Online Vertex-Weighted Bipartite Matching: Beating 1 1 with Random Arrivals

Transcription

1 Onlin Vrtx-Wightd Bipartit Matching: Bating with Random Arrivals Zhiyi Huang Dpartmnt of Computr Sicnc, Th Univrsity of Hong Kong, Hong Kong Zhihao Gavin Tang 2 Dpartmnt of Computr Sicnc, Th Univrsity of Hong Kong, Hong Kong zhtang@cs.hku.hk Xiaowi Wu 3 Dpartmnt of Computing, Th Hong Kong Polytchnic Univrsity, Hong Kong wxw7@gmail.com Yuhao Zhang Dpartmnt of Computr Sicnc, Th Univrsity of Hong Kong, Hong Kong yhzhang2@cs.hku.hk Abstract W introduc a wightd vrsion of th ranking algorithm by Karp t al. (STOC 99), and prov a comptitiv ratio of.6534 for th vrtx-wightd onlin bipartit matching problm whn onlin vrtics arriv in random ordr. Our rsult shows that random arrivals hlp bating th -/ barrir vn in th vrtx-wightd cas. W build on th randomizd primal-dual framwork by Dvanur t al. (SODA 23) and dsign a two dimnsional gain sharing function, which dpnds not only on th rank of th offlin vrtx, but also on th arrival tim of th onlin vrtx. To our knowldg, this is th first comptitiv ratio strictly largr than -/ for an onlin bipartit matching problm achivd undr th randomizd primal-dual framwork. Our algorithm has a natural intrprtation that offlin vrtics offr a largr portion of thir wights to th onlin vrtics as tim gos by, and ach onlin vrtx matchs th nighbor with th highst offr at its arrival. 22 ACM Subjct Classification Thory of computation Approximation algorithms analysis, Thory of computation Onlin algorithms, Thory of computation Linar programming Kywords and phrass Vrtx Wightd, Onlin Bipartit Matching, Randomizd Primal-Dual Digital Objct Idntifir.423/LIPIcs.ICALP Rlatd Vrsion A full vrsion of th papr can b found at Acknowldgmnts Th first author would lik to thank Nikhil Dvanur, Ankit Sharma, and Mohit Singh with whom h mad an initial attmpt to rproduc th rsults of Mahdian and Yan using th randomizd primal-dual framwork. Partially supportd by th Hong Kong RGC undr th grant HKU7225E. 2 Partially supportd by his suprvisor Hubrt Chan s Hong Kong RGC grant Part of th work was don whn th author was a postdoc at th Univrsity of Hong Kong. Zhiyi Huang, Zhihao Tang, Xiaowi Wu, and Yuhao Zhang; licnsd undr Crativ Commons Licns CC-BY 45th Intrnational Colloquium on Automata, Languags, and Programming (ICALP 28). Editors: Ioannis Chatzigiannakis, Christos Kaklamanis, Dánil Marx, and Donald Sannlla; Articl No. 79; pp. 79: 79:4 Libniz Intrnational Procdings in Informatics Schloss Dagstuhl Libniz-Zntrum für Informatik, Dagstuhl Publishing, Grmany EATC S

2 79:2 Onlin Vrtx-Wightd Bipartit Matching: Bating with Random Arrivals Introduction With a wid rang of applications, Onlin Bipartit Matching and its variants ar a focal point in th onlin algorithms litratur. Considr a bipartit graph G(L R, E) on vrtics L R, whr th st L of offlin vrtics is known in advanc and vrtics in R arriv onlin. On th arrival of an onlin vrtx, its incidnt dgs ar rvald and th algorithm must irrvocably ithr match it to on of its unmatchd nighbors or lav it unmatchd. In a sminal papr, Karp t al. [9] proposd th Ranking algorithm, which picks at th bginning a random prmutation ovr th offlin vrtics L, and matchs ach onlin vrtx to th first unmatchd nighbor according to th prmutation. Thy provd a tight comptitiv ratio of Ranking, whn onlin vrtics arriv in an arbitrary ordr. Th analysis has bn simplifid in a sris of subsqunt works [4, 5, 2]. Furthr, th Ranking algorithm has bn xtndd to othr variants of th Onlin Bipartit Matching problm, including th vrtx-wightd cas [2], th random arrival modl [8, 2], and th Adwords problm [23, 7, ]. As a natural gnralization, Onlin Vrtx-Wightd Bipartit Matching was considrd by Aggarwal t al. [2]. In this problm, ach offlin vrtx v L has a non-ngativ wight, and th objctiv is to maximiz th total wight of th matchd offlin vrtics. A wightd vrsion of th Ranking algorithm was proposd in [2] and shown to b ( )- comptitiv, matching th problm hardnss in th unwightd vrsion. Thy fix a nonincrasing prturbation function ψ : [, ] [, ], and draw a rank y v [, ] uniformly and indpndntly for ach offlin vrtx v L. Th offlin vrtics ar thn sortd in dcrasing ordr of th prturbd wight ψ(y v ). Each onlin vrtx matchs th first unmatchd nighbor on th list upon its arrival. It is shown that by choosing th prturbation function ψ(y) := y, th wightd Ranking algorithm achivs a tight comptitiv ratio. In a subsqunt work, Dvanur t al. [2] simplifid th analysis undr th randomizd primal-dual framwork and gav an altrnativ intrprtation of th algorithm: ach offlin vrtx v maks an offr of valu ( g(y v )) as long as it is not matchd, whr g(y) := y, and ach onlin vrtx matchs th nighbor that offrs th highst. Motivatd by th practical importanc of Onlin Bipartit Matching and its applications for onlin advrtismnts, anothr lin of rsarch sks for a bttr thortical bound byond th worst-cas hardnss rsult providd by Karp t al. [9]. Onlin Bipartit Matching with random arrivals was considrd indpndntly by Karand t al. [8] and Mahdian t al. [2]. Thy both studid th prformanc of Ranking assuming that onlin vrtics arriv in a uniform random ordr and provd comptitiv ratios.653 and.696 rspctivly. On th ngativ sid, Karand t al. [8] xplicitly constructd an instanc for which Ranking prforms no bttr than.727, which is latr improvd to.724 by Chan t al. [9]. In trms of problm hardnss, Manshadi t al. [22] showd that no algorithm can achiv a comptitiv ratio largr than.823. Th natural nxt stp is thn to considr Onlin Vrtx-Wightd Bipartit Matching with random arrivals. Do random arrivals hlp bating vn in th vrtx-wightd cas? Arbitrary Arrivals Random Arrivals Unwightd.632 [9, 5, 2, 4].696 [2] Vrtx-wightd.632 [2, 2].6534 (this papr)

