Weighted Random Popular Matchings

W eigh ted Random P opular Matc hings Toshiy a Itoh Osamu W a t anabe titoh@dac.gsic.titech.ac.jp w atanab e@is.titech.ac.jp Global Sci. Info rm. and Comput. Center Dept. of Math. and Co mput. Sys. T oky o Institute of T echnology T oky o Institute of T echnology Meguro-ku, T okyo 15 2-8550 , Japan Meguro-ku, T okyo 15 2-8552 , Japan Abstract: F or a set A of n applican ts and a set I of m items, w e consider a problem of computing a matc h - ing of applican ts to items, i.e., a fu nctio n M mapping A to I ; here w e assume that eac h applican t x ∈ A pro vides a pr e fer enc e list on items in I . W e say that an applican t x ∈ A pr efers an item p than an item q if p is located at a higher p ositio n than q in its preference list, and w e sa y that x pr efers a mat c hing M o v er a matc hing M ′ if x pr efers M ( x ) o v er M ′ ( x ). F or a g iv en matc hing problem A , I , and preference lists, w e sa y that M is mor e p opular than M ′ if the n umber of applicants p referring M ov er M ′ is larger than that of applican ts preferrin g M ′ o v er M , and M is calle d a p opular matching if th ere is no other matc h ing that is more p opular than M . Here w e consider the situation that A is partitioned in to A 1 , A 2 , . . . , A k , and that eac h A i is assigned a w eigh t w i > 0 such that w 1 > w 2 > · · · > w k > 0. F or suc h a matc h ing problem, w e sa y that M is mor e p opular than M ′ if the total weigh t of applican ts p referring M o v er M ′ is larger than that of applican ts pr e ferring M ′ o v er M , and w e call M an k -weighte d p opular matching if there is no other matc hing that is more p opular than M . Mahdian [ In Proc. of the 7th A CM Con f erence on Elec- tronic Commerce, 2 006] show ed t hat if m > 1 . 42 n , t hen a r a ndom instance of the (non weig ht ed) matc hing problem has a p opu la r matc hing with high probabilit y . In this pap er, w e analyze the 2-wei gh ted matc hin g problem, and we sho w that (lo wer b ound) if m/n 4 / 3 = o (1), then a random instance of the 2-w eigh ted matc h ing pr ob lem with w 1 ≥ 2 w 2 has a 2-w eigh ted p opular matc h ing with probabilit y o (1); and (upp er b ound) if n 4 / 3 /m = o (1), then a random instance o f the 2-we igh ted matc h ing p roblem with w 1 ≥ 2 w 2 has a 2-weig h ted p opular matc hing with probability 1 − o (1 ). Key W ords: Random Popular Matc hings, W eigh ted Popular Matc h ings, W ell-F ormed Matc hings. 1 In tro du cti on F or a set A of n applican ts and a set I of m items, w e consider the problem of computing a certain matc hing of applicants to it ems, i.e. , a fu nctio n M mapping A to I . Here we assume that eac h applican t x ∈ A pro- vides its pr ef er enc e list defined on a subset J x ⊆ I . A preference list ~ ℓ x of eac h app lic an t x ma y con tain ties among the items and it ranks subsets J h x ’s of J x ; that is, J x is partitioned in to J 1 x , J 2 x , . . . , J d x , where J h x is a set of the h th preferred items. W e sa y that a n applican t x pr efers p ∈ J x than q ∈ J x if p ∈ J i x and q ∈ J h x for i < h . F or any matc hings M and M ′ , w e say that an applican t x p r efers M o v er M ′ if the app lic an t x prefers M ( x ) o v er M ′ ( x ), and we sa y th at M is mor e p opular than M ′ if the total num b er of applican ts preferring M o ver M ′ is larger than that of app licants preferring M ′ o v er M . M is called a p opular match- ing [6] if there is no other matc h ing that is more p opular than M . The p opular matching pr oblem is to compute this p opular matc hing f or giv en A , I , and preference lists. Th is problem has applications in the real world, e.g., mail-based D VD rent al systems such as NetFlix [1]. Here we consider the (general) situation that th e set A of applican ts is partitioned in to seve ral cate- gories A 1 , A 2 , . . . , A k , and that eac h category A i is assigned a weigh t w i > 0 su c h that w 1 > w 2 > · · · > w k . This setting can b e regarded as a case where the applican ts in A 1 are platin um members, t he applicant s i n A 2 are gold mem b ers, the applican ts in A 3 are silv er members, the applican ts in A 4 are regular mem b ers, etc. In a w ay similar to the ab o ve , we d efine the k -weighte d p opular matching pr oblem [8], wh ere the goal is to compute a p opular matc hin g M in the sense that for an y other matc hing M ′ , the total w eigh t of applican ts preferr ing M is larger than that of applican ts pr e ferring M ′ . Notice that th e original p opular 1 matc h ing problem, whic h we w ill call the single c ate gory p op u lar matc h ing problem, is the 1-w eigh ted p opular matc hing problem. W e sa y that a preference list ~ ℓ x of a n applican t x is c omplete if J x = I , that is, x sho w s its preferences on all items, and a k -we igh ted p opular matc hin g problem ( A, I , { ~ ℓ x } x ∈ A ) is called c omplete if ~ ℓ x is complete for ev ery app li can t x ∈ A . W e also s a y that a pr eference list ~ ℓ x of an applican t x is strict if | J h x | = 1 for eac h h , that is, x prefers eac h item in J x differen tly , and a k -w eigh ted p opular matc h ing problem is calle d strict if ~ ℓ x is strict for ev ery applican t x ∈ A . 