1 A lower bound. Lecture notes: Online bipartite matching algorithms

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1 Cornll Univrsity, Fall 217 Lctur nots: Onlin bipartit matching algorithms CS 682: Algorithms 6 8 Spt In practic whn dsigning algorithms, it is oftn th cas that th input is a stram of data arriving ovr a span of tim, and th algorithm nds to mak dcisions in ral-tim, bfor th ntir input stram has bn obsrvd. This is th subjct of onlin algorithms. In ths lctur nots w xplor a spcific and archtypical xampl in th dsign and analysis of onlin algorithms: th onlin bipartit matching problm. In th onlin bipartit matching problm, thr is a bipartit graph G = (V, E) whos vrtx st V is partitiond into two sids L, R, known as th lft and right or offlin and onlin sids; vry dg of G has on ndpoint in L and th othr ndpoint in R. Th contnts of th st L ar known to th algorithm at initialization tim (t = ), whras th rmaining information about G is rvald at tims t = 1, 2,..., n = R, by xposing on vrtx of R at ach tim stp. Whn vrtx j R arrivs, all of its incidnt dgs ar rvald. Th algorithm is thn allowd to tak on of th following actions: slct on of th dgs (i, j) that was rvald in th currnt stp; or do nothing. Th st of slctd dgs is rquird to b a matching; thus, if vrtx i L blongs to a prviously slctd dg, thn (i, j) may not b slctd in th currnt tim stp. Th algorithm s objctiv is to maximiz th numbr of dgs slctd. A variation of this problm is th onlin bipartit fractional matching problm, in which th input squnc is th sam, but th algorithm s output at tim j is a tupl of numbrs (x ij ) i L satisfying: x ij = whn (i, j) E. i L x ij 1. for all i L, j R x ij 1. In othr words, th matrix of valus (x ij ) (i,j) L R vntually computd by th algorithm must blong to th fractional matching polytop of G. Fractional matching is of intrst as a problm in its own right, and also as a window into th dsign of randomizd onlin bipartit matching algorithms. From any such randomizd algorithm, on can dfin a corrsponding dtrministic onlin fractional bipartit matching algorithm, obtaind by stting x ij to b th unconditional probability that th onlin algorithm slcts dg (i, j). (Not that this unconditional probability can b computd at th tim whn vrtx j arrivs i.., it dos not dpnd on any information to b rvald in th futur which is th rason why th fractional matching algorithm is a valid onlin algorithm.) Not that thr is no obvious way to invrt this transformation; in othr words, givn a dtrministic fractional onlin matching algorithm, thr is no obvious way to obtain a randomizd onlin matching algorithm whos xpctd bhavior yilds th dsignatd fractional algorithm. 1 A lowr bound What should w hop to achiv in an onlin bipartit matching algorithm? If w ar unrasonably optimistic, w might hop to dsign an algorithm that is guarantd to output a maximum cardinality matching. Th following xampl shows that this is hoplss. Suppos L = {i 1, i 2 } and R = {j 1, j 2 }. Considr two possibl input squncs. In both of thm, vrtx j 1 arrivs at