3 Z. Huang, Z. Tang, X. Wu, and Y. Zhang 79:3. Our Rsults and Tchniqus W answr this affirmativly by showing that a gnralizd vrsion of th Ranking algorithm achivs a comptitiv ratio Thorm. Thr xists a.6534-comptitiv algorithm for th vrtx-wightd Onlin Bipartit Matching with random arrivals. Intrstingly, w do not obtain our rsult by gnralizing xisting works that brak th barrir on th unwightd cas [8, 2] to th vrtx-wightd cas. Instad, w tak a totally diffrnt path, and build our analysis on th randomizd primal-dual tchniqu introducd by Dvanur t al. [2], which was usd to provid a mor unifid analysis of th algorithms for th Onlin Bipartit Matching with arbitrary arrival ordr and its xtnsions. W first brifly rviw th proof of Dvanur t al. [2]. Th randomizd primal-dual tchniqu can b viwd as a charging argumnt for sharing th gain of ach matchd dg btwn its two ndpoints. Rcall that in th algorithm of [2, 2], ach unmatchd offlin vrtx offrs a valu of ( g(y v )) to onlin vrtics, and ach onlin vrtx matchs th nighbor that offrs th highst at its arrival. Whnvr an dg (u, v) is addd to th matching, whr v L is an offlin vrtx and u R is an onlin vrtx, imagin a total gain of bing shard btwn u and v such that u gts ( g(y v )) and v gts g(y v ). Sinc g is non-dcrasing, th smallr th rank of v, th smallr shar it gts. Thy showd that by fixing g(y) = y, for any dg (u, v) and any fixd ranks of offlin vrtics othr than v, th xpctd gains of u and v (from all of thir incidnt dgs) combind is at last ( ) ovr th randomnss of y v, which implis th comptitiv ratio. Now w considr th problm with random arrivals. Analogous to th offlin vrtics, as th onlin vrtics arriv in random ordr, in th gain sharing procss, it is natural to giv an onlin vrtx u a smallr shar if u arrivs arly (as it is mor likly b gt matchd), and a largr shar whn u arrivs lat. Thus w considr th following vrsion of th wightd Ranking algorithm. Lt y u b th arrival tim of onlin vrtx u R, which is chosn uniformly at random from [, ]. Analogous to th ranks of th offlin vrtics, w also call y u th rank of u R. Fix a function g : [, ] 2 [, ] that is non-dcrasing in th first dimnsion and non-incrasing in th scond dimnsion. On th arrival of u R, ach unmatchd nighbor v L of u maks an offr of valu ( g(y v, y u )), and u matchs th nighbor with th highst offr. This algorithm straightforwardly lads to a gain sharing rul for dual assignmnts: whnvr u R matchs v L, lt th gain of u b ( g(y v, y u )) and th gain of v b g(y v, y u ). It suffics to show that, for an appropriat function g, th xpctd gain of u and v combind is at last.6534 ovr th randomnss of both y u and y v. Th main difficulty of th analysis is to giv a good charactrization of th bhavior of th algorithm whn w vary th ranks of both u R and v L, whil fixing th ranks of all othr vrtics arbitrarily. Th prvious analysis for th unwightd cas with random arrivals [8, 2] havily rlis on a symmtry btwn th random ranks of offlin vrtics and onlin vrtics: Proprtis dvlopd for th offlin vrtics in prvious work dirctly translat to thir onlin countrparts. Unfortunatly, th onlin and offlin sids ar no longr symmtric in th vrtx-wightd cas. In particular, for th offlin vrtx v, an important proprty is that for any givn rank y u of th onlin vrtx u, w can dfin a uniqu marginal rank θ such that v will b matchd if and only if its rank y v < θ. Howvr, it is not possibl to dfin such a marginal rank for th onlin vrtx u in th vrtx-wightd cas: As its arrival tim changs, its matching status may chang back and forth. Th most important tchnical ingrdint of our analysis is an appropriat lowr bound on th xpctd gain I C A L P 2 8