1.1 Kno wn Results F or the strict single category p opular matc hing pr oblem, Abraham, et al. [2] presented a deterministic O ( n + m ) time algorithm th a t outp uts a p opular matc hing if it exists; th ey also show ed, for the single catego ry popu la r matc hing problem with ties, a deterministic O ( √ nm ) time algorithm. T o deriv e these al- gorithms, Abraham, et al. introdu ce d the notions of f -items (the first ite ms) and s -items (the second items), and c h arac terized p opular matc hings by f -items and s -items. Mestre [8] generalized those results to the k -w eighte d popu la r matc hing p roblem, and he sh o we d a deterministic O ( n + m ) time algorithm for the strict case , where it outpu ts a k -weig h ted p opular ma tc hing if any , and a dete rministic O (min( k √ n, n ) m ) time algorithm f o r the case with ties. In general, some instances of the co mplete and strict single category p opular matc hing problem do n ot ha v e a p opular matc hing. Answering to a question of when a rand om instance of the complete and strict single catego ry popu la r matc hing p roblem has a p opu lar matc hing, Mahd ia n [7] sh o we d that if m > 1 . 42 n , then a random instance of the p opu la r matc hing problem has a p opular matc hing w it h probabilit y 1 − o (1); he also sho wed that if m < 1 . 42 n , then a random instance o f th e p opular matc h in g problem has a p opular matc h ing with probab ility o (1). 1.2 Main Results In this pap er, w e consider the complete and strict 2-w eigh ted p opular matc hing p roblem, and inv estigate when a random instance of the complete and strict 2-w eigh ted p opular matc hing p roblem has a 2-w eigh ted p opular matc hing. Our results are su mmarized as f o llo ws. Theorem 4.1: I f m/n 4 / 3 = o (1), then a random ins tance of the complete and strict 2-wei gh ted p opu la r matc h ing p r oblem with w 1 ≥ 2 w 2 has a 2-w eigh ted p opular matc hing with probabilit y o (1). Theorem 5.1: I f n 4 / 3 /m = o (1), then a random in sta nce of the complete and strict 2-w eigh ted p opular matc h ing p r oblem with w 1 ≥ 2 w 2 has a 2-w eigh ted p opular matc hing with probabilit y 1 − o (1). F or an in sta nce of the single category p opular matc hing problem, it suffices to c onsider only a s et F of f -items and a set S of s -it ems [7]. F or an instance o f the 2 -w eigh ted p opular matc hing problem, ho w ev er, w e need to sep ar ately consider f 1 -items, s 1 -items, f 2 -items, and s 2 -items; let F 1 , S 1 , F 2 , and S 2 denote these item sets. Some careful a nalysis is necessary , in particular, b eca use in general, w e may hav e the situation S 1 ∩ F 2 6 = ∅ , whic h mak es our probabilistic analysis m uch h a rder than (and quite d ifferent from) the single category case. 2 Preliminaries In the rest of this pap er, w e consider the complete and strict 2-wei gh ted p opular matc h ing p roblem. L e t A b e the set of n applican ts and I b e the set of m items. W e assum e t hat A is p a rtitioned in to A 1 and A 2 , and w e refer to A 1 (resp. A 2 ) as the first (resp. the second) category . F or any constan t 0 < δ < 1, w e also assume that | A 1 | = δ | A | = δn and | A 2 | = (1 − δ ) | A | = (1 − δ ) n . Let w 1 > w 2 > 0 b e weigh ts of th e first categ ory A 1 and th e second category A 2 , resp ectiv ely . 2 W e defin e f -items and s -items [2 , 8] as follo ws: F or eac h applican t x ∈ A 1 , let f 1 ( x ) b e the most preferred item in its p reference list ~ ℓ x , and w e call it an f 1 -item of x . W e u se F 1 to denote the set of all f 1 -items of a pplicant s x ∈ A 1 . F or ea c h app licant x ∈ A 1 , le t s 1 ( x ) b e th e most p referred item in its preference list ~ ℓ x that is not in F 1 , and w e u se S 1 to denote the set of all s 1 -items of ap p lic an ts x ∈ A 1 . Similarly , for eac h a pplican t y ∈ A 2 , le t f 2 ( y ) and s 2 ( y ) b e th e most preferred ite m in its preference list ~ ℓ y that is not in F 1 and not in F 1 ∪ F 2 , resp ecti v ely , where w e u se F 2 and S 2 to denote the set of a ll f 2 -items and s 2 -items, resp ectiv ely . F rom this d efinition, w e ha v e that F 1 ∩ S 1 = ∅ , F 1 ∩ F 2 = ∅ , and F 2 ∩ S 2 = ∅ ; on the other hand , w e ma y ha v e that S 1 ∩ F 2 6 = ∅ or S 1 ∩ S 2 6 = ∅ . F or charac terizing the existence of k -w eight ed p opular matc hing, Mestre [8] defined the n ot ion of “w ell- formed matc hing,” whic h generaliz es well-fo rmed matc hing for the single category p opular m atching p rob- lem [2]. W e recall this charact erization here. Belo w we consider any instance ( A, I , { ~ ℓ x } x ∈ A ) of the strict (not