2 tim t = 1 and rvals that it is connctd to both i 1 and i 2. At tim t = 2, vrtx j 2 arrivs and rvals that it has only on nighbor: in Input 1 this nighbor is i 1 ; in Input 2 it is i 2. Notic that th maximum matching has siz 2 in both of ths inputs: j 2 can b matchd to its only nighbor, whras j 1 can b matchd to th rmaining lmnt of L. Also notic that in both cass, this is th uniqu matching of siz 2. Thrfor, an onlin algorithm that sks to slct th maximum matching facs an insurmountabl prdicamnt: at tim t = 1 it must match j 1 to on of its nighbors, thr is a uniqu choic that is consistnt with picking th maximum matching, and thr is no way to know which choic this is until tim t = 2. Thus, for vry dtrministic onlin algorithm, w can find an input instanc that causs th algorithm to slct a matching of siz at most 1, whil th maximum matching has siz 2. On can plac this impossibility rsult in th broadr contxt of comptitiv analysis of onlin algorithms, which valuats algorithms according to th following critrion. Dfinition 1. An onlin algorithm for a maximization problm is c-comptitiv if thr xists a constant b such that for all input squncs, c ALG + b OPT, whr ALG and OPT dnot th valus of th algorithm s solution and th optimum on, rspctivly. It is strictly c-comptitiv if b = in th abov bound. A randomizd algorithm is c-comptitiv (against an oblivious advrsary) if th abov holds with E[ALG] in plac of ALG. Our analysis of th two four-vrtx input squncs abov implis that dtrministic onlin matching algorithms cannot b strictly c-comptitiv for any c < 2. By considring inputs comprising an arbitrarily long squnc of disjoint copis of ithr Input 1 or Input 2, w can liminat th word strictly and conclud that dtrministic algorithms cannot b c-comptitiv for any c < 2. Our abov discussion of th rlationship btwn randomizd and fractional algorithms shows that a lowr bound on th comptitiv ratio of dtrministic fractional onlin algorithms implis th sam lowr bound on th comptitiv ratio of randomizd onlin algorithms. In particular, th comptitiv ratio of fractional (and hnc randomizd) onlin matching algorithms can b boundd blow by 4/3, by an asy analysis of th sam st of input squncs that furnishd th lowr bound of 2 for dtrministic algorithms. 2 Th grdy algorithm It turns out that th xampl prsntd in Sction 1 is th worst possibl for dtrministic algorithms, from th standpoint of comptitiv analysis. Thr is a strictly 2-comptitiv dtrministic onlin algorithm. In fact, a comptitiv ratio of 2 is achivd by th most naïv algorithm: th grdy algorithm that matchs ach nw vrtx j to an arbitrary unmatchd nighbor, i, whnvr an unmatchd nighbor xists. This fact follows dirctly from two simpl lmmas. Lmma 1. Lt G b any graph, M a maximum matching in G, and M a maximal matching in G (i.., on that is not a propr subst of any othr matching). Th cardinalitis of M and M satisfy 2 M M. Proof. Construct a function f from M to M as follows. For vry dg in M, dfin f() to b any dg in M that has an ndpoint in common with. Thr must b at last on such dg

3 in M, bcaus othrwis M {} would b a matching, contradicting our hypothsis that M is maximal. For vry dg = (i, j) M, th st f 1 ( ) has at most two lmnts. (At most on with ndpoint i, and at most on with ndpoint j.) Th inquality 2 M M follows immdiatly. Lmma 2. Th grdy onlin bipartit matching algorithm always slcts a maximal matching in G. Proof. Lt M dnot th matching slctd by th grdy algorithm. For vry dg = (i, j) that dos not blong to M, considr th tim stp in which vrtx j arrivd. Eithr j was matchd to a vrtx othr than i at that tim, or j was not matchd to any vrtx bcaus all of its nighbors (including i) wr alrady matchd in M. In both cass, M contains an dg having ithr i or j as an ndpoint, and thrfor M {} is not a matching. 3 Onlin fractional matching: th watrfilling algorithm It turns out that onlin fractional matching algorithms can achiv comptitiv ratios significantly bttr than 2, as w will s in this sction. First, a usful bit of trminology: w will rfr to th sum j R x ij as th fractional dgr of vrtx i in fractional matching x. For a vrtx j R th fractional dgr is dfind similarly. Prhaps th most natural ida for onlin fractional matching is to hav ach vrtx j balanc load qually among its nighbors. In othr words, if a nw vrtx j arrivs and has d nighbors, thn for ach nighbor i w st th valu of x ij to b 1/d, unlss that would violat th dgr constraint of vrtx i (th constraint that j x ij 1) in which cas w mrly incras x ij as much as possibl givn th dgr constraint. Howvr, this statlss balancing algorithm fails to b bttr than 2-comptitiv. To construct a countrxampl, w tak th xampl form Sction 1 and blow up ach vrtx into n vrtics, carfully modifying th dg st to caus th algorithm to mak catastrophic dcisions. Th st L now has 2n vrtics, which w will labl as a 1, a 2,..., a n, b 1, b 2,..., b n, and th st R has 2n vrtics labld c 1, c 2,..., c n, d 1, d 2,..., d n. Each vrtx c j has n + 1 nighbors: it is connctd to a j and also to b 1, b 2,..., b n. Each vrtx d j has only on nighbor, namly b j. Th maximum matching in this graph has siz 2n: it matchs (a i, c i ) and (b i, d i ) for i = 1,..., n. If th vrtics c 1,..., c n, d 1,..., d n arriv in that ordr, th statlss balancing algorithm will first 1 assign a valu of to ach dg incidnt to c n+1 1,..., c n. Thus, whn d 1,..., d n start arriving, ach of thm has a uniqu nighbor and th fractional dgr of that nighbor is alrady n, so n+1 d j can contribut only 1 additional units to th siz of th fractional matching. Thus, whn n+1 th algorithm is finishd procssing th ntir graph, th total siz of its fractional matching is n + n, only slightly mor than half of th optimum. n+1 What wnt wrong in this algorithm? Th vrtics b 1,..., b n ar mor highly dmandd than a 1,..., a n and it was unwis for vrtics c 1,..., c n to us up almost all of th capacity of b 1,..., b n whil using almost non of a 1,..., a n. Th first vrtx, c 1, can b forgivn for making this mistak sinc all of its nighbors lookd indistinguishabl whn it arrivd. But latr on, w should hav known bttr: w had alrady sn that th capacitis of b 1,..., b n wr bing dpltd and should hav takn masurs to consrv that capacity. In short, thr was nothing vidntly wrong with th load-balancing ida, but it was silly to do statlss load-balancing; instad, w should hav kpt track of th currnt stat (th amount of load alrady placd on ach vrtx in L) and adjustd our load-balancing dcisions to corrct for imbalancs in th currnt load vctor.