4 79:4 Onlin Vrtx-Wightd Bipartit Matching: Bating with Random Arrivals which allows us to partially charactriz th worst-cas scnario (in th sns of minimizing th lowr bound on th xpctd gain). Furthr, th worst-cas scnario dos admit simpl marginal ranks vn for th onlin vrtx u. This allows us to dsign a symmtric gain sharing function g and complt th comptitiv analysis of Othr Rlatd Works Thr is a vast litratur on problms rlatd to Onlin Bipartit Matching. For spac rasons, w only list som of th most rlatd hr. Ksslhim t al. [2] considrd th dg-wightd Onlin Bipartit Matching with random arrivals, and proposd a -comptitiv algorithm. Th comptitiv ratio is tight as it matchs th lowr bound on th classical scrtary problm [8]. Wang and Wong [24] considrd a diffrnt modl of Onlin Bipartit Matching with both sids of vrtics arriving onlin (in an arbitrary ordr): A vrtx can only activly match othr vrtics at its arrival; if it fails to match at its arrival, it may still gt matchd passivly by othr vrtics latr. Thy showd a.526-comptitiv algorithm for a fractional vrsion of th problm. Rcntly, Cohn and Wajc [] considrd th Onlin Bipartit Matching (with arbitrary arrival ordr) on rgular graphs, and providd a ( O( log d/d))-comptitiv algorithm, whr d is th dgr of vrtics. Vry rcntly, Huang t al. [6] proposd a fully onlin matching modl, in which all vrtics of th graph arriv onlin (in an arbitrary ordr). Extnding th randomizd primal-dual tchniqu, thy obtaind comptitiv ratios abov.5 for both bipartit graphs and gnral graphs. Similar but diffrnt from th Onlin Bipartit Matching with random arrivals, in th stochastic Onlin Bipartit Matching, th onlin vrtics arriv according to som known probability distribution (with rptition). Comptitiv ratios braking th barrir hav bn achivd for th unwightd cas [3, 4, 6] and th vrtx-wightd cas [5, 7, 6]. Th Onlin Bipartit Matching with random arrivals is closly rlatd to th oblivious matching problm [3, 9, ] (on bipartit graphs). It can b asily shown that Ranking has quivalnt prformanc on th two problms. Thus comptitiv ratios abov [8, 2] dirctly translat to th oblivious matching problm. Gnralizations of th problm to arbitrary graphs hav also bn considrd, and comptitiv ratios abov half ar achivd for th unwightd cas [3, 9] and vrtx-wightd cas []. 2 Prliminaris W considr th Onlin Vrtx-Wightd Bipartit Matching with random arrival ordr. Lt G(L R, E) b th undrlying graph, whr vrtics in L ar givn in advanc and vrtics in R arriv onlin in random ordr. Each offlin vrtx v L is associatd with a non-ngativ wight. Without loss of gnrality, w assum th arrival tim y u of ach onlin vrtx u R is drawn indpndntly and uniformly from [, ]. Mahdian and Yan [2] us anothr intrprtation for th random arrival modl. Thy dnot th ordr of arrival of onlin vrtics by a prmutation π and assum that π is drawn uniformly at random from th prmutation group S n. It is asy to s th quivalnc btwn two intrprtations 4. 4 Mapping from an arrival tim vctor to a prmutation is immdiat. Givn a prmutation π, w indpndntly draw n random variabls uniformly from [, ] and assign ths valus to b th arrival tims of all vrtics according to th prmutation π, from th smallst to th largst.

5 Z. Huang, Z. Tang, X. Wu, and Y. Zhang 79:5 Wightd Ranking. Fix a function g : [, ] 2 [, ] such that g(x,y) x and g(x,y) y. Each offlin vrtx v L draws indpndntly a random rank y v [, ] uniformly at random. Upon th arrival of onlin vrtx u R, u is matchd to its unmatchd nighbor v with maximum ( g(y v, y u )). Rmark. In th advrsarial modl, Aggarwal t al. s algorithm [2] can b intrprtd as choosing g(y v, y u ) := yv in our algorithm. Our algorithm is a dirct gnralization of thirs to th random arrival modl. For simplicity, for ach u R, w also call its arrival tim y u th rank of u. W us y : L R [, ] to dnot th vctor of all ranks. Considr th linar program rlaxation of th bipartit matching problm and its dual. max : (u,v) E w v x uv min : u V α u s.t. v:(u,v) E x uv u L R s.t. α u + α v (u, v) E x uv (u, v) E α u u L R Randomizd Primal-Dual. Our analysis builds on th randomizd primal-dual tchniqu by Dvanur t al. [2]. W st th primal variabls according to th matching producd by Ranking, i.. x uv = if and only if u is matchd to v by Ranking, and st th dual variabls so that th dual objctiv quals th primal. In particular, w split th gain of ach matchd dg (u, v) btwn vrtics u and v; th dual variabl for ach vrtx thn quals th shar it gts. Givn primal fasibility and qual objctivs, th usual primal-dual tchniqus would furthr sk to show approximat dual fasibility, namly, α u + α v F for vry dg (u, v), whr F is th targt comptitiv ratio. Obsrv that th abov primal and dual assignmnts ar thmslvs random variabls. Dvanur t al. [2] claimd that th primal-dual argumnt gos through givn approximat dual fasibility in xpctation. W formulat this insight in th following lmma and includ a proof for compltnss. Lmma 2. Ranking is F -comptitiv if w can st (non-ngativ) dual variabls such that (u,v) E x uv = u V α u; and E y [α u + α v ] F for all (u, v) E. Proof. W can st a fasibl dual solution α u := E y [α u ] /F for all u V. It s fasibl bcaus w hav α u + α v = E y [α u + α v ] /F for all (u, v) E. Thn by duality w know that th dual solution is at last th optimal primal solution PRIMAL, which is also at last th optimal offlin solution of th problm: u V α u PRIMAL OPT. Thn by th first assumption, w hav OPT u V α u = E y [α u] u V F = F E [ [ y u V α u] = ] (u,v) E x uv E [ALG], which implis an F comptitiv ratio. F E y = F In th rst of th papr, w st g(x, y) = 2( h(x) + h(y) ), x, y [, ] whr h : [, ] [, ] is a non-dcrasing function (to b fixd latr) with h (x) h(x) for all x [, ]. Obsrv that g(x,y) x = 2 h (x) and g(x,y) y = 2 h (y). By dfinition of g, w hav g(x, y) + g(y, x) =. Morovr, for any x, y [, ], w hav th following fact that will b usful for our analysis. Claim 2.. Proof. g(x,y) y g(x, y). g(x,y) y = 2 h (y) 2 h(y) 2 (h(x) + h(y)) = g(x, y). I C A L P 2 8