n ec essarily complete) 2-w eigh ted p opular matc h ing pr oblem. Definition 2.1 A matching M is we ll-fo rmed if by M (1) e ach x ∈ A 1 is matche d to f 1 ( x ) or s 1 ( x ); (2) e ach e ach y ∈ A 2 is mat che d to f 2 ( y ) or s 2 ( y ); (3) e ach p ∈ F 1 is mat che d to some x ∈ A 1 such th at p = f 1 ( x ); and (4) e ach q ∈ F 2 is matche d to some y ∈ A 2 such that q = f 2 ( y ) . Mestre [8] show ed that th e existe nce of a 2-w eigh ted p opular matc hin g is almost equiv alent to that of a w ell- formed m a tc hing. Precisely , he prov ed the follo wing c h aracterization. Prop os ition 2.1 ([8]) L et ( A, I , { ~ ℓ x } x ∈ A ) b e an instanc e of the strict 2 - weig h te d p opula r matching pr ob- lem. Any 2 -weighte d p opular matching of ( A, I , { ~ ℓ x } x ∈ A ) is a wel l-forme d matching, and if w 1 ≥ 2 w 2 , then any wel l-forme d matching of ( A, I , { ~ ℓ x } x ∈ A ) is a 2 -weighte d p opular matching. Consider an instance ( A, I , { ~ ℓ x } x ∈ A ) of the strict (not necessarily complete) 2-w eigh ted p opular matc h- ing problem with we igh ts w 1 ≥ 2 w 2 . As s ho wn ab o ve, the existence of a 2-we igh ted p opular matc hing is c haracterized b y that of a we ll-formed matc hing, which is d et ermined b y the stru cture of f 1 -, f 2 -, s 1 -, an d s 2 -items. Here we in tr oduce a graph G = ( V , E ) for in v estigating this structure, and in the follo wing dis- cussion, w e will mainly use this graph. T h e graph G = ( V , E ) is defin ed by a set V = F 1 ∪ S 1 ∪ F 2 ∪ S 2 of v ertices, and the follo win g set E of edges. E = { ( f 1 ( x ) , s 1 ( x )) : x ∈ A 1 } ∪ { ( f 2 ( y ) , s 2 ( y )) : y ∈ A 2 } . W e use E 1 and E 2 to denote the sets of edges defin ed for applicant s in A 1 and A 2 , resp ectiv ely , i.e., the former and th e la tter sets of the ab o v e. In the follo wing, the graph G = ( V , E ) defined ab o v e is called an fs-r elation gr aph for ( A, I , { ~ ℓ x } x ∈ A ). Note that this f s-rela tion graph G = ( V , E ) consists o f M = | V | ≤ m v ertices and n = | A | ed g es. If e 1 ∈ E 1 and e 2 ∈ E 2 are incident to the same verte x p ∈ V , th e n we hav e either p ∈ S 1 ∩ F 2 or p ∈ S 1 ∩ S 2 . This s itu ation mak es the analysis of the 2-we igh ted p opular matc hing problem harder than and differen t from th e one for th e sin gle category case. W e no w c haracterize the existence of a well- formed matc hing as follo ws. Lemma 2.1 An instanc e ( A, I , { ~ ℓ x } x ∈ A ) of the strict 2 -weighte d p opular matching pr oblem has a wel l- forme d matching iff its fs-r elation gr aph G = ( V , E ) has an orientation O on e dges such that (a) e ach p ∈ V has at most one inc oming e dge in E 1 ∪ E 2 ; (b) e ach p ∈ F 1 has one inc oming e dge in E 1 ; and (c) e ach q ∈ F 2 has one inc oming e dge in E 2 . Pro of: Con s ider any ins tance ( A, I , { ~ ℓ x } x ∈ A ) of the strict 2-weig h ted p opular matc hing problem, where A = A 1 ∪ A 2 , and let G = ( V , E ) b e its fs-relation grap h . First assume that this instance h as a w ell-formed matc hing M . De fine an orient ation O on edges of th e graph G = ( V , E ) as follo ws: F or eac h applican t a ∈ A i , orient an edge e a = ( f i ( a ) , s i ( a )) ∈ E i to ward M ( a ). Since M is a matc hing b et wee n A and I , w e ha ve that eac h p ∈ V has at most one in coming edge. F r om the c ondition (3) of Definition 2.1, it f o llo ws that eac h p ∈ F 1 has one incoming edge in E 1 , and from 3 the condition (4) of Definition 2.1, it follo ws that eac h q ∈ F 2 has one incoming edge in E 2 . Thus the ori- en tation O on edges of G = ( V , E ) satisfies the cond it ions (a), (b), and (c). Assume that the graph G = ( V , E ) has an orienta tion O o n edges satisfying the cond it ions (a), (b), and (c). Then we defin e a matc hing M as follo ws: F or eac h x ∈ A 1 , its f 1 -item f 1 ( x ) (resp . s 1 -item s 1 ( x )) is matc h ed to x if O orien ts th e edge e x = ( f 1 ( x ) , s 1 ( x )) ∈ E 1 b y f 1 ( x ) ← s 1 ( x ) (resp . f 1 ( x ) → s 1 ( x )), and for eac h y ∈ A 2 , its f 2 -item f 2 ( y ) (resp. s 2 -item s 2 ( y )) is m atched to y if O orients the edge e y = ( f 2 ( y ) , s 2 ( y )) ∈ E 2 b y f 2 ( y ) ← s 2 ( y ) (resp . f 2 ( y ) → s 2 ( y )). F rom the cond it ion (a) of the orien tation O , it is immediate to see that M is a matc hin g for ( A, I , { ~ ℓ x } x ∈ A ). F r om the definition of the graph G = ( V , E ), w e ha ve that M s a tisfies th e conditions (1) a nd (2) o f Defin it ion 2.1. The condition (b) of the orien tation O imp lie s that eac h p ∈ F 1 is matc hed to x ∈ A 1 b y M , w here f 1 ( x ) = p , and the cond ition (c) of the orien tation O guarantees that eac h q ∈ F 2 is matc hed to y ∈ A 2 b y M , wh er e f 2 ( y ) = q . Thus the matc h ing M for ( A, I , { ~ ℓ x } x ∈ A ) satisfies the conditions (1), (2), (3), and (4) of Definition 2.1. 