4 This bring us to th watrfilling algorithm. It kps track of a watr lvl for ach i L rprsnting th currnt fractional dgr d(i) = j x ij, summing ovr all vrtics j R that hav arrivd in th past. Whn a nw vrtx j arrivs, it allocats its on unit of fractional dgr among its nighbors by finding th nighbors with th lowst watr lvl and continuously raising thir watr lvl until ithr on unit of watr has bn pourd into th graph, or th watr lvl of all nighbors rachs 1, whichvr coms first. In lss mtaphorical trms, th algorithm finds th uniqu numbr ˆl(j) such that max{ˆl(j), d(i)} = 1 + d(i), whr N(j) rprsnts th st of all nighbors of j. It thn sts l(j) = min{ˆl(j), 1} x ij = max{l(j), d(i)} d(i) (i, j) E and it updats d(i) to d(i) + x ij for all i. W will analyz th watrfilling algorithm using th primal-dual mthod. This mans that w ll us th fractional matching LP max s.t. i,j x ij j x ij 1 i i x ij 1 j x ij i, j and its dual min i α i + j β j s.t. α i + β j 1 (i, j) E α i, β j i, j In particular, w dfin a dual solution (α i ) i L, (β j ) j R by spcifying that α i = g(d(i)) i (1) β j = 1 g(l(j)) j, (2) whr g(y) = y 1 1. Th choic of this spcific function g will mak mor sns latr in th analysis. Th vital proprtis of g that ar ndd in th analysis ar: 1. g is an incrasing function. 2. g() = 3. g(1) = g(t) + g (t) = for all t. First, lt s obsrv that th dual solution dfind by (1)-(2) is fasibl. This is bcaus at th tim w finish procssing vrtx j, th inquality d(i) l(j) is satisfid by all nighboring vrtics i. Sinc th valu d(i) will not subsquntly dcras, w also hav d(i) l(j) at trmination. Furthrmor, sinc g is an incrasing function, w hav α i + β j = g(d(i)) + 1 g(l(j)) g(l(j)) + 1 g(l(j)) = 1,

