OcycC on the edges of the cycle C in one of the clockwise

and counterclockwise orientations to meet the conditions (a), (b), and (c) of Lemma 2.1.Next we orient forest edges. For any tree T and any vertex v in T , we define the standard

tree orientation started from v to be a simple orientation starting from v as a root nodetowards the leaves of T . Now we consider trees T ∈ FJ. From the assumption that G doesnot contain G3, it follows that each T ∈ FJ is incident with a unique cycle CT at some vertexvT . Then from the assumption that G does not contain G2, we have that the standard treeorientation started from vT satisfies the desired conditions on all vertices in T , where theconditions of Lemma 2.1 on the vertex vT (if any applicable ones exist) are also satisfiedtogether with the orientation Ocyc

CTof CT .

We then consider trees in FNJ, and for this we partition FNJ as follows:

FNJk = {T ∈ FNJ : there exist k ≥ 0 vertices v ∈ S1 ∩ F2 of T}.

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We say that a tree Tv rooted at v is critical if there exists a path P = v0 = v, v1, v2, . . . , vt

from the root v to a leaf of Tv such that vi ∈ F1, vi+1 ∈ S1 ∩ F2, and vi+2 ∈ S2 for some0 ≤ i ≤ t − 2. Note that for a noncritical tree Tv rooted at v, the standard tree orientationstarted from v satisfies the conditions of Lemma 2.1 on all vertices (except for v) of Tv. Byinduction on the number k ≥ 0 of vertices in S1 ∩ F2 of T ∈ FNJk , we show that T has anorientation satisfying the conditions of Lemma 2.1 on all vertices of T .

For each T ∈ FNJ0 , we choose any vertex u ∈ S1 ∪ S2 of T (such u certainly exists)

and define an orientation ONJ0T on T as the standard tree orientation started from u. Then

since u ∈ F1 ∪ F2, this clearly meets the conditions of Lemma 2.1 on all vertices in T(including u).

For any k ≥ 1 and each 0 ≤ � ≤ k − 1, assume that T ∈ FNJ� has an orientation ONJ�T

satisfying the conditions of Lemma 2.1 on all vertices of T . For T ∈ FNJk , choose vertexvT ∈ S1 ∩ F2, and let NF1(vT ) be the set of f1-items adjacent with vT and NS2(vT ) be theset of s2-items adjacent with vT . For each p ∈ NF1(vT ) and q ∈ NS2(vT ), delete the edges(vT , p) and (vT , q) from T , and let Tp and Tq be the separated subtrees of T rooted at p andq, respectively. Since G does not contain G1

1, we have that there exist no critical trees in{Tp : p ∈ NF1(vT )} and there exists at most one critical tree in {Tq : q ∈ NS2(vT )}. For the

case where there exist no critical trees in {Tq : q ∈ NS2(vT )}, we define an orientation ONJkT

as follows (here choose any vertex r ∈ NS2(vT ) and let it be fixed):

1. For each p ∈ NF1(vT ), orient Tp by the standard tree orientation started from p, andorient the edge (vT , p) by vT → p.

2. For each q ∈ NS2(vT ) − {r}, orient Tq by the standard tree orientation started from q,and orient the edge (vT , q) by vT → q.

3. Orient Tr by the standard tree orientation started from r, and orient the edge (vT , r)by vT ← r.

From the fact that there exist no critical trees in {Tp : p ∈ NF1(vT )}, it follows that foreach p ∈ NF1(vT ), the standard tree orientation started from p together with the orientationof the edge (vT , p) meets the conditions of Lemma 2.1 on all vertices of Tp. From theassumption that there exist no critical trees in {Tq : q ∈ NS2(vT )}, it follows that for eachq ∈ NS2(vT ) − {r}, the standard tree orientation started from q together with the orientationof the edge (vT , q) meets the conditions of Lemma 2.1 on all vertices of Tq. The standardtree orientation started from r ∈ NS2(vT ) together with the orientation of the edge (vT , r)guarantees that vT ∈ S1 ∩ F2 has an incoming edge in E2 and r ∈ S2 has no incoming edge.This implies that the orientation ONJk

T meets the conditions of Lemma 2.1 on all verticesof T . Let us consider the case where there exists a critical tree Tr ∈ {Tq : q ∈ NS2(vT )}.In this case, it is obvious that Tr ∈ FNJ� for some 0 ≤ � ≤ k − 1, and by the inductionhypothesis, we have that Tr has an orientation ONJ�

Trsatisfying the conditions of Lemma 2.1

on all vertices of Tr . Then we define an orientation ONJkT as follows:

1. For each p ∈ NF1(vT ), orient Tp by the standard tree orientation started from p, andorient the edge (vT , p) by vT → p.