6 79:6 Onlin Vrtx-Wightd Bipartit Matching: Bating with Random Arrivals 3 A Simpl Lowr Bound In this sction, w prov a slightly smallr comptitiv ratio, , as a warm-up of th latr analysis. W rintrprt our algorithm as follows. As tim t gos, ach unmatchd offlin vrtx v L is dynamically pricd at g(y v, t). Sinc g is non-incrasing in th scond dimnsion, th prics do not incras as tim gos by. Upon th arrival of u R, u can choos from its unmatchd nighbors by paying th corrsponding pric. Th utility of u drivd by choosing v quals g(y v, y u ). Thn u chooss th on that givs th highst utility. Rcall that g is non-dcrasing in th first dimnsion. Thus, u prfrs offlin vrtics with smallr ranks, as thy offr lowr prics. This lads to th following monotonicity proprty as in prvious works [2, 2]. Fact 3. (Monotonicity). For any y, if v L is unmatchd whn u R arrivs, thn whn y v incrass, v rmains unmatchd whn u arrivs. Equivalntly, if v L is matchd whn u R arrivs, thn whn y v dcrass, v rmains matchd whn u arrivs. Gain Sharing. Th abov intrprtation inducs a straightforward gain sharing rul: whnvr u R is matchd to v L, lt α v := g(y v, y u ) and α u := ( g(y v, y u )) = g(y u, y v ). Not that th gain of an offlin vrtx is largr if it is matchd arlir, i.., bing matchd arlir is mor bnficial for offlin vrtics (α v is largr). Howvr, th fact dos not hold for onlin vrtics. For ach onlin vrtx u R, th arlir u arrivs (smallr y u is), th mor offrs u ss. On th othr hand, th prics of offlin vrtics ar highr whn u coms arlir. Thus, it is not guarantd that arlir arrival tim y u inducs largr α u. This is whr our algorithm dviats from prvious ons [2, 2], in which th prics of offlin vrtics ar static (indpndnt of tim). Th abov obsrvation is crucial and ncssary for braking th barrir in th random arrival modl. To apply Lmma 2, w considr a pair of nighbors v L and u R. W fix an arbitrary assignmnt of ranks to all vrtics but u, v. Our goal is to stablish a lowr bound of E [α u + α v ], whr th xpctation is simultanously takn ovr y u and y v. Lmma 3. For ach y [, ], thr xist thrsholds θ(y) β(y) such that whn u arrivs at tim y u = y, if y v < β(y), v is matchd whn u arrivs; if y v (β(y), θ(y)), v is matchd to u; if y v > θ(y), v is unmatchd aftr u s arrival. Morovr, β(y) is a non-dcrasing function and if θ(x) = for som x [, ], thn θ(x ) = for all x x. Proof. Considr th momnt whn u arrivs. By Fact 3., thr xists a thrshold β(y u ) such that v is matchd whn u arrivs iff y v < β(y u ). Now suppos y v > β(y u ), in which cas v is unmatchd whn u arrivs. Thus v is pricd at g(y v, y u ) and u can gt utility g(y u, y v ) by choosing v. Rcall that g(y u, y v ) is non-incrasing in trms of y v. Lt θ(y u ) β(y u ) b th minimum valu of y v such that v is not chosn by u. In othr words, whn β(y u ) < y v < θ(y u ), v is matchd to u and whn y v > θ(y u ), v is unmatchd aftr u s arrival. Nxt w show that β is a non-dcrasing function of y u. By dfinition, if y v < β(y u ), thn v is matchd whn u arrivs. Straightforwardly, whn y u incrass to y u (arrivs vn

7 Z. Huang, Z. Tang, X. Wu, and Y. Zhang 79:7 y v y v θ(y u ) γ γ θ(y u ) β(y u ) β(y u ) τ y u τ y u Figur θ(y u) and β(y u) (lft hand sid); truncatd θ(y u) and β(y u) (right hand sid). latr), v would rmain matchd. Hnc, w hav β(y u) β(y u ) for all y u > y u, i.. β is non-dcrasing (rfr to Figur ). Finally, w show that if θ(x) = for som x [, ], thn θ(x ) = for all x x. Assum for th sak of contradiction that θ(x ) < for som x > x. In othr words, whn y u = x and y v =, v is unmatchd whn u arrivs, but u chooss som vrtx z v, such that w z g(x, y z ) > g(x, ). Now considr th cas whn u arrivs at tim y u = x. Rcall that w hav θ(x) =, which mans that u matchs v whn y u = x and y v =. By our assumption, both v and z ar unmatchd whn u arrivs at tim x. Thus whn u arrivs at an arlir tim x, both v and z ar unmatchd. Morovr, choosing z inducs utility w z g(x, y z ) = w z g(x, y z ) g(x, y z) g(x, y z ) > g(x, ) g(x, y z) g(x, y z ) = g(x, ) h(x) + h(y z) h(x ) + h(y z ) g(x h(x) + h(), ) h(x ) + h() = g(x g(x, ), ) g(x, ) = g(x, ), whr th scond inquality holds sinc h is a non-dcrasing function and x < x. This givs a contradiction, sinc whn y u = x and y v =, u chooss v, whil choosing z givs strictly highr utility. Rmark. Obsrv that th function θ is not ncssarily monoton. This coms from th fact that u may prfr v to z whn u arrivs at tim t but prfr z to v whn u arrivs latr at tim t > t. Not that this happns only whn th offlin vrtics hav gnral wights: for th unwightd cas, it is asy to show that θ must b non-dcrasing. W dfin τ, γ [, ], which dpnd on th input instanc, as follows. If θ(y) < for all y [, ], thn lt τ = ; othrwis lt τ b th minimum valu such that θ(τ) =. Lt γ := β(). Not that it is possibl that γ {, }. Sinc β is non-dcrasing, w dfin β (x) := sup{y : β(y) = x} for all x γ. In th following, w stablish a lowr bound for E [α u + α v ]. Lmma 4 (Main Lmma). For ach pair of nighbors u R and v L, w hav { τ } E [α u + α v ] min ( τ) ( γ) + g(x, τ)dx + g(x, γ)dx. γ,τ I C A L P 2 8