5 which vrifis dual fasibility. W claim that th fractional matching and th dual solution computd by our algorithm satisfy x ij α i + β j. (3) 1 (i,j) E i L j R By th wak duality, th sum on th right sid is an uppr bound on th siz of any fractional matching in G, and thrfor (3) implis that th watrfilling algorithm is ( ) -comptitiv. To prov (3), w compar β j with a paramtr β j dfind as follows. For t [, 1] lt n j (t) dnot th numbr of dgs (i, j) E such that th inquality d(i) t hld at th tim whn j arrivd. Not that n j (t) dt = 1 providd that l(j) < 1, bcaus in that cas vrtx j contributd on unit of watr and th intgrand dnots th rat at which watr was filling th systm as w incrasd th watr lvl l from t to t + dt. Now, dfin β j = (1 g(t)) n j (t) dt. Th inquality β j β j always holds: whn l(j) = 1 this is bcaus β j =, and whn l(j) < 1 it is bcaus 1 g(t) is a dcrasing function of t and thrfor (1 g(t)) n j (t) dt > (1 g(l(j)) n j (t) dt = 1 g(l(j)) = β j. Ltting d(i) dnot th dgr of a vrtx i L bfor th arrival of vrtx j, th amount by which th dual objctiv incrass whn procssing j is: β j + [g(l(j)) g(d(i))] = 1 g(l(j)) + = 1 g(l(j)) + = 1 = 1 d(i) g (t) dt g (t) n j (t) dt [1 g(t) + g (t)] n j (t) dt n j (t) dt hnc th incras in th dual objctiv is at most tims th incras in th primal objctiv. Sinc th primal and dual objctivs both start out at zro, this mans that th dual objctiv at trmination is at most tims th primal objctiv, crtifying inquality (3) and complting th proof that th watrfilling algorithm is ( ) -comptitiv. x ij,

6 4 Randomizd onlin matching: Th ranking algorithm (Most of this sction is an xcrpt from th papr Randomizd Primal-Dual Analysis of ranking for Onlin Bipartit Matching by N. Dvanur, K. Jain, and R. Klinbrg, 212.) Givn an onlin fractional matching algorithm, it is tmpting to try constructing a randomizd onlin matching algorithm whos probability of choosing dg (i, j) is qual to th valu x ij computd by th fractional matching algorithm. If such a transformation wr possibl, it would yild a randomizd onlin matching algorithm whos comptitiv ratio is xactly th sam as that of th givn fractional matching algorithm. Unfortunatly, such a transformation is not possibl in gnral. (For xampl, thr is no randomizd matching algorithm whos probability of slcting ach dg (i, j) is xactly qual to th valu assignd to that dg by th watrfilling algorithm. It is quit instructiv to try proving this.) Howvr, thr is a randomizd onlin matching algorithm, known as ranking, that achivs xactly th sam comptitiv ratio as th watrfilling algorithm, namly. Problm (1c) on this wk s homwork asks you to prov that ranking achivs th bst possibl comptitiv ratio for randomizd onlin matching algorithms. Th ranking algorithm is actually vry asy to dscrib: at initialization tim, it sampls a uniformly random total ordring of th vrtics in L. Subsquntly, as ach vrtx j R arrivs, if j has an unmatchd nighbor in L thn w choos th unmatchd nighbor i that coms arlist in th random ordring, and w add (i, j) to th matching. To analyz th ranking algorithm, w bgin with a rintrprtation of th algorithm in a way that is conduciv to our analysis. Instad of picking a random total ordring of th vrtics in L, ach vrtx in L picks a random numbr in [, 1] and a vrtx j R, upon its arrival, is assignd to th unmatchd nighbor who pickd th lowst numbr. Th algorithm is prsntd as Algorithm 1 blow. Algorithm 1: Th ranking algorithm. forach i L do Pick Y i [, 1] uniformly at random forach j R do Whn j arrivs, lt N(j) dnot th st of unmatchd nighbors of j; if N(j) = thn j rmains unmatchd ls Match j to arg min{y i : i N(j)} To analyz th algorithm, w not th standard LP rlaxation for matching and its dual. maximiz x ij s.t. minimiz α i + β j s.t. (i,j) E i L j R i V, x ij 1. (i, j) E, α i + β j 1. j:(i,j) E (i, j) E, x ij. i, j, α i, β j. Our analysis constructs a dual solution which is also randomizd. Th dual variabls w construct may not always b fasibl; in othr words, thy may violat th constraint α i +β j 1