2. For each q ∈ NS2(vT ) − {r}, orient Tq by the standard tree orientation started from q,and orient the edge (vT , q) by vT → q.

3. Orient Tr by ONJ�Tr

and orient the edge (vT , r) by vT ← r.

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484 ITOH AND WATANABE

In a way similar to the case where there exist no critical trees in {Tq : q ∈ NS2(vT )}, itis obvious that for each p ∈ NF1(vT ), the standard tree orientation started from p togetherwith the orientation of the edge (vT , p) meets the conditions of Lemma 2.1 on all verticesof Tp; for each q ∈ NS2(vT ) − {r}, the standard tree orientation started from q together withthe orientation of the edge (vT , q) meets the conditions of Lemma 2.1 on all vertices ofTq. The orientation ONJ�

Trtogether with the orientation of the edge (vT , r) guarantees that

vT ∈ S1 ∩ F2 has an incoming edge in E2; r ∈ S2 has at most on incoming edge in E1 ∪ E2.This implies that the orientation ONJk

T meets the conditions of Lemma 2.1 on all verticesof T .

The desired orientation is obtained by combining all these defined orientations.

From Proposition 2.1 and Theorem 3.1, we immediately have the following corollary:

Corollary 3.1. Any instance (A, I , {��x}x∈A) of the strict two-weighted popular matchingproblem with w1 ≥ 2w2 has a two-weighted popular matching iff its fs-relation graphG = (V , E) contains none of the graphs G1

1, G21, G3

1, G2, nor G3 of Fig. 1.

Let us consider a random instance of the complete and strict two-weighted popularmatching problem. Roughly speaking, a natural uniform distribution is considered here.That is, given a set A = A1 ∪ A2 of n applicants and a set I of m items, and we consideran instance obtained by defining a random preference list ��x for each applicant x ∈ A,which is a permutation on I that is chosen independently and uniformly at random. But asdiscussed above for the two-weighted case, the situation is completely determined by thecorresponding fs-relation graph that depends only on the first and second items of applicants.Thus, instead of considering a random instance of the problem, we simply define the firstand second items as follows, and discuss with the fs-relation graph G = (V , E) obtained byf1-, s1-, f2-, and s2-items.

1. For each x ∈ A1, assign an item p ∈ I as an f1-item f1(x) independently and uniformlyat random, and let F1 be the set of all f1-items;

2. For each x ∈ A1, assign an item p ∈ I − F1 as an s1-item s1(x) independently anduniformly at random, and let S1 be the set of all s1-items;

3. For each x ∈ A2, assign an item p ∈ I − F1 as an f2-item f2(x) independently anduniformly at random, and let F2 be the set of all f2-items; and

4. For each x ∈ A2, assign an item p ∈ I − (F1 ∪ F2) as an s2-item s2(x) independentlyand uniformly at random, and let S2 be the set of all s2-items.

It is easy to see that this choice of the first and second items is the same as defining thefirst and second items from a random instance of the complete and strict 2-weighted popularmatching problem.

4. LOWER BOUNDS FOR THE TWO-WEIGHTED POPULARMATCHINBG PROBLEM

Let n be the number of applicants and m be the number of items. Assume that m is largeenough so that m−n ≥ m/c for some constant c > 1, i.e., m ≥ cn/(c−1). For any constant0 < δ < 1, let n1 = δn and n2 = (1 − δ)n be the numbers of applicants in A1 and A2,

Random Structures and Algorithms DOI 10.1002/rsa

WEIGHTED RANDOM POPULAR MATCHINGS 485

Fig. 2. The simplest “Bad” subgraphs G′1.

respectively. In this section, we show a lower bound for m such that a random instance of thecomplete and strict two-weighted popular matching problem has a two-weighted popularmatching with low probability.

Theorem 4.1. A random instance of the complete and strict two-weighted popularmatching problem with w1 ≥ 2w2 has a two-weighted popular matching with probabilityP2

L(n, m) = O(m3/n4). Thus if m/n4/3 = o(1), then P2L(n, m) = o(1).