8 79:8 Onlin Vrtx-Wightd Bipartit Matching: Bating with Random Arrivals W prov Lmma 4 by th following thr lmmas. Obsrv that for any y u [, ], if y v (β(y u ), θ(y u )), u, v ar matchd to ach othr, which implis α u + α v =. Hnc w hav th following lmma immdiatly. Lmma 5 (Cornr Gain). E [(α u + α v ) (y u > τ, y v > γ)] = ( τ) ( γ). Now w giv a lowr bound for th gain of v whn y v < γ, i.., α v (y v < γ), plus th gain of u whn y v < γ and y u > τ, i.., α u (y v < γ, y u > τ). Th ky to prov th lmma is to show that for all y v < γ, no mattr whn u arrivs, w always hav α v g(y v, β (y v )). Lmma 6 (v s Gain). E [α v (y v < γ) + α u (y v < γ, y u > τ)] g(x, τ)dx. Proof. Fix y v = x < γ. W first show that for all y u [, ], α v g(x, β (x)). By dfinition, w hav β (x) <. Hnc whn y u > β (x), v is alrady matchd whn u arrivs. Suppos v is matchd to som z R, thn w hav y z β (x) and hnc α v g(x, β (x)). Now considr whn u arrivs at tim y < β (x). If y > y z, thn v is still matchd to z whn u arrivs, and α v g(x, β (x)) holds. Now suppos y < y z. W compar th two procsss, namly whn y u > β (x) and whn y u = y. W show that for ach vrtx w L, th tim it is matchd is not latr in th scond cas (compard to th first cas). In othr words, w show that dcrasing th rank of any onlin vrtx is not harmful for all offlin vrtics. Suppos othrwis, lt w b th first vrtx in L that is matchd latr whn y u = y than whn y u > β (x). I.. among all ths vrtics, w s matchd nighbor arrivs th arlist whn y u > β (x). Lt u b th vrtx w is matchd to whn y u > β (x) and u 2 b th vrtx w is matchd to whn y u = y. By assumption, w hav y u2 > y u. Considr whn y u = y and th momnt whn u arrivs, w rmains unmatchd but is not chosn by u. Howvr, w is th first vrtx that is matchd latr than it was whn y u > β (x), w know that at u s arrival, th st of unmatchd nighbor of u is a subst of that whn y u > β (x). This lads to a contradiction, sinc w givs th highst utility, but is not chosn by u. In particular, this proprty holds for vrtx v, i.. v is matchd arlir or at th arrival of z and hnc α v g(x, y z ) g(x, β (x)), as claimd. Obsrv that for y v < γ and y u (τ, β (y v )), w hav α u + α v =. Thus for y v = x < γ, w lowr bound E yu [α v (y v < γ) + α u (y v < γ, y u > τ)] by f(x, β (x)) := g(x, β (x)) + max{, β (x) τ} ( g(x, β (x))). It suffics to show that f(x, β (x)) g(x, τ). Considr th following two cass.. If β (x) < τ, thn f(x, β (x)) = g(x, β (x)) g(x, τ), sinc g(x,y) y. 2. If β (x) τ, thn f(x, β (x)) is non-dcrasing in th scond dimnsion, sinc f(x, β (x)) β (x) = g(x, β (x)) β (x) + g(x, β (x)) (β (x) τ) g(x, β (x)) β (x), whr th inquality follows from Claim 2. and g(x,β (x)) β (x) f(x, β (x)) f(x, τ) = g(x, τ).. Thrfor, w hav Hnc for vry fixd y v = x < γ w hav E yu [α v (y v < γ) + α u (y v < γ, y u > τ)] g(x, τ). Taking intgration ovr x (, γ) concluds th lmma. Nxt w giv a lowr bound for th gain of u whn y u < τ, i.., α u (y u < τ), plus th gain of v whn y u < τ and y v > γ, i.., α v (y u < τ, y v > γ). Th following proof is in th