7 for som dgs (i, j). Howvr, th xpctd valus of th dual variabls will constitut a fasibl dual solution. Th comptitiv ratio of will follow from th fact that th valu of th dual solution is always tims th siz of th matching found, and that th xpctation of th dual variabls constituts a fasibl dual solution (whos valu, of cours, is also tims th xpctd siz of th matching found). Our construction of th duals dpnds on a monoton non-dcrasing function h that is closly rlatd to th function g that cam up in th analysis of th watrfilling algorithm in Sction 3. Th formula for h is h(y) = y 1 and its rlvant proprtis ar: 1. h is an incrasing function; 2. h(1) = 1; θ 3. θ [, 1] h(y) dy + 1 h(θ) =. Not th similarity btwn th intgral quation in proprty 3 of h and th diffrntial quation in proprty 4 of th function g in Sction 3; not also, howvr, that if w diffrntiat both sids of th intgral quation dfining h w crtainly don t gt th diffrntial quation dfining g. Whnvr i is matchd to j, lt α i = h(y i), β j = (1 h(y i)). For all unmatchd i and j, st α i = β j =. It will b usful to intrprt th algorithm as follows: on matching i to j, w gnrat a valu of 1 for th primal, which translats to a valu of for th dual. Each unmatchd vrtx i L that is a nighbor of j offrs (1 h(y i)) of this valu to j (to b assignd to β j ), whil kping th rst to itslf (to b assignd to α i ). Thn j is matchd to th vrtx that maks th highst offr. Bfor w show that th xpctation of th duals is fasibl, w nd crtain proprtis of th algorithm spcifid by th following two lmmas. Lt (i, j) E b any dg in th graph. Considr an instanc of th algorithm on G \ {i}, with th sam choic of Y i for all othr i L. Lt y c b th valu of Y i for th i that is matchd to j. Dfin y c to b 1 if j is not matchd. Lt βj c b th valu of β j in this run, i.. βj c = (1 h(yc )). Lmma 3 (Dominanc Lmma). Givn Y i for all othr i L, i gts matchd if Y i < y c. Proof. Suppos i is not matchd whn j arrivs. This mans that th run of th algorithm until thn is idntical to th run without i. From th dfinition of y c, in th run without i, j is matchd to i such that Y i = y c. Sinc Y i < y c, j is matchd to i. Lmma 4 (Monotonicity Lmma). Givn Y i for all othr i L, for all choics of Y i, β j β c j. Proof. Considr xcuting th algorithm on graphs G and G\{i} in paralll. At th start of vry stp of th two paralll xcutions, th unmatchd vrtics in L for th G xcution constitut a suprst of th unmatchd vrtics in L for th G\{i} xcution. This statmnt is asily provn by induction: givn that it holds at th start of on stp, th only way it could b violatd at th start of th nxt stp is if th G xcution chooss a vrtx i L that is also unmatchd, but is not chosn, in th G \ {i} xcution. Instad th G \ {i} xcution must choos som othr vrtx i such that Y i < Y i. By our induction hypothsis i was also unmatchd in th G xcution, contradicting th fact that th algorithm chos i instad. Whn nod j arrivs, its unmatchd nighbors in th G xcution form a suprst of its unmatchd nighbors in th G \ {i} xcution, so in th both xcutions j has an unmatchd nighbor whos Y -valu is y c. If th algorithm instad chooss anothr nighbor of j, its Y -valu can b at most y c and hnc, by th monotonicity of h, w hav β j β c j.

8 W now show that th abov proprtis of h imply a comptitiv ratio of Lmma 5. ranking is -comptitiv. for ranking. Proof. Whnvr i is matchd to j, α i + β j =. Thrfor th ratio of th dual solution to th primal is always. W show that th dual is fasibl in xpctation. In particular, w show that for all (i, j) E, E Yi [α i + β j ] 1 for all choics of Y i for all i i L. By th Dominanc Lmma (Lmma 3) i is matchd whnvr Y i y c. Hnc E Yi [α i ] y c h(y) dy. 1 By th Monotonicity Lmma (Lmma 4), β j βj c = (1 h(yc )) for all choics of Y i. Th lmma now follows from th intgral quation listd abov as proprty 3 of h.

cycle that does not cross any edges (including its own), then it has at least

cycle that does not cross any edges (including its own), then it has at least W prov th following thorm: Thorm If a K n is drawn in th plan in such a way that it has a hamiltonian cycl that dos not cross any dgs (including its own, thn it has at last n ( 4 48 π + O(n crossings Th