Proof. Consider a random fs-relation graph G = (V , E). As shown in Corollary 3.1,it suffices to show that G = (V , E) contains one of the graphs G1

1, G21, G3

1, G2, and G3

of Fig. 1 with high probability. But here we focus on one simple such graph, namely, G′1

given in Fig. 2, and in the following, we argue that the probability that G = (V , E) containsG′

1 is high if m/n4/3 = o(1).Let F1 and F2 be the sets of the first items, S1 and S2 be the sets of the second items,

respectively, for applicants in A1 and A2. By the definitions of F1, F2, S1, and S2, we havethat F1 ∩ S1 = ∅, F1 ∩ F2 = ∅, F1 ∩ S2 = ∅, and F2 ∩ S2 = ∅. On the other hand, we mayhave that S1 ∩F2 = ∅ or S1 ∩S2 = ∅. Let R1 = I −F1 and R2 = R1 −F2 = I −(F1 ∪F2). It isobvious that 1 ≤ |F1| ≤ δn and 1 ≤ |F2| ≤ (1−δ)n, which implies that m−δn ≤ |R1| ≤ mand m − n ≤ |R2| ≤ m.

Fix a total order on the set A1 arbitrarily. For any pair of x1, x2 ∈ A1 such that x1 < x2

and any pair of y1, y2 ∈ A2 such that y1 = y2, we simply use �v to denote (x1, x2, y1, y2), andW to denote the set of all such �v’s. Since n1 = δn = |A1| and n2 = (1 − δ)n = |A2|, wehave that for sufficiently large n,

|W | =(

n1

2

)n2(n2 − 1) ≥ δ2(1 − δ)2

3n4. (1)

For each �v = (x1, x2, y1, y2) ∈ W , define a random variable Z�v to be Z�v = 1 if x1, x2, y1,and y2 form the bad graph G′

1 in Fig. 2; Z�v = 0 otherwise. Let Z = ∑�v∈V Z�v. Then from

Chebyshev’s Inequality [6, Theorem 3.3], it follows that

Pr[Z = 0] ≤ Pr[|Z − E[Z]| ≥ E[Z]]= Pr

[|Z − E[Z]| ≥ E[Z]

σZσZ

]≤ σ 2

Z

E2[Z] = Var[Z]E2[Z] . (2)

Random Structures and Algorithms DOI 10.1002/rsa

486 ITOH AND WATANABE

To derive the lower bound for Pr[Z > 0], we estimate the upper bound for Var[Z]/E2[Z].We first consider E[Z]. For each �v ∈ W , it is easy to see that

Pr[Z�v = 1] ≥ 1

m·(

1

m

)2

= 1

m3;

Pr[Z�v = 1] ≤ 1

m·(

1

m − n1

)2

≤ 1

m·(

1

m − n

)2

= c2

m3, (3)

where Inequality (3) follows from the assumption that m − n1 ≥ m − n ≥ m/c for someconstant c > 1. Thus from the estimations for Pr[Z�v = 1], it follows that

E[Z] = E

[∑�v∈W

Z�v

]=

∑�v∈W

E[Z�v] =∑�v∈W

Pr[Z�v = 1] ≥ |W |m3

; (4)

E[Z] = E

[∑�v∈W

Z�v

]=

∑�v∈W

E[Z�v] =∑�v∈W

Pr[Z�v = 1] ≤ c2|W |m3

. (5)

We then consider Var[Z]. From the definition of Var[Z], it follows that

Var[Z] = E

(∑

�v∈W

Z�v

)2 −

(E

[∑�v∈W

Z�v

])2

= E

�v∈W

Z2�v +

∑�v∈W

∑�w∈W−{�v}

Z�vZ�w

(E

[∑�v∈W

Z�v

])2

= E

[∑�v∈W

Z�v

]−

(E

[∑�v∈W

Z�v

])2

+∑�v∈W

∑�w∈W−{�v}

E[Z�vZ�w]

= E[Z] − E2[Z] +∑�v∈W

∑�w∈W−{�v}

E[Z�vZ�w]. (6)

In the following, we estimate the last term of Equality (6). For each �v = (x1, x2, y1, y2) ∈ Wand each integer 0 ≤ h ≤ 2, we say that �w = (x′

1, x′2, y′

1, y′2) ∈ W − {�v} is h-common to �v if

|{x1, x2} ∩ {x′1, x′

2}| = h. For any �w = (x′1, x′

2, y′1, y′

2) ∈ W that is two-common to �v, we havethat x1 = x′

1 and x2 = x′2, because if x1 = x′

2 and x2 = x′1, then x1 = x′

2 > x′1 = x2, which

contradicts the assumption that x1 < x2. For each �v ∈ W , we use W2(�v) to denote the setof �w ∈ W − {�v} that is two-common to �v; W1(�v) to denote the set of �w ∈ W − {�v} that isone-common to �v; W0(�v) to denote the set of �w ∈ W − {�v} that is zero-common to �v. Thenfrom the assumption that m − n ≥ m/c, it follows that

∑�v∈W

∑�w∈W2(�v)

E[Z�vZ�w] ≤{

c4(1 − δ)2

m5n2 + 2c3(1 − δ)

m4n

}|W |; (7)