3 Characterization for the 2-W eigh ted P opular Matc hing Problem In this section, w e presen t necessary and sufficien t conditions for an instance of the strict 2-w eighte d p opular mat c hing problem to ha ve a 2-w eigh ted p opular matc hing. F or an instance ( A, I , { ~ ℓ x } x ∈ A ) of the strict 2-weig h ted p opular matc h ing problem, let G = ( V , E ) b e its fs-relation graph, and consider the sub - graphs G 1 , G 2 , and G 3 of the graph G = ( V , E ) as in Figure 1. • • • • • • v i 1 ∈ S 2 v i 2 ∈ S 1 ∩ F 2 v i 3 ∈ F 1 v i k ∈ S 2 v i k − 1 ∈ S 1 ∩ F 2 v i k − 2 ∈ F 1 E 1 E 1 E 2 E 2 • • • • C v i k v i 3 v i 2 ∈ S 1 ∩ F 2 v i 1 E 1 E 2 • • C 1 C 2 (a) Subgraph G 1 (b) Subgraph G 2 (c) Su bgraph G 3 Figure 1: (a) a path P = v i 1 , v i 2 , . . . , v i k that has vertice s v i 2 , v i k − 1 ∈ S 1 ∩ F 2 suc h that ( v i 2 , v i 3 ) ∈ E 1 and ( v i k − 2 , v i k − 1 ) ∈ E 1 ; (b) a cycle C and a path P = v i 1 , v i 2 , . . . , v i k inciden t to C at v i k that has a v ertex v i 2 ∈ S 1 ∩ F 2 suc h th at ( v i 2 , v i 3 ) ∈ E 1 ; (c) a connected comp onen t includ ing cycles C 1 and C 2 . Theorem 3.1 An instanc e ( A, I , { ~ ℓ x } x ∈ A ) of the strict 2 -weighte d p opular matching pr oblem has a wel l- forme d matching iff i ts fs-r elation g r aph G = ( V , E ) c ontains none of the su b gr aphs G 1 , G 2 , nor G 3 in Figure 1 . Pro of: Assume that the graph G = ( V , E ) contai ns one of th e su b graphs G 1 , G 2 , a nd G 3 in Figure 1 . F or the case where the graph G con tains the sub graph G 1 , if the edge ( v i 2 , v i 3 ) ∈ E 1 is orien ted b y v i 2 ← v i 3 , then the edge ( v i 1 , v i 2 ) ∈ E 2 is oriente d by v i 1 ← v i 2 to satisfy the condition (a) of Lemma 2.1. Ho w ev er, this do es not meet th e condition (c) of Lemma 2.1, since the v ertex v i 2 ∈ S 1 ∩ F 2 ⊆ F 2 has no incoming edges in E 2 . S o the edge ( v i 2 , v i 3 ) ∈ E 1 m ust b e orien ted by v i 2 → v i 3 . It is also the case f o r the edge ( v i k − 2 , v i k − 1 ) ∈ E 1 , that is, ( v i k − 2 , v i k − 1 ) ∈ E 1 m ust b e orien ted by v i k − 1 → v i k − 2 . These f a cts imp ly that 4 there exi sts 2 < j < k − 1 such that t he v ertex v i j ∈ V h a s at least t w o incoming edges, whic h vio lates the condition (a) of Lemma 2.1. Thus if the graph G con tains the subgraph G 1 , then the in s t ance d oes n o t ha v e a w ell-formed m a tc hing. Similarly w e can sho w that if the graph G con tains the subgraph G 2 , then the in sta nce d oes n o t ha ve a well-formed matc hing. Th e case where th e graph G con tains the s u bgraph G 3 can b e argued in a w a y similar to the pro of by Mahdian [7, Lemma 2]. Assume that the graph G = ( V , E ) do es n o t con tain an y of the subgraph G 1 , G 2 , or G 3 and let { C i } i ≥ 1 b e the set of cycles in G . W e first o rient cycles { C i } i ≥ 1 . Since the graph G do es not con tain the subgraph G 1 , w e ca n orien t eac h cycle C i in one of the clockwise and counte rclo c kwise orien tations to meet th e conditions (a), (b), and (c) of Lemma 2.1. F rom th e assumption that the graph G do es not con tain the subgraph G 3 , the remaining edges can b e cat egorized as follo ws: E cyc tree = the s et of edges in subtrees of G that are incident to some cycle C ∈ { C i } i ≥ 1 , and E tree = the set of edges in su btrees of G that are not inciden t to any cyc le C ∈ { C i } i ≥ 1 . Since the graph G do es not co n tain the su bgraphs G 1 and G 2 , w e can orien t edges in E cyc tree a w a y from the cycles to meet the conditions (a), (b), and (c) of Lemma 2.1. Notice that edges in E tree form sub tr e es of G . F or eac h s uc h T , let E 2 T b e the set of edges ( v , u ) that is assigned to some applican t in A 2 and u ∈ S 1 ∩ F 2 . F or eac h edge e = ( v , u ) ∈ E 2 T , we first orien t the edge e by v → u and then the remaining edges in E 2 T are orient ed a w a y from eac h u ∈ S 1 ∩ F 2 . By the assumption that the graph G do es not con tain the su bgraph G 1 , su c h an orienta tion meets th e conditions (a), (b), and (c) of Lemma 2.1 for eac h v ∈ T . F r om Prop osition 2.1 and Theorem 3.1, we immediately ha v e the follo wing corollary: Corollary 3.1 Any instanc e ( A, I , { ~ ℓ x } x ∈ A ) of the strict 2 -weighte d p opular matching pr oblem with w 1 ≥ 2 w 2 has a 2 -weighte d p opular match ing iff its fs-r elation gr aph G = ( V , E ) c ontains none of the sub gr aphs G 1 , G 2 , nor G 3 in Figure 1 . Let us consider a r an d om instance of the complete and strict 2-w eigh ted p opular matc h ing problem. Roughly sp eaking, a n a tural uniform d istribution is consid ered here. Th at is, giv en a set A = A 1 ∪ A 2 of n applican ts and a set I of m items, and we consider an instance obtained b y defin ing a rand om p reference list ~ ℓ x for eac h applican t x ∈ A , whic h is a p erm utation on I that is chose n indep endent ly and u n iformly at random. But as discussed ab o v e for the 2-w eigh ted case, the situation is completely d et ermined by the corresp onding