9 Z. Huang, Z. Tang, X. Wu, and Y. Zhang 79:9 sam spirit as in th proof of Lmma 6, although th ranks of offlin vrtics hav diffrnt maning from th ranks (arrival tims) of onlin vrtics. Similar to th proof of Lmma 6, th ky is to show that for all y u < τ, no mattr what valu y v is, th gain of α u is always at last g(y u, θ(y u )). Lmma 7 (u s Gain). E [α u (y u < τ) + α v (y u < τ, y v > γ)] τ g(x, γ)dx. Proof. Fix y u = x < τ. By dfinition w hav θ(x) <. Th analysis is similar to th prvious. W first show that for all y v [, ], w hav α u g(x, θ(x)). W us θ to dnot th valu that is arbitrarily clos to, but largr than θ(x). By dfinition, whn y v = θ, u matchs som vrtx othr than v. Thus w hav α u g(x, θ(x)). Hnc, whn y v > θ, i.. v has a highr pric, u would choos th sam vrtx as whn y v = θ, and α u g(x, θ(x)) still holds. Now considr th cas whn y v = y < θ. As in th analysis of Lmma 6, w compar two procsss, whn y v = θ and whn y v = y < θ. W show that for ach vrtx w R (including u) with y w x = y u, th utility of w whn y v = y is not wors than its utility whn y v = θ. Suppos othrwis, lt w b such a vrtx with arlist arrival tim. Lt v b th vrtx that is matchd to w whn y v = θ. Thn w know that (whn y v = y) at w s arrival, w chooss a vrtx that givs lss utility comparing to v. Hnc, at this momnt v is alrady matchd to som w with y w < y w. This implis that whn y v = θ, v (which is matchd to w) is unmatchd whn w arrivs, but not chosn by w. Thrfor, w has lowr utility whn y v = y compard to th cas whn y v = θ, which contradicts th assumption that w is th first such vrtx. Obsrv that whn y v (γ, θ(x)), w hav α u + α v =. Thus for any fixd y u = x < τ, w lowr bound E yv [α u (y u < τ) + α v (y u < τ, y v > γ)] by f(x, θ(x)) := g(x, θ(x)) + max{, θ(x) γ} ( g(x, θ(x))). In th following, w show that f(x, θ(x)) g(x, γ). Considr th following two cass.. If θ(x) γ, thn f(x, θ(x)) = g(x, θ(x)) g(x, γ), sinc g(x,y) y. 2. If θ(x) > γ, thn f(x, θ(x)) θ(x) = g(x, θ(x)) θ(x) + g(x, θ(x)) (θ(x) γ) g(x, θ(x)) θ(x), whr th inquality follows from Claim (2.) and g(x,θ(x)) θ(x) f(x, θ(x)) f(x, γ) = g(x, γ). Finally, tak intgration ovr x (, τ) concluds th lmma.. Thrfor, w hav Proof of Lmma 4. Obsrv that α u + α v = (α u + α v ) (y u > τ, y v > γ) + α v (y v < γ) + α u (y v < γ, y u > τ) + α u (y u < τ) + α v (y u < τ, y v > γ). Combing Lmma 5, 6 and 7 finishs th proof immdiatly. Thorm 8. Fix h(x) = min{, x.5 }. For any pair of nighbors u and v, and any fixd ranks of vrtics in L R \ {u, v}, w hav E yu,y v [α u + α v ] Proof. It suffics to show that th RHS of Lmma 4 is at last Sinc th xprssion is symmtric for τ and γ, w assum τ γ without loss of gnrality. I C A L P 2 8

10 79: Onlin Vrtx-Wightd Bipartit Matching: Bating with Random Arrivals Lt f(τ, γ) b th trm on th RHS of Lmma 4 to b minimizd. By our choic of g, f(τ, γ) = τ γ + τ γ + 2 Obsrv that f(τ, γ) τ ( ) h(x) + h(τ) dx + 2 = τ 2 ( + h(γ)) γ 2 ( + h(τ)) + τ γ + 2 = γ 2 ( + h(γ)) γ 2 h (τ) + 2 h(τ). τ h(x)dx + 2 ( h(x) + h(γ) ) dx τ h(x)dx. It is asy to chck that γ 2 h(γ) whn γ 2 ; and γ 2 h(γ) > whn γ > 2. Hnc whn γ f(τ,γ) 2, w hav τ, which mans that th minimum is attaind whn τ =. Not that whn γ 2, w hav f(, γ) = 2 ( h(γ)) + 2 h(x)dx + 2 h(x)dx, which attains its minimum at γ = (sinc h (γ) = h(γ) for γ 2 ): f(, ) = 2 (.5 ) + 2 ( ) = Whn τ γ > f(τ,γ) 2, w hav τ = γ 2h(γ) >. Hnc th minimum is attaind whn τ = γ, which is f(γ, γ) = 2γ + γ 2 + h(x)dx. Obsrv that df(γ, γ) dγ = 2 + 2γ + h(γ) =. Th minimum is attaind whn γ = 2, which quals f( 2, 2 ) = Improving th Comptitiv Ratio Obsrv that in Lmma 4, w rlax th total gain of α u + α v into two parts: () whn y u τ and y v γ, α u + α v =. (2) for othr ranks y u, y v, w lowr bound α u and α v by g(y u, γ) and g(y v, τ) rspctivly. For th scond part, th inqualitis usd in th proof of Lmma 6 and 7 ar tight only if β, θ ar two stp functions (rfr to Figur ). On th othr hand, givn ths β, θ, whn y u τ and y v γ, w actually hav α u + α v =, which is strictly largr than our stimation (g(y u, γ) + g(y v, τ)). With this obsrvation, it is natural to xpct an improvd bound if w can rtriv this part of gain (vn partially). In this sction, w prov an improvd comptitiv ratio.6534, using a rfind lowr bound for E [α u + α v ] (compard to Lmma 4) as follows. Lmma 9 (Improvd Bound). For any pair of nighbors u R and v L, w hav E [α u + α v ] { min ( τ)( γ) + ( τ) γ,τ τ + { min g(x, θ) + θ γ θ g(x, τ)dx g(y, x)dy + θ } } g(y, τ)dy dx. Proof. Lt γ and τ b dfind as bfor, i.., γ = β() and τ = min{x : θ(x) = }.