∑�v∈W

∑�w∈W1(�v)

E[Z�vZ�w] ≤{

4c4δ(1 − δ)2

m6n3 + 4c3δ(1 − δ)

m5n2 + 4c3δ

m5n

}|W |; (8)

∑�v∈W

∑�w∈W0(�v)

E[Z�vZ�w] ≤ E2[Z] +{

2c4δ2(1 − δ)

m6n3 + c4δ2

m6n2

}|W |. (9)

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WEIGHTED RANDOM POPULAR MATCHINGS 487

The proofs of Inequalities (7), (8), and (9) are shown in Appendices A.1, A.2, and A.3,respectively. Thus from Inequalities (5), (6), (7), (8), and (9), it follows that

Var[Z] = E[Z] − E2[Z] +∑�v∈W

∑�w∈W−{�v}

E[Z�vZ�w]

= E[Z] − E2[Z] +∑�v∈W

∑�w∈W2(�v)

E[Z�vZ�w]

+∑�v∈W

∑�w∈W1(�v)

E[Z�vZ�w] +∑�v∈W

∑�w∈W0(�v)

E[Z�vZ�w]

≤ c2|W |m3

+{

c4(1 − δ)2

m5n2 + 2c3(1 − δ)

m4n

}|W |

+{

4c4δ(1 − δ)2

m6n3 + 8c3δ(1 − δ)

m5n2 + 4c3δ

m5n

}|W |

+{

2c4δ2(1 − δ)

m6n3 + c4δ2

m6n2

}|W |

≤ c2|W |m3

{1 + c2(1 − δ)2

m2n2 + 2c(1 − δ)

mn + 4c2δ(1 − δ)2

m3n3

+ 8cδ(1 − δ)

m2n2 + 4cδ

m2n + 2c2δ2(1 − δ)

m3n3 + c2δ2

m3n2

}

≤ c2|W |m3

{1 + (c − 1)2(1 − δ)2 + 2(c − 1)(1 − δ) + 4(c − 1)3δ(1 − δ)2

c

+ 8(c − 1)2δ(1 − δ)

c+ 4(c − 1)δ

m+ 2(c − 1)3δ2(1 − δ)

c+ (c − 1)2δ2

m

},

(10)

where Inequality (10) follows from the assumption that m − n ≥ m/c, i.e., cn/m ≤ c − 1.Thus, it follows that Var[Z] ≤ d|W |/m3 for some constant d that is determined by theconstants 0 < δ < 1 and c > 1. Then from Inequalities (1), (2), and (4), we finally havethat

Pr[Z = 0] ≤ Var[Z]E2[Z] ≤ d|W |

m3· m6

|W |2 = dm3

|W | ≤ 3dm3

δ2(1 − δ)2n4= O

(m3

n4

),

which implies that Pr[Z = 0] = o(1) for any m ≥ n with m/n4/3 = o(1). Therefore, ifm/n4/3 = o(1), then with probability 1 − o(1), we have Z > 0, that is, G = (V , E) containsG′

1 as a subgraph.

5. UPPER BOUNDS FOR THE TWO-WEIGHTED POPULARMATCHING PROBLEM

As shown in Theorem 4.1, a random instance of the complete and strict two-weightedpopular matching problem has a two-weighted popular matching with probability o(1) if

Random Structures and Algorithms DOI 10.1002/rsa

488 ITOH AND WATANABE

m/n4/3 = o(1). Here we consider roughly opposite case, i.e., n4/3/m = o(1), and prove thata random instance has a two-weighted popular matching with probability 1 − o(1).

First, we show the following lemma that will greatly simplify our later analysis.

Lemma 5.1. A random instance G = (V , E) of the fs-relation graphs contains a cycle asa subgraph with probability P2

C(n, m) = O(n2/m2). Thus if n/m = o(1), then P2C(n, m) =

o(1).

Proof. For each � ≥ 2, let E cyc� be the event that a random fs-relation graph G = (V , E)

contains a cycle of length �. We estimate Pr[E cyc� ] for each � ≥ 2. For any pair p, q of distinct

items, let P1(a, p, q) be the probability that an applicant a ∈ A1 chooses p as its f1-item andq as its s1-item and P2(b, p, q) be the probability that an applicant b ∈ A2 chooses p as itsf2-item and q as its s2-item. Then we have that

P1(a, p, q) ≤ 1

(m − n)2; (11)

P2(b, p, q) ≤ 1

(m − n)2. (12)

We state a derivation for the bound (11) below (the bound (12) is similar and omitted here).For this, consider any applicant a ∈ A1, and let f1,a (respectively s1,a) be a random variabledenoting an f1-item (respectively an s1-item) that a chooses. Also let F ′