fs-relation graph that d epend s only on the first and second items of applican ts. Thus, instead of considering a random in s t ance of the pr o blem, w e simply define the fi rst an d second items as follo ws , and discuss with the fs-relation graph G = ( V , E ) obtained defin ed b y f 1 -, s 1 -, f 2 -, and s 2 -items. (1) F or eac h x ∈ A 1 , assign an item p ∈ I as a f 1 -item f 1 ( x ) ind e p enden tly and uniformly at random, and let F 1 b e the set of all f 1 -items; (2) F or eac h x ∈ A 1 , assign an i tem p ∈ I − F 1 as a s 1 -item s 1 ( x ) indep endently and un ifo rmly at random, and let S 1 b e the set of all s 1 -items; (3) F or eac h x ∈ A 2 , assign an i tem p ∈ I − F 1 as a f 2 -item f 2 ( x ) indep endent ly and uniformly at random, and let F 2 b e the set of all f 2 -items; and (4) F or eac h x ∈ A 2 , assign an item p ∈ I − ( F 1 ∪ F 2 ) as a s 2 -item s 2 ( x ) indep endently and uniformly at random, an d let S 2 b e the set of all s 2 -items. It is easy to see that this c hoice of first and second items is th e same as defi n ing first and second items from a r a ndom instance of the complete and strict 2-w eigh ted p opular matc hing p roblem. 4 Lo w er Bounds for the 2-W eigh ted P opular Matc hing Problem Let n b e the num b er of applican ts and m b e th e num b er of items. Assume that m is large enough so th a t m − n ≥ m /c for some constan t c > 1, i.e., m ≥ cn/ ( c − 1). F or any constan t 0 < δ < 1, let n 1 = δn 5 and n 2 = (1 − δ ) n b e the num b ers of applicant s in A 1 and A 2 , resp ectiv ely . In this section, w e sho w a lo wer b ound for m su c h that a random instance of the complete and strict 2-w eigh ted p opular matc h ing problem has a 2-w eigh ted p opular matc hing with lo w probabilit y . Theorem 4.1 If m /n 4 / 3 = o (1) , then a r ando m instanc e of the c omplete and strict 2 -weighte d p opular matching pr oblem with w 1 ≥ 2 w 2 has a 2 -weighte d p opular matching with pr ob ability o (1) . Pro of: Consider a random fs-relatio n graph G = ( V , E ). As s h o wn in Corollary 3.1, it suffices to pro v e that G = ( V , E ) con tains one of the graphs G 1 , G 2 , a nd G 3 of Figure 1 with high probabilit y . But h ere w e fo cus on one simple such graph, namely , G ′ 1 giv en Figure 2, and in the follo wing, we argue that the p r obabil- it y that G = ( V , E ) con tains G ′ 1 is high if m/n 4 / 3 = o (1). • • • y 1 ∈ A 2 x 2 ∈ E 1 y 2 ∈ A 2 x 1 ∈ A 1 q ∈ S 1 ∩ F 2 r ∈ S 1 ∩ F 2 p ∈ F 1 Figure 2: T he Simplest “Bad” S u bgraphs G ′ 1 Let F 1 and F 2 b e the s ets of t he first items, S 1 and S 2 b e the sets of the second items, r espectiv ely , for applican ts in A 1 and A 2 . By the definitions of F 1 , F 2 , S 1 , and S 2 , w e ha v e that F 1 ∩ S 1 = ∅ , F 1 ∩ F 2 = ∅ , F 1 ∩ S 2 = ∅ , and F 2 ∩ S 2 = ∅ . On the other hand, w e ma y ha v e that S 1 ∩ F 2 6 = ∅ or S 1 ∩ S 2 6 = ∅ . Let R 1 = I − F 1 and R 2 = R 1 − F 2 = I − ( F 1 ∪ F 2 ). It is ob vious that 1 ≤ | F 1 | ≤ δ n and 1 ≤ | F 2 | ≤ (1 − δ ) n , whic h imp lie s that m − δ n ≤ | R 1 | ≤ m and m − n ≤ | R 2 | ≤ m . F or any p ai r of x 1 , x 2 ∈ A 1 suc h that x 1 < x 2 and an y pair of y 1 , y 2 ∈ A 2 suc h th a t y 1 6 = y 2 , w e simply use ~ v to denote ( x 1 , x 2 , y 1 , y 2 ), and T to denote the set of all such ~ v ’s. Since n 1 = δ n = | A 1 | and n 2 = (1 − δ ) n = | A 2 | , we hav e that for su fficie nt ly large n , | T | = n 1 2 ! n 2 ( n 2 − 1) ≥ δ 2 (1 − δ ) 2 3 n 4 . (1) F or eac h ~ v = ( x 1 , x 2 , y 1 , y 2 ) ∈ T , define a rand o m v ariable Z ~ v to b e Z ~ v = 1 if x 1 , x 2 , y 1 , and y 2 form the bad subgraph G ′ 1 in Figure 2; Z ~ v = 0 otherwise. L et Z = P ~ v ∈ V Z ~ v . Then from Cheb yshev’s Inequalit y [9, Th eo rem 3.3], it follo ws that Pr [ Z = 0 ] ≤ Pr [ | Z − E [ Z ] | ≥ E [ Z ]] = Pr  | Z − E [ Z ] | ≥ E [ Z ] σ Z σ Z  ≤ σ 2 Z E 2 [ Z ] = V ar [ Z ] E 2 [ Z ] . (2) T o derive the lo w er b ound f o r Pr[ Z > 0 ], we estimate the upp er b ound for V ar [ Z ] / E 2 [ Z ]. W e fi rst consider E [ Z ]. F or eac h ~ v ∈ T , it is easy to see that Pr [ Z ~ v = 1] ≥ 1 m ·  1 m  2 = 1 m 3 ; Pr [ Z ~ v = 1] ≤ 1 m ·  1 m − n 1  2 ≤ 1 m ·  1 m − n  2 = c 2 m 3 , (3) 6 where In e qualit y (3) follo ws from the assu m ptio n th a t m − n 1 ≥ m − n ≥ m/c for some constant c > 1. Th us from the estimations for Pr[ Z ~ v = 1], it follo ws that E [ Z ] = E   X ~ v ∈ T Z ~ v   = X ~ v ∈ T E [ Z ~ v ] = X ~ v ∈ T Pr [ Z ~ v = 1] ≥ | T | m 3 ; (4) E [ Z ] = E   X ~ v ∈ T Z ~ v   = X ~ v ∈ T E [ Z ~ v ] = X ~ v ∈ T Pr [ Z ~ v = 1] ≤ c 2 | T | m 3 . (5) W e then consider V ar [ Z ]. F rom the defin ition of V ar [ Z ], it follo ws that V ar [ Z ] = E      X ~ v ∈ T Z ~ v   2    −   E   X ~ v ∈ T Z ~ v     2 = E   X ~ v ∈ T Z 2 ~ v + X ~ v ∈ T X ~ w ∈ T −{ ~ v } Z ~ v Z ~ w   −   E   X ~ v ∈ T Z ~ v     2 = E   X ~ v ∈ T Z ~ v   −   E   X ~ v ∈ T Z ~ v     2 + X ~ v ∈ T X ~ w ∈ T −{ ~ v } E [ Z ~ v Z ~ w ] = E [ Z ] − E 2 [ Z ] + X ~ v ∈ T X ~ w ∈ T −{ ~ v } E [ Z ~ v Z ~ w ] . (6) In the f ollo wing, w e estimate the last term of Equalit y (6). F or eac h ~ v = ( x 1 , x 2 , y 1 , y 2 ) ∈ T and eac h 0 ≤ h ≤ 2, w e sa y that ~ w = ( x ′ 1 , x ′ 2 , y ′ 1 , y ′ 2 ) ∈ T − { ~ v } is h -common to ~ v if |{ x 1 , x 2 } ∩ { x ′ 1 , x ′ 2 }| = h . F or a n y ~ w = ( x ′ 1 , x ′ 2 , y ′ 1 , y ′ 2 ) ∈ T that is 2-common to ~ v , w e ha ve that x 1 = x ′ 1 and x 2 = x ′ 2 , b eca use if x 1 = x ′ 2 and x 2 = x ′ 1 , then x 1 = x ′ 2 > x ′ 1 = x 2 , whic h c on tradicts the assumption that x 1 < x 2 . F or eac h ~ v ∈ T , w e use T 2 ( ~ v ) to denote the set of ~ w ∈ T − { ~ v } that is 2-common to ~ v ; T 1 ( ~ v ) to denote the set of ~ w ∈ T − { ~ v } that is 1-common to ~ v ; T 0 ( ~ v ) t o denote the set of ~ w ∈ T − { ~ v } that is 0-common to ~ v . Then f rom the assumption that m − n ≥ m/c , it follo w s that X ~ v ∈ T X ~ w ∈ T 2 ( ~ v ) E [ Z ~ v Z ~ w ] ≤ ( c 4 (1 − δ ) 2 m 5 n 2 + 2 c 3 (1 − δ ) m 4 n ) | T | ; (7) X ~ v ∈ T X ~ w ∈ T 1 ( ~ v ) E [ Z ~ v Z ~ w ] ≤ ( 4 c 4 δ (1 − δ ) 2 m 6 n 3 + 4 c 3 δ (1 − δ ) m 5 n 2 + 4 c 3 δ m 5 n ) | T | ; (8) X ~ v ∈ T X ~ w ∈ T 0 ( ~ v ) E [ Z ~ v Z ~ w ] ≤ E 2 [ Z ] + ( 2 c 4 δ 2 (1 − δ ) m 6 n 3 + c 4 δ 2 m 6 n 2 ) | T | . (9) The p roofs of Inequalities (7), (8), and (9) are sho w n in Su b sect ions A.1, A.2, and A.3, respectiv ely . Thus from In equalit ies (5), (6), (7 ), (8), and (9), it follo ws that V ar [ Z ] ≤ E [ Z ] − E 2 [ Z ] + X ~ v ∈ T X ~ w ∈ T −{ ~ v } E [ Z ~ v Z ~ w ] = E [ Z ] − E 2 [ Z ] + X ~ v ∈ T X ~ w ∈ T 2 ( ~ v ) E [ Z ~ v Z ~ w ] + X ~ v ∈ T X ~ w ∈ T 1 ( ~ v ) E [ Z ~ v Z ~ w ] + X ~ v ∈ T X ~ w ∈ T 0 ( ~ v ) E [ Z ~ v Z ~ w ] ≤ c 2 | T | m 3 + ( c 4 (1 − δ ) 2 m 5 n 2 + 2 c 3 (1 − δ ) m 4 n ) | T | 7 + ( 4 c 4 δ (1 − δ ) 2 m 6 n 3 + 4 c 3 δ (1 − δ ) m 5 n 2 + 4 c 3 δ m 5 n ) | T | + ( 2 c 4 δ 2 (1 − δ ) m 6 n 3 + c 4 δ 2 m 6 n 2 ) | T | ≤ c 2 | T | m 3 ( 1 + c 2 (1 − δ ) 2 m 2 n 2 + 2 c (1 − δ ) m n + 4 c 2 δ (1 − δ ) 2 m 3 n 3 + 4 cδ (1 − δ ) m 2 n 2 + 4 cδ m 2 n + 2 c 2 δ 2 (1 − δ ) m 3 n 3 + c 2 δ 2 m 3 n 2 ) ≤ c 2 | T | m 3 ( 1 + ( c − 1) 2 (1 − δ ) 2 + 2( c − 1)(1 − δ ) + 4( c − 1) 3 δ (1 − δ ) 2 c + 4( c − 1) 2 δ (1 − δ ) c + 4( c − 1) δ m + 2( c − 1) 3 δ 2 (1 − δ ) c + ( c − 1) 2 δ 2 m ) , (10) where In equ a lit y (10) follo ws from the assum ption that m − n ≥ m/c , i.e., cn/m ≤ c − 1. Thus it follo ws that V ar [ Z ] ≤ d | T | /m 3 for some constan t d that is determined by the constan ts 0 < δ < 1 and c > 1. Then from Inequalities (1), (2), and (4), we fin al ly ha v e that Pr [ Z = 0] ≤ V ar [ Z ] E 2 [ Z ] ≤ d | T | m 3 · m 6 | T | 2 = dm 3 | T | ≤ 3 dm 3 δ 2 (1 − δ ) 2 n 4 = O m 3 n 4 ! , whic h implies that Pr[ Z = 0] = o (1) for any m ≥ n with m/n 4 / 3 = o (1). Therefore, if m/n 4 / 3 = o (1), then with p robabilit y 1 − o (1), we ha ve Z > 0, th a t is, G = ( V , E ) con tains G ′ 1 as a subgraph . 5 Upp er B ounds for the 2-W eigh ted P op ula r Matc hing Problem As sho wn in Theorem 4.1, a random instance of the complete an d strict 2-w eigh ted p opular matc hing prob- lem has a 2-w eighte d p opular matc hing w ith pr obabilit y o (1) if m/n 4 / 3 = o (1). Here we co nsider roughly opp osite case, i.e., n 4 / 3 /m = o (1), and pr o v e that a rand om instance has a 2-w eigh ted p opular matc hing with p robabilit y 1 − o (1). First w e sh o w the follo win g lemma that will greatly simp lify our later analysis. Lemma 5.1 If n/m = o ( 1) , then a r ando m instanc e G = ( V , E ) of the fs-r elation gr aphs c ontains a cycle as a sub gr aph with pr ob ability o (1) . Pro of: F or eac h ℓ ≥ 2, let C ℓ b e a cycle with ℓ v ertices and ℓ edges, a nd E cyc ℓ b e the ev en t that a r andom fs-relation graph G = ( V , E ) con tains a cycle C ℓ . Then f r om the assu mption that m − n ≥ m/c for some constan t c > 1, it follo ws that Pr [ G con tains a cycle] = Pr   [ ℓ ≥ 2 E cyc ℓ   ≤ X ℓ ≥ 2 Pr[ E cyc ℓ ] ≤ X ℓ ≥ 2 ( 1 2 ℓ ℓ ! m ℓ ! ℓ ! n ℓ !  1 m − n  2 ℓ ) ≤ X ℓ ≥ 2 ( 1 2 ℓ m ℓ n ℓ  c m  2 ℓ ) = X ℓ ≥ 2 1 2 ℓ c 2 n m ! ℓ ≤ X ℓ ≥ 2 c 2 n m ! ℓ = c 4 n 2 m 2 X h ≥ 0 c 2 n m ! h = O n 2 m 2 ! , where the last equalit y follo ws from the assumption that n/m = o (1) and c > 1 is a constant. Th us it fo l- lo w s that if n/m = o (1), then a rand om fs-relation graph G = ( V , E ) con tains a cycle as a subgraph with probabilit y o (1). 