11 Z. Huang, Z. Tang, X. Wu, and Y. Zhang 79: W divid E [α u + α v ] into thr parts, namly () whn y u > τ and y v > γ; (2) whn y u > τ and y v < γ; and (3) whn y u < τ: E [α u + α v ] = E [(α u + α v ) (y u > τ, y v > γ)] + E [(α u + α v ) (y u > τ, y v < γ)] + E [(α u + α v ) (y u < τ)]. As shown in Lmma 5, th first trm is at last ( τ) ( γ), as w hav α u +α v = for all y u > τ and y v > γ. Thn w considr th scond trm, th xpctd gain of α u + α v whn y v < γ and y u > τ. For any y v < γ, as w hav shown in Lmma 6, α v g(y v, β (y v )) for all y u > τ. Morovr, whn y u < β (y v ), w hav α u + α v =. Hnc th scond trm can b lowr boundd by ( ( τ) g(y v, β (y v )) + max{, β (y v ) τ} ( g(y v, β (y v )) )) dy v. Now w considr th last trm and fix a y u < τ. As w hav shown in Lmma 7, for all y v [, ], α u g(y u, θ(y u )). Considr th cas whn θ(y u ) > γ, thn for y v (, γ), α v g(y v, y u ); for y v (γ, θ(y u )), α u + α v =. Thus th xpctd gain of α u + α v (takn ovr th randomnss of y v ) can b lowr boundd by ( ) g(y u, θ(y u )) + g(y v, y u )dy v + (θ(y u ) γ) ( g(y u, θ(y u ))). As w hav shown in Lmma 7, th partial drivativ ovr θ(y u ) is non-ngativ, thus for th purpos of lowr bounding E [α u + α v ], w can assum that θ(y u ) γ for all y u < τ. Givn that θ(y u ) γ, w hav α v g(y v, y u ) whn y v (,, θ(y u )); and α v g(y v, β (y v )) whn y v (θ(y u ), γ). Hnc th third trm can b lowr boundd by τ ( g(y u, θ(y u )) + θ(yu) g(y v, y u )dy v + g(y v, β (y v ))dy v )dy u θ(y u) Putting th thr lowr bounds togthr and taking th partial drivativ ovr β (y v ), for thos β (y v ) > τ, w hav a non-ngativ drivativ as follows: g(y v, β (y v )) β (y v ) + g(y v, β (y v )) (β (y v ) τ) g(y v, β (y v )) β (y v ) Thus for lowr bounding E [α u + α v ], w assum β (y v ) τ for all y v < γ. Hnc { E [α u + α v ] min ( τ)( γ) + ( τ) g(y v, τ)dy v γ,τ + τ ( g(y u, θ(y u )) + θ(yu) g(y v, y u )dy v + θ(y u). g(y v, τ)dy v ) dy u }. Taking th minimum ovr θ(y u ) concluds Lmma 9. I C A L P 2 8

12 79:2 Onlin Vrtx-Wightd Bipartit Matching: Bating with Random Arrivals Obsrv that for any θ γ, w hav g(x, θ) + θ g(y, x)dy + g(y, τ)dy g(x, γ) + g(y, τ)dy. θ Thus th lowr bound givn by Lmma 9 is not wors than Lmma 4. Thorm. Fix h(x) = min{, 2 x }. For any pair of nighbors u and v, and any fixd ranks of vrtics in L R \ {u, v}, w hav E yu,y v [α u + α v ] ln W giv a proof sktch and dfr th complt analysis to th full vrsion of th papr. Proof Sktch. For h(x) = min{, 2 x }, w hav h (x) = h(x) whn x < ln(2), and h (x) =, h(x) = whn x > ln(2). Lt f(τ, γ) b th xprssion on th RHS to b minimizd in Lmma 9. W first show that for any fixd τ and γ, th minimum (ovr θ) of f(τ, γ) is obtaind whn θ = min{ln 2, γ}. Hnc w can lowr bound f(τ, γ) by ( τ)( γ)+ γ 2 ( h(τ))+ τ ln 2 ( h(γ))+ 2 2 τ h(τ)+ h(y)dy + ln 2 τ h(x)dx. 2 2 Thn w show that f(τ, γ) ln for all τ, γ [, ]. W show that thr xists τ.3574 (solution for + h(τ) 2τ = ) such that for τ τ, f(τ,γ) γ. Thus f(τ, γ) f(τ, ). Furthr mor, w show that f(τ,) τ, which implis f(τ, γ) f(τ, ) f(, ) = 2 ( h()) + 2 h(y)dy = ln For any fixd τ > τ, w show that th minimum (ovr γ) of f(τ, γ) is attaind whn γ = ln 2. Hnc for τ > τ f(τ,ln 2) w hav f(τ, γ) f(τ, ln 2). Finally, w show that τ < whn τ < τ ; and > whn τ > τ, whr τ , which implis f(τ,ln 2) τ f(τ, γ) f(τ, ln 2) f(τ, ln 2) =( τ )( ln 2) + ln 2 4 (2 τ + τ τ ) ln 2 ( τ ).6557 > ln Thus for all τ, γ [, ], w hav f(τ, γ) ln 2 2, as claimd. 5 Conclusion In this papr, w show that comptitiv ratios abov can b obtaind undr th randomizd primal-dual framwork whn quippd with a two dimnsional gain sharing function. Th ky of th analysis is to lowr bound th xpctd combind gain of vry pair of nighbors (u, v), ovr th randomnss of th rank y v of th offlin vrtx, and th arrival tim y u of th onlin vrtx. Rfrring to Figur, it can b shown that th comptitiv ratio F f(y u)dy u, whr f(y u ) := ( θ(y u ) + β(y u )) g(y u, θ(y u )) + θ(y u ) β(y u ) + β(yu) g(y v, β (y v ))dy v + θ(y u) g(y v, β (y v ))dy v.