1 be a random variabledenoting the set of f1-items that applicants in A1 other than a choose. For any X ⊆ I , letPr(s1,a = q : f1,a = p, F ′

1 = X) be the probability that s1,a = q under the condition that achooses p as its f1-item and F ′

1 = X . Thus, we have that

P1(a, p, q) =∑X⊆I

Pr(f1,a = p, F ′1 = X) · Pr(s1,a = q | f1,a = p, F ′

1 = X)

=∑X⊆I

Pr(f1,a = p) · Pr(F ′1 = X) · Pr(s1,a = q | f1,a = p, F ′

1 = X)

≤ Pr(f1,a = p) · 1

m − n×

∑X⊆I

Pr(F ′1 = X)

≤ 1

m· 1

m − n×

∑X⊆I

Pr(F ′1 = X) = 1

(m − n)2.

Fix a cycle C� of length � ≥ 2 arbitrarily and assume that the cycle C� consists of � distinctapplicants aj1 , aj2 , . . . , aj� and � distinct items ph1 , ph2 , . . . , ph�

. For each applicant aji anda pair of items phi , phi+1 that aji is incident with, we have that phi is an f -item and phi+1 isan s-item or that phi is an s-item and phi+1 is an f -item. Then the probability that a randomfs-relation graph G = (V , E) contains the cycle C� is at most 2�/(m − n)2�.

We count the number Nn,m(�) of distinct cycles of length � ≥ 2. Choose � distinct itemsph1 , ph2 , . . . , ph�

from the set I of m items and � distinct applicants aj1 , aj2 , . . . , aj� from theset A of n applicants, and construct a cycle of length �, which is formed by the verticesph1 , ph2 , . . . , ph�

and the edges aj1 , aj2 , . . . , aj� . By identifying those cycles by rotations andreflections, it is obvious that

Random Structures and Algorithms DOI 10.1002/rsa

WEIGHTED RANDOM POPULAR MATCHINGS 489

Nn,m(�) =

(m

)�!

(n

)if � = 2;

1

2��!

(m

)�!

(n

)if � ≥ 3,

and this implies that Nn,m(�) ≤ m�n� for each � ≥ 2. So we have that Pr[E cyc� ] ≤

2�m�n�/(m − n)2� for each � ≥ 2. Thus, it immediately follows that

Pr[G contains a cycle] = Pr

[⋃�≥2

E cyc�

]≤

∑�≥2

Pr[E cyc

] ≤∑�≥2

2�m�n�

(m − n)2�≤

∑�≥2

(2c2n

m

)�

= 4c4n2

m2

∑h≥0

(2c2n

m

)h

= O

(n2

m2

),

where the last inequality follows from the the assumption that m − n ≥ m/c for someconstant c > 1 and the last equality follows from the assumption that n/m = o(1) andc > 1 is a constant. Thus if n/m = o(1), then a random fs-relation graph G = (V , E)

contains a cycle as a subgraph with probability o(1).

Theorem 5.1. A random instance of the complete and strict two-weighted popularmatching problem with w1 ≥ 2w2 has a two-weighted popular matching with probabilityP2

U(n, m) = 1 − O(n4/m3). Thus if n4/3/m = o(1), then P2U(n, m) = 1 − o(1).

Proof. Consider a random fs-relation graph G = (V , E) corresponding to a randominstance of the complete and strict two-weighted popular matching problem. By Lemma5.1 and the assumption that n4/3/m = o(1), we know that the fs-relation graph G = (V , E)

contains any of the graphs G21, G3

1, G2, and G3 of Fig. 1 with vanishing probability o(1).Thus in the remainder of the proof, we estimate the probability that G = (V , E) containsG1

1 of Fig. 1.For any � ≥ 4, let Bpath

� be the event that G = (V , E) contains G11 of length � ≥ 4 and

Epath� be the event that G = (V , E) contains a path of length � ≥ 4. Since G1

1 of length � is apath of length �, we have that Pr[Bpath

� ] ≤ Pr[Epath� ]. For each � ≥ 4, we estimate Pr[Epath

� ].Fix a path P� of length � ≥ 4 arbitrarily. In a way similar to the proof of Lemma 5.1, we canshow that the probability that a random fs-relation graph G = (V , E) contains the path P� isat most 2�/(m−n)2�. We count the number Mn,m(�) of distinct path of length � ≥ 4. Choose� + 1 distinct items ph1 , ph2 , . . . , ph�+1 from the set I of m items and � distinct applicantsaj1 , aj2 , . . . , aj� from the set A of n applicants, and construct a path of length �, which isformed by the vertices ph1 , ph2 , . . . , ph�+1 and the edges aj1 , aj2 , . . . , aj� . By identifying thosepaths by reverses, it is obvious that for each � ≥ 4,

Mn,m(�) = 1

2(� + 1)!