8 Theorem 5.1 If n 4 / 3 /m = o (1) , then a r andom instanc e of the c omplete and strict 2 -weighte d p opular matching pr oblem with w 1 ≥ 2 w 2 has a 2 -weighte d p opular matching with pr ob ability 1 − o (1) . Pro of: Consider a random fs-relation graph G = ( V , E ) corresp onding to a r an d om in sta nce of the com- plete and s tr ic t 2-we igh ted p opular matc hing problem. By Lemma 5.1 and the assumption th at n 4 / 3 /m = o (1), w e know that the fs-relation graph G = ( V , E ) contai ns bad subgraphs G 2 or G 3 of Figure 1 with v an- ishing probabilit y o (1). Th us in the r est of the p roof, we estimate the p robabilit y that the graph G = ( V , E ) con tains a b ad subgraph G 1 of Figure 1. F or any ℓ ≥ 4, l et P ℓ b e a path with ℓ + 1 ve rtices and ℓ edges, and E path ℓ b e the ev en t that G = ( V , E ) con tains a path P ℓ . It is obvio us that a path P ℓ is a bad subgraph G 1 for eac h ℓ ≥ 4. Th en from the as- sumption that m − n ≥ m/c for some constant c > 1, it follo ws that Pr [ G con tains a bad subgraph G 1 ] = Pr   [ ℓ ≥ 4 E path ℓ   ≤ X ℓ ≥ 4 Pr h E path ℓ i ≤ X ℓ ≥ 4 ( 1 ( m − n ) 2 ℓ ( ℓ + 1)! m ℓ + 1 ! ℓ ! n ℓ !) ≤ X ℓ ≥ 4 (  c m  2 ℓ m ℓ +1 n ℓ ) = c 8 n 4 m 3 X h ≥ 0 c 2 n m ! h = O n 4 m 3 ! , where the last equalit y follo ws from the assump tio n that n 4 / 3 /m = o (1) and th at c > 1 is a constan t. Notice that n/m = o (1) if n 4 / 3 /m = o (1). Thus fr o m Lemma 5.1 and C o rollary 3.1, it follo ws that if n 4 / 3 /m = o (1), then a random instance of the complete and strict 2-w eight ed p opular matc hing p roblem has a 2- w eigh ted p opular matc hing with probabilit y 1 − o (1). 6 Concluding Remarks In this pap er, w e ha v e analyzed the 2-w eigh ted matc hing problem, and ha v e sho wn that (Theorem 4.1) if m/n 4 / 3 = o (1), th e n a random instance of th e complete and strict 2-w eigh ted p opular matc h in g problem with w 1 ≥ 2 w 2 has a 2-w eigh ted p opular matching w it h probabilit y o (1); (Theorem 5.1) if n 4 / 3 /m = o (1), then a random instance of the complete and str ic t 2-w eigh ted p opular matc hing p roblem with w 1 ≥ 2 w 2 has a 2-weig h ted p opular matc hing with pr o babilit y 1 − o (1). Th ese results imply that there exists a thresh- old m ≈ n 4 / 3 to admit 2-w eigh ted p opular matc hings, whic h is quite differen t from the case f o r the single catego ry p opular matc h ing problem due to Mahdian [7]. Theorem 4.1 can b e trivially generalized to an y multiple category case; that is, with the same pr oof, w e h a v e the follo win g b ound. Theorem 6.1 F or a ny inte ger k > 2 , i f m/n 4 / 3 = o (1) , then a r andom instanc e of the c omplete and strict k -weighte d p opular matching pr oblem with w i ≥ 2 w i +1 (1 ≤ i ≤ k − 1 ) has a k -weighte d p opular matching with pr ob ability o (1) . Then an in teresting problem is to sho w some upp er b ound resu lt b y generalizing Th eo rem 5.1 for an y in- teger k > 2, maybe un der the condition that w i ≥ 2 w i +1 for all i , 1 ≤ i ≤ k − 1. References [1] D.J. Abraham, N. Chen, V. Kumar, and V. Mirrokni. Assignment Pr o blems in Renta l Market s. I n Pr o c. Internet and Network Ec onomics , Lecture Notes in C omputer Science 4286, pp.198-213 , 2006. 9 [2] D.J. Abr a ham, R.W. Irving, T. Ka vitha, and K. Mehlhorn. P opular Matc h ings. In Pr o c. of the 16th Annual ACM-SIAM Symp osium on D iscr ete A lgorithms , pp.424-432, 2005. [3] D.J. Abraham and T. K avitha. Dynamic Matc hing Markets and V oting Pa ths. In Pr o c. of the 10th Sc andinavian Workshop on Algorithm The ory , Lecture Not es in Computer Science 4059 , pp.65-76, 2006. [4] N. Alon and J. S p encer. The Pr ob abilistic M eth o d . John Wiley & Sons, 2000. [5] B. Bollob´ as. R andom Gr aphs . Cambridge Universit y Press, 2001. [6] P . Gardenfors. Matc h Making: Ass ignm e nt Ba sed on Bilateral Pr e ferences. Behaviourial Scienc es , 20:166 -173, 1975. [7] M. Mahdian. Random P opular Matc h ings. In Pr o c. of the 7th ACM Confer enc e on Ele ctr onic Com- mer c e , pp.238-242, 2006. [8] J. Mestre. W eigh ted Popular Matc hings. In P r o c. of the 33r d International Col lo quiu m on Automa ta, L anguages, and Pr o gr amming (Part I) , Lecture Notes in Computer Science 4051, pp.715-726 , 2006. [9] R. Mot w ani and P . Ragha v an. R andomize d Algorith ms . Cambridge Univ ersit y Press, 1995. 10 A Pro ofs of Inequalities A.1 Pro of of I n equalit y (7) Let ~ v = ( x 1 , x 2 , y 1 , y 2 ) ∈ T . F or eac h ~ w = ( x ′ 1 , x ′ 2 , y ′ 1 , y ′ 2 ) ∈ T 2 ( ~ v ), let us consid e r the follo wing ca ses: (case-0) |{ y 1 , y 2 } ∩ { y ′ 1 , y ′ 2 }| = 0; (case-1 ) |{ y 1 , y 2 } ∩ { y ′ 1 , y ′ 2 }| = 1. Let T 0 2 ( ~ v ) = { ~ w ∈ T 2 ( ~ v ) : |{ y 1 , y 2 } ∩ { y ′ 1 , y ′ 2 }| = 0 } ; T 1 2 ( ~ v ) = { ~ w ∈ T 2 ( ~ v ) : |{ y 1 , y 2 } ∩ { y ′ 1 , y ′ 2 }| = 1 } . F or eac h ~ v ∈ T , it is imm e diate to see that T 0 2 ( ~ v ) , T 1 2 ( ~ v ) i s the partition of T 2 ( ~ v ), and from th e defin it ions of T 0 2 ( ~ v ) and T 1 2 ( ~ v ), w e ha v e that | T 0 2 ( ~ v ) | ≤ n 2 2 ; | T 1 2 ( ~ v ) | ≤ 2 n 2 . So f rom th e assumption that m − n ≥ m/c for some constan t c > 1 , it f ollo ws th at for eac h ~ v ∈ T , X ~ w ∈ T 0 2 ( ~ v ) E [ Z ~ v Z ~ w ] ≤ X ~ w ∈ T 0 2 ( ~ v ) 1 m  1 m − n 1  4 ≤ X ~ w ∈ T 0 2 ( ~ v ) 1 m  1 m − n  4 ≤ X ~ w ∈ T 0 2 ( ~ v ) 1 m  c m  4 = c 4 m 5 | T 0 2 ( ~ v ) | ≤ c 4 m 5 n 2 2 = c 4 (1 − δ ) 2 m 5 n 2 ; (11) X ~ w ∈ T 1 2 ( ~ v ) E [ Z ~ v Z ~ w ] ≤ X ~ w ∈ T 1 2 ( ~ v ) 1 m  1 m − n 1  3 ≤ X ~ w ∈ T 1 2 ( ~ v ) 1 m  1 m − n  3 ≤ X ~ w ∈ T 1 2 ( ~ v ) 1 m  c m  3 = c 3 m 4 | T 1 2 ( ~ v ) | ≤ 2 c 3 m 4 n 2 = 2 c 3 (1 − δ ) m 4 n. (12) Th us from Inequalities (11) and (12 ), we finally h a v e that X ~ v ∈ T X ~ w ∈ T 2 ( ~ v ) E [ Z ~ v Z ~ w ] = X ~ v ∈ T X ~ w ∈ T 0 2 ( ~ v ) E [ Z ~ v Z ~ w ] + X ~ v ∈ T X ~ w ∈ T 1 2 ( ~ v ) E [ Z ~ v Z ~ w ] ≤ X ~ v ∈ T ( c 4 (1 − δ ) 2 m 5 n 2 + 2 c 3 (1 − δ ) m 4 n ) = ( c 4 (1 − δ ) 2 m 5 n 2 + 2 c 3 (1 − δ ) m 4 n ) | T | . A.2 Pro of of I n equalit y (8) Let ~ v = ( x 1 , x 2 , y 1 , y 2 ) ∈ T . F or eac h ~ w = ( x ′ 1 , x ′ 2 , y ′ 1 , y ′ 2 ) ∈ T 1 ( ~ v ), we ha ve the follo win g cases: (case-0) |{ y 1 , y 2 } ∩ { y ′ 1 , y ′ 2 }| = 0; (case-1) |{ y 1 , y 2 } ∩ { y ′ 1 , y ′ 2 }| = 1; (case-2) |{ y 1 , y 2 } ∩ { y ′ 1 , y ′ 2 }| = 2. Let T 0 1 ( ~ v ) = { ~ w ∈ T 1 ( ~ v ) : |{ y 1 , y 2 } ∩ { y ′ 1 , y ′ 2 }| = 0 } ; T 1 1 ( ~ v ) = { ~ w ∈ T 1 ( ~ v ) : |{ y 1 , y 2 } ∩ { y ′ 1 , y ′ 2 }| = 1 } ; T 2 1 ( ~ v ) = { ~ w ∈ T 1 ( ~ v ) : |{ y 1 , y 2 } ∩ { y ′ 1 , y ′ 2 }| = 2 } . F or eac h ~ v ∈ T , it is immediate that T 0 1 ( ~ v ) , T 1 1 ( ~ v ) , T 2 1 ( ~ v ) is the partition of T 1 ( ~ v ), and f rom the definitions of T 0 1 ( ~ v ), T 1 1 ( ~ v ), and T 2 1 ( ~ v ), w e ha v e that | T 0 1 ( ~ v ) | ≤ 4 n 1 n 2 2 ; | T 1 1 ( ~ v ) | ≤ 4 n 1 n 2 ; | T 2 1 ( ~ v ) | ≤ 4 n 1 . S o from the 11 assumption that m − n ≥ m / c for some constant c > 1, it follo ws that for eac h ~ v ∈ T , X ~ w ∈ T 0 1 ( ~ v ) E [ Z ~ v Z ~ w ] ≤ X ~ w ∈ T 0 1 ( ~ v ) 1 m 2  1 m − n 1  4 ≤ X ~ w ∈ T 0 1 ( ~ v ) 1 m 2  1 m − n  4 ≤ X ~ w ∈ T 0 1 ( ~ v ) 1 m 2  c m  4 = c 4 m 6 | T 0 1 ( ~ v ) | ≤ 4 c 4 m 6 n 1 n 2 2 = 4 c 4 δ (1 − δ ) 2 m 6 n 3 ; (13) X ~ w ∈ T 1 1 ( ~ v ) E [ Z ~ v Z ~ w ] ≤ X ~ w ∈ T 1 1 ( ~ v ) 1 m 2  1 m − n 1  3 ≤ X ~ w ∈ T 1 1 ( ~ v ) 1 m 2  1 m − n  3 ≤ X ~ w ∈ T 1 1 ( ~ v ) 1 m 2  c m  3 = c 3 m 5 | T 1 1 ( ~ v ) | ≤ 4 c 3 m 5 n 1 n 2 = 4 c 3 δ (1 − δ ) m 5 n 2 ; (14) X ~ w ∈ T 2 1 ( ~ v ) E [ Z ~ v Z ~ w ] ≤ X ~ w ∈ T 2 1 ( ~ v ) 1 m 2  1 m − n 1  3 ≤ X ~ w ∈ T 2 1 ( ~ v ) 1 m 2  1 m − n  3 ≤ X ~ w ∈ T 2 1 ( ~ v ) 1 m 2  c m  3 = c 3 m 5 | T 2 1 ( ~ v ) | ≤ 4 c 3 m 5 n 1 = 4 c 3 δ m 5 n. (15) Th us from Inequalities (13), (14), and (15), we finally ha v e that X ~ v ∈ T X ~ w ∈ T 1 ( ~ v ) E [ Z ~ v Z ~ w ] = X ~ v ∈ T X ~ w ∈ T 0 1 ( ~ v ) E [ Z ~ v Z ~ w ] + X ~ v ∈ T X ~ w ∈ T 1 1 ( ~ v ) E [ Z ~ v Z ~ w ] + X ~ v ∈ T X ~ w ∈ T 2 1 ( ~ v ) E [ Z ~ v Z ~ w ] ≤ X ~ v ∈ T ( 4 c 4 δ (1 − δ ) 2 m 6 n 3 + 4 c 3 δ (1 − δ ) m 5 n 2 + 4 c 3 δ m 5 n ) = ( 4 c 4 δ (1 − δ ) 2 m 6 n 3 + 4 c 3 δ (1 − δ ) m 5 n 2 + 4 c 3 δ m 5 n ) | T | . A.3 Pro of of I n equalit y (9) Let ~ v = ( x 1 , x 2 , y 1 , y 2 ) ∈ T . F or eac h ~ w = ( x ′ 1 , x ′ 2 , y ′ 1 , y ′ 2 ) ∈ T 0 ( ~ v ), we ha ve the follo win g cases: (case-0) |{ y 1 , y 2 } ∩ { y ′ 1 , y ′ 2 }| = 0; (case-1) |{ y 1 , y 2 } ∩ { y ′ 1 , y ′ 2 }| = 1; (case-2) |{ y 1 , y 2 } ∩ { y ′ 1 , y ′ 2 }| = 2. Let T 0 0 ( ~ v ) = { ~ w ∈ T 0 ( ~ v ) : |{ y 1 , y 2 } ∩ { y ′ 1 , y ′ 2 }| = 0 } ; T 1 0 ( ~ v ) = { ~ w ∈ T 0 ( ~ v ) : |{ y 1 , y 2 } ∩ { y ′ 1 , y ′ 2 }| = 1 } ; T 2 0 ( ~ v ) = { ~ w ∈ T 0 ( ~ v ) : |{ y 1 , y 2 } ∩ { y ′ 1 , y ′ 2 }| = 2 } . F or eac h ~ v ∈ T , it is immediate that T 0 0 ( ~ v ) , T 1 0 ( ~ v ) , T 2 0 ( ~ v ) is the partition of T 0 ( ~ v ). F or an y ~ w ∈ T 0 0 ( ~ v ), it is ob vious that Pr[ Z ~ v = 1 ∧ Z ~ w = 1] = Pr [ Z ~ v = 1] × Pr[ Z ~ w = 1], whic h imp lie s that X ~ v ∈ T X ~ w ∈ T 0 0 ( ~ v ) E [ Z ~ v Z ~ w ] = X ~ v ∈ T X ~ w ∈ T 0 0 ( ~ v ) Pr[ Z ~ v = 1 ∧ Z ~ w = 1] 12 = X ~ v ∈ T X ~ w ∈ T 0 0 ( ~ v ) Pr[ Z ~ v = 1] × Pr[ Z ~ w = 1] = X ~ v ∈ T Pr[ Z ~ v = 1] X ~ w ∈ T 0 0 ( ~ v ) Pr[ Z ~ w = 1] ≤ X ~ v ∈ T Pr[ Z ~ v = 1] X ~ w ∈ T Pr[ Z ~ w = 1] = E 2 [ Z ] . (16 ) F r om the definitions of T 1 0 ( ~ v ) an d T 2 0 ( ~ v ), we hav e that | T 1 0 ( ~ v ) | ≤ 2 n 2 1 n 2 ; | T 2 0 ( ~ v ) | ≤ n 2 1 . Th en from the as- sumption that m − n ≥ m/c for some constant c > 1, it follo ws that for eac h ~ v ∈ T , X ~ w ∈ T 1 0 ( ~ v ) E [ Z ~ v Z ~ w ] = X ~ w ∈ T 1 0 ( ~ v ) 1 m 2  1 m − n 1  4 ≤ X ~ w ∈ T 1 0 ( ~ v ) 1 m 2  1 m − n  4 ≤ X ~ w ∈ T 1 0 ( ~ v ) 1 m 2  c m  4 = c 4 m 6 | T 1 0 ( ~ v ) | ≤ 2 c 4 m 6 n 2 1 n 2 = 2 c 4 δ 2 (1 − δ ) m 6 n 3 ; (17) X ~ w ∈ T 2 0 ( ~ v ) E [ Z ~ v Z ~ w ] = X ~ w ∈ T 2 0 ( ~ v ) 1 m 2  1 m − n 1  4 ≤ X ~ w ∈ T 2 0 ( ~ v ) 1 m 2  1 m − n  4 ≤ X ~ w ∈ T 2 0 ( ~ v ) 1 m 2  c m  4 = c 4 m 6 | T 2 0 ( ~ v ) | ≤ c 4 m 6 n 2 1 = c 4 δ 2 m 6 n 2 . (18) Th us from Inequalities (16), (17), and (18), we finally ha v e that X ~ v ∈ T X ~ w ∈ T 0 ( ~ w ) E [ Z ~ v Z ~ w ] = X ~ v ∈ T X ~ w ∈ T 0 0 ( ~ w ) E [ Z ~ v Z ~ w ] + X ~ v ∈ T X ~ w ∈ T 1 0 ( ~ w ) E [ Z ~ v Z ~ w ] + X ~ v ∈ T X ~ w ∈ T 2 0 ( ~ w ) E [ Z ~ v Z ~ w ] ≤ E 2 [ Z ] + X ~ v ∈ T ( 2 c 4 δ 2 (1 − δ ) m 6 n 3 + c 4 δ 2 m 6 n 2 ) = E 2 [ Z ] + ( 2 c 4 δ 2 (1 − δ ) m 6 n 3 + c 4 δ 2 m 6 n 2 ) | T | . 13