13 Z. Huang, Z. Tang, X. Wu, and Y. Zhang 79:3 Not that hr w assum β (y v ) = for all y v γ, and g(x, ) = for all x [, ]. For vry fixd g, thr xist thrshold functions θ and β that minimiz th intgration. Thus th main difficulty is to find a function g such that th intgration has a larg lowr bound for all functions θ and β (which dpnd on th input instanc). W hav shown that thr xists a choic of g such that th minimum is attaind whn θ and β ar stp functions, basd on which w can giv a lowr bound on th comptitiv ratio. It is thus an intrsting opn problm to know how much th comptitiv ratio can b improvd by (fixing an appropriat function g and) giving a tightr lowr bound for th intgration. W bliv that it is possibl to giv a lowr bound vry clos to (or vn bttr than) th.696 comptitiv ratio obtaind for th unwightd cas [2]. Rfrncs Mlika Abolhassani, T.-H. Hubrt Chan, Fi Chn, Hossin Esfandiari, MohammadTaghi Hajiaghayi, Hamid Mahini, and Xiaowi Wu. Bating ratio.5 for wightd oblivious matching problms. In ESA, volum 57 of LIPIcs, pags 3: 3:8. Schloss Dagstuhl - Libniz-Zntrum fur Informatik, Gagan Aggarwal, Gagan Gol, Chinmay Karand, and Aranyak Mhta. Onlin vrtxwightd bipartit matching and singl-bid budgtd allocations. In Procdings of th Twnty-Scond Annual ACM-SIAM Symposium on Discrt Algorithms, SODA 2, San Francisco, California, USA, January 23-25, 2, pags , 2. doi:.37/ Jonathan Aronson, Martin Dyr, Alan Friz, and Stphn Sun. Randomizd grdy matching. ii. Random Struct. Algorithms, 6():55 73, 995. doi:.2/rsa Bahman Bahmani and Michal Kapralov. Improvd bounds for onlin stochastic matching. In ESA (), volum 6346 of Lctur Nots in Computr Scinc, pags 7 8. Springr, 2. 5 Bnjamin Birnbaum and Clair Mathiu. On-lin bipartit matching mad simpl. ACM SIGACT Nws, 39():8 87, Brian Brubach, Karthik Abinav Sankararaman, Aravind Srinivasan, and Pan Xu. Nw algorithms, bttr bounds, and a novl modl for onlin stochastic matching. In ESA, volum 57 of LIPIcs, pags 24: 24:6. Schloss Dagstuhl - Libniz-Zntrum fur Informatik, Niv Buchbindr, Kamal Jain, and Josph Naor. Onlin primal-dual algorithms for maximizing ad-auctions rvnu. In ESA, volum 4698 of Lctur Nots in Computr Scinc, pags Springr, Niv Buchbindr, Kamal Jain, and Mohit Singh. Scrtary problms via linar programming. Math. Opr. Rs., 39():9 26, T.-H. Hubrt Chan, Fi Chn, Xiaowi Wu, and Zhichao Zhao. Ranking on arbitrary graphs: Rmatch via continuous lp with monoton and boundary condition constraints. In Procdings of th Twnty-Fifth Annual ACM-SIAM Symposium on Discrt Algorithms, SODA 24, Portland, Orgon, USA, January 5-7, 24, pags SIAM, 24. doi:.37/ Ilan Ruvn Cohn and David Wajc. Randomizd onlin matching in rgular graphs. In SODA, pags SIAM, 28. Nikhil R. Dvanur and Kamal Jain. Onlin matching with concav rturns. In STOC, pags ACM, Nikhil R. Dvanur, Kamal Jain, and Robrt D. Klinbrg. Randomizd primal-dual analysis of RANKING for onlin bipartit matching. In SODA, pags 7. SIAM, 23. I C A L P 2 8

14 79:4 Onlin Vrtx-Wightd Bipartit Matching: Bating with Random Arrivals 3 Jon Fldman, Aranyak Mhta, Vahab S. Mirrokni, and S. Muthukrishnan. Onlin stochastic matching: Bating -/. In FOCS, pags IEEE Computr Socity, Gagan Gol and Aranyak Mhta. Onlin budgtd matching in random input modls with applications to adwords. In SODA, pags , 28. URL: citation.cfm?id= Brnhard Hauplr, Vahab S. Mirrokni, and Mortza Zadimoghaddam. Onlin stochastic wightd matching: Improvd approximation algorithms. In WINE, volum 79 of Lctur Nots in Computr Scinc, pags 7 8. Springr, 2. 6 Zhiyi Huang, Ning Kang, Zhihao Gavin Tang, Xiaowi Wu, Yuhao Zhang, and Xu Zhu. How to match whn all vrtics arriv onlin. CoRR (to appar in STOC 28), abs/82.395, 28. arxiv: Patrick Jaillt and Xin Lu. Onlin stochastic matching: Nw algorithms with bttr bounds. Math. Opr. Rs., 39(3): , Chinmay Karand, Aranyak Mhta, and Pushkar Tripathi. Onlin bipartit matching with unknown distributions. In Procdings of th 43rd ACM Symposium on Thory of Computing, STOC 2, San Jos, CA, USA, 6-8 Jun 2, pags , 2. doi:.45/ Richard M. Karp, Umsh V. Vazirani, and Vijay V. Vazirani. An optimal algorithm for on-lin bipartit matching. In Procdings of th 22nd Annual ACM Symposium on Thory of Computing, May 3-7, 99, Baltimor, Maryland, USA, pags , 99. doi:.45/ Thomas Ksslhim, Klaus Radk, Andras Tönnis, and Brthold Vöcking. An optimal onlin algorithm for wightd bipartit matching and xtnsions to combinatorial auctions. In ESA, volum 825 of Lctur Nots in Computr Scinc, pags Springr, Mohammad Mahdian and Qiqi Yan. Onlin bipartit matching with random arrivals: an approach basd on strongly factor-rvaling LPs. In Procdings of th 43rd ACM Symposium on Thory of Computing, STOC 2, San Jos, CA, USA, 6-8 Jun 2, pags , 2. doi:.45/ Vahidh H. Manshadi, Shayan Ovis Gharan, and Amin Sabri. Onlin stochastic matching: Onlin actions basd on offlin statistics. Math. Opr. Rs., 37(4): , Aranyak Mhta, Amin Sabri, Umsh V. Vazirani, and Vijay V. Vazirani. Adwords and gnralizd onlin matching. J. ACM, 54(5):22, Yajun Wang and Sam Chiu-wai Wong. Two-sidd onlin bipartit matching and vrtx covr: Bating th grdy algorithm. In ICALP (), volum 934 of Lctur Nots in Computr Scinc, pags 7 8. Springr, 25.