(m

� + 1

)�!

(n

)≤ m�+1n�,

Random Structures and Algorithms DOI 10.1002/rsa

490 ITOH AND WATANABE

and this implies that Pr[Epath� ] ≤ 2�m�+1n�/(m − n)2� for each � ≥ 4. Thus it follows that

Pr[G contains a subgraph G11] = Pr

[⋃�≥4

Bpath�

]≤

∑�≥4

Pr[Bpath

] ≤∑�≥4

Pr[Epath

]

≤∑�≥4

2�m�+1n�

(m − n)2�≤

∑�≥4

(c

m

)2�

2�m�+1n� = m∑�≥4

(2c2n

m

)�

= c8n4

m3

∑h≥0

(2c2n

m

)h

= O

(n4

m3

),

where the last inequality follows from the the assumption that m − n ≥ m/c for someconstant c > 1 and the last equality follows from the assumption that n4/3/m = o(1) andc > 1 is a constant. Notice that n/m = o(1) if n4/3/m = o(1). Thus from Lemma 5.1and Corollary 3.1, it follows that if n4/3/m = o(1), then a random instance of the completeand strict two-weighted popular matching problem has a two-weighted popular matchingwith probability 1 − o(1).

6. CONCLUDING REMARKS

In this article, we have analyzed the two-weighted matching problem, and have shown that(Theorem 4.1) if m/n4/3 = o(1), then a random instance of the complete and strict two-weighted popular matching problem with w1 ≥ 2w2 has a two-weighted popular matchingwith probability o(1); (Theorem 5.1) if n4/3/m = o(1), then a random instance of thecomplete and strict two-weighted popular matching problem with w1 ≥ 2w2 has a two-weighted popular matching with probability 1 − o(1). These results imply that there existsa threshold m ≈ n4/3 to admit two-weighted popular matchings, which is quite differentfrom the case for the single category popular matching problem due to Mahdian [4].

Theorem 4.1 can be trivially generalized to any multiple category case; that is, with thesame proof, we have the following bound.

Theorem 6.1. For any integer k > 2, a random instance of the complete and strict k-weighted popular matching problem with wi ≥ 2wi+1 (1 ≤ i ≤ k − 1) has a k-weightedpopular matching with probability Pk

L(n, m) = O(m3/n4). Thus if m/n4/3 = o(1), thenPk

L(n, m) = o(1).

Then an interesting problem is to show some upper bound result by generalizingTheorem 5.1 for any integer k > 2, maybe under the condition that wi ≥ 2wi+1 for alli, 1 ≤ i ≤ k − 1.

ACKNOWLEDGMENTS

The authors wish to thank anonymous referees for their valuable comments and insightfulsuggestions on an earlier version of the article.

Random Structures and Algorithms DOI 10.1002/rsa

WEIGHTED RANDOM POPULAR MATCHINGS 491

APPENDIX A. PROOFS OF INEQUALITIES

A.1. Proof of Inequality (7)

Let �v = (x1, x2, y1, y2) ∈ W . For each �w = (x′1, x′

2, y′1, y′

2) ∈ W2(�v), let us consider thefollowing cases: (case-0) |{y1, y2} ∩ {y′

1, y′2}| = 0; (case-1) |{y1, y2} ∩ {y′

1, y′2}| = 1. Let

W 02 (�v) = {�w ∈ W2(�v) : |{y1, y2} ∩ {y′

1, y′2}| = 0};

W 12 (�v) = {�w ∈ W2(�v) : |{y1, y2} ∩ {y′

1, y′2}| = 1}

For each �v ∈ W , it is immediate to see that W 02 (�v), W 1

2 (�v) is the partition of W2(�v), and fromthe definitions of W 0

2 (�v) and W 12 (�v), we have that |W 0

2 (�v)| ≤ n22; |W 1

2 (�v)| ≤ 2n2. So fromthe assumption that m − n ≥ m/c for some constant c > 1, it follows that for each �v ∈ W ,

∑�w∈W0

2 (�v)E[Z�vZ�w] ≤

∑�w∈W0

2 (�v)

1

m

(1

m − n1

)4

≤∑

�w∈W02 (�v)

1

m

(1

m − n

)4

≤∑

�w∈W02 (�v)

1

m

(c

m

)4

= c4

m5|W 0

2 (�v)| ≤ c4

m5n2

2

= c4(1 − δ)2

m5n2; (13)

∑�w∈W1

2 (�v)E[Z�vZ�w] ≤

∑�w∈W1

2 (�v)

1

m

(1

m − n1

)3

≤∑

�w∈W12 (�v)

1

m

(1

m − n

)3

≤∑

�w∈W12 (�v)

1

m

(c

m

)3

= c3

m4|W 1

2 (�v)| ≤ 2c3

m4n2

= 2c3(1 − δ)

m4n. (14)

Thus from Inequalities (13) and (14), we finally have that

∑�v∈W

∑�w∈W2(�v)

E[Z�vZ�w] =∑�v∈W

∑�w∈W0

2 (�v)E[Z�vZ�w] +

∑�v∈W

∑�w∈W1

2 (�v)E[Z�vZ�w]

≤∑�v∈W

{c4(1 − δ)2

m5n2 + 2c3(1 − δ)

m4n

}

={

c4(1 − δ)2

m5n2 + 2c3(1 − δ)

m4n

}|W |.

A.2. Proof of Inequality (8)

Let �v = (x1, x2, y1, y2) ∈ W . For each �w = (x′1, x′

2, y′1, y′

2) ∈ W1(�v), we have the fol-lowing cases: (case-0) |{y1, y2} ∩ {y′

1, y′2}| = 0; (case-1) |{y1, y2} ∩ {y′

1, y′2}| = 1; (case-2)

|{y1, y2} ∩ {y′1, y′

2}| = 2. Let

Random Structures and Algorithms DOI 10.1002/rsa

492 ITOH AND WATANABE

W 01 (�v) = {�w ∈ W1(�v) : |{y1, y2} ∩ {y′

1, y′2}| = 0};

W 11 (�v) = {�w ∈ W1(�v) : |{y1, y2} ∩ {y′

1, y′2}| = 1};

W 21 (�v) = {�w ∈ W1(�v) : |{y1, y2} ∩ {y′

1, y′2}| = 2}.

For each �v ∈ W , it is immediate that W 01 (�v), W 1

1 (�v), W 21 (�v) is the partition of W1(�v), and

from the definitions of W 01 (�v), W 1

1 (�v), and W 21 (�v), we have that |W 0

1 (�v)| ≤ 4n1n22; |W 1

1 (�v)| ≤8n1n2; |W 2

1 (�v)| ≤ 4n1. So from the assumption that m − n ≥ m/c for some constantc > 1, it follows that for each �v ∈ W ,

∑�w∈W0

1 (�v)E[Z�vZ�w] ≤

∑�w∈W0

1 (�v)

1

m2

(1

m − n1

)4

≤∑

�w∈W01 (�v)

1

m2

(1

m − n

)4

≤∑

�w∈W01 (�v)

1

m2

(c

m

)4

= c4

m6|W 0

1 (�v)| ≤ 4c4

m6n1n2

2

= 4c4δ(1 − δ)2

m6n3; (15)

∑�w∈W1

1 (�v)E[Z�vZ�w] ≤

∑�w∈W1

1 (�v)

1

m2

(1

m − n1

)3

≤∑

�w∈W11 (�v)

1

m2

(1

m − n

)3

≤∑

�w∈W11 (�v)

1

m2

(c

m

)3

= c3

m5|W 1

1 (�v)| ≤ 8c3

m5n1n2

= 8c3δ(1 − δ)

m5n2; (16)

∑�w∈W2

1 (�v)E[Z�vZ�w] ≤

∑�w∈W2

1 (�v)

1

m2

(1

m − n1

)3

≤∑

�w∈W21 (�v)

1

m2

(1

m − n

)3

≤∑

�w∈W21 (�v)

1

m2

(c

m

)3

= c3

m5|W 2

1 (�v)| ≤ 4c3

m5n1

= 4c3δ

m5n. (17)

Thus from Inequalities (15), (16), and (17), we finally have that

∑�v∈W

∑�w∈W1(�v)

E[Z�vZ�w] =∑�v∈W

∑�w∈W0

1 (�v)E[Z�vZ�w] +

∑�v∈W

∑�w∈W1

1 (�v)E[Z�vZ�w] +

∑�v∈W

∑�w∈W2

1 (�v)E[Z�vZ�w]

≤∑�v∈W

{4c4δ(1 − δ)2

m6n3 + 8c3δ(1 − δ)

m5n2 + 4c3δ

m5n

}

={

4c4δ(1 − δ)2

m6n3 + 8c3δ(1 − δ)

m5n2 + 4c3δ

m5n

}|W |.

Random Structures and Algorithms DOI 10.1002/rsa

WEIGHTED RANDOM POPULAR MATCHINGS 493

A.3. Proof of Inequality (9)

Let �v = (x1, x2, y1, y2) ∈ W . For each �w = (x′1, x′

2, y′1, y′

2) ∈ W0(�v), we have the fol-lowing cases: (case-0) |{y1, y2} ∩ {y′

1, y′2}| = 0; (case-1) |{y1, y2} ∩ {y′

1, y′2}| = 1; (case-2)

|{y1, y2} ∩ {y′1, y′

2}| = 2. Let

W 00 (�v) = {�w ∈ W0(�v) : |{y1, y2} ∩ {y′

1, y′2}| = 0};

W 10 (�v) = {�w ∈ W0(�v) : |{y1, y2} ∩ {y′

1, y′2}| = 1};

W 20 (�v) = {�w ∈ W0(�v) : |{y1, y2} ∩ {y′

1, y′2}| = 2}.

For each �v ∈ W , it is immediate that W 00 (�v), W 1

0 (�v), W 20 (�v) is the partition of W0(�v). For any

�w ∈ W 00 (�v), it is obvious that Pr[Z�v = 1 ∧ Z�w = 1] = Pr[Z�v = 1] × Pr[Z�w = 1], which

implies that

∑�v∈W

∑�w∈W0

0 (�v)E[Z�vZ�w] =

∑�v∈W

∑�w∈W0

0 (�v)Pr[Z�v = 1 ∧ Z�w = 1]

=∑�v∈W

∑�w∈W0

0 (�v)Pr[Z�v = 1] × Pr[Z�w = 1]

=∑�v∈W

Pr[Z�v = 1]∑

�w∈W00 (�v)

Pr[Z�w = 1]

≤∑�v∈W

Pr[Z�v = 1]∑�w∈W

Pr[Z�w = 1]

= E2[Z]. (18)

From the definitions of W 10 (�v) and W 2

0 (�v), we have that |W 10 (�v)| ≤ 2n2

1n2; |W 20 (�v)| ≤ n2

1.Then from the assumption that m − n ≥ m/c for some constant c > 1, it follows that foreach �v ∈ W ,

∑�w∈W1

0 (�v)E[Z�vZ�w] =

∑�w∈W1

0 (�v)

1

m2

(1

m − n1

)4

≤∑

�w∈W10 (�v)

1

m2

(1

m − n

)4

≤∑

�w∈W10 (�v)

1

m2

(c

m

)4

= c4

m6|W 1

0 (�v)| ≤ 2c4

m6n2

1n2

= 2c4δ2(1 − δ)

m6n3; (19)

∑�w∈W2

0 (�v)E[Z�vZ�w] =

∑�w∈W2

0 (�v)

1

m2

(1

m − n1

)4

≤∑

�w∈W20 (�v)

1

m2

(1

m − n

)4

≤∑

�w∈W20 (�v)

1

m2

(c

m

)4

= c4

m6|W 2

0 (�v)| ≤ c4

m6n2

1

= c4δ2

m6n2. (20)

Random Structures and Algorithms DOI 10.1002/rsa

494 ITOH AND WATANABE

Thus from Inequalities (18), (19), and (20), we finally have that∑�v∈W

∑�w∈W0(�w)

E[Z�vZ�w] =∑�v∈W

∑�w∈W0

0 (�w)

E[Z�vZ�w] +∑�v∈W

∑�w∈W1

0 (�w)

E[Z�vZ�w] +∑�v∈W

∑�w∈W2

0 (�w)

E[Z�vZ�w]

≤ E2[Z] +∑�v∈W

{2c4δ2(1 − δ)

m6n3 + c4δ2

m6n2

}

= E2[Z] +{

2c4δ2(1 − δ)

m6n3 + c4δ2

m6n2

}|W |.

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[2] D. J. Abraham, R. W. Irving, T. Kavitha, and K. Mehlhorn, Popular matchings, In Proceedingsof the 16th Annual ACM-SIAM Symposium on Discrete Algorithms, 2005, pp. 424–432.

[3] P. Gardenfors, Match making: Assignment based on bilateral preferences, Behav Sci 20 (1975),166–173.

[4] M. Mahdian, Random popular matchings, In Proceedings of the 7th ACM Conference onElectronic Commerce, Ann Arbor, MI, 2006, pp. 238–242.

[5] J. Mestre, Weighted popular matchings, In Proceeding of the 33rd International Colloquiumon Automata, Languages, and Programming (Part I), Lecture notes in computer science 4051,Springer, Berlin/Heidelberg, 2006, pp. 715–726.

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Random Structures and Algorithms DOI 10.1002/rsa