The Advice Complexity of a Class of Hard Online Problems

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1 The Advie Complexiy of a Class of Hard Online Problems Joan Boyar, Lene M. Favrhold, Chrisian Kudahl, and Jesper W. Mikkelsen Deparmen of Mahemais and Compuer Siene Universiy of Souhern Denmark July 1, 2015 Absra. The advie omplexiy of an online problem is a measure of how muh knowledge of he fuure an online algorihm needs in order o ahieve a erain ompeiive raio. We deermine he advie omplexiy of a number of hard online problems inluding independen se, verex over, dominaing se and several ohers. These problems are hard, sine a single wrong answer by he online algorihm an have devasaing onsequenes. For eah of hese problems, we show ha log n = Θn/ bis of advie are neessary and suffiien up o an addiive erm of Olog n o ahieve a ompeiive raio of. The resuls are obained by inroduing a new sring guessing problem relaed o hose of Emek e al. TCS 2011 and Bökenhauer e al. TCS I urns ou ha his gives a powerful bu easyo-use mehod for providing boh upper and lower bounds on he advie omplexiy of an enire lass of online problems. Previous resuls of Halldórsson e al. TCS 2002 on online independen se, in a relaed model, imply ha he advie omplexiy of he problem is Θn/. Our resuls improve on his by providing an exa formula for he higher-order erm. For online disjoin pah alloaion, Bökenhauer e al. ISAAC 2009 gave a lower bound of Ωn/ and an upper bound of On log / on he advie omplexiy. We improve on he upper bound by a faor of log. For he remaining problems, no bounds on heir advie omplexiy were previously known. 1 Inroduion An online problem is an opimizaion problem in whih he inpu is divided ino small piees, usually alled requess, arriving sequenially. An online algorihm mus serve eah reques wihou any knowledge of fuure requess, and he deisions made by he online algorihm are irrevoable. The goal is o minimize or maximize some objeive funion. Tradiionally, he qualiy of an online algorihm is measured by he ompeiive raio, whih is an analog of he approximaion raio for approximaion algorihms: The soluion produed by he online algorihm is ompared o he soluion produed by an opimal offline algorihm, Op, whih knows he enire reques sequene in advane, and only he wors ase is onsidered. For some online problems, i is impossible o ahieve a good ompeiive raio. As an example, onsider he lassial problem of finding a maximum independen se in a graph. Suppose ha, a some poin, an online algorihm deides o inlude a verex v in is soluion. I hen urns ou ha all forhoming veries in he graph are onneed o v, bu no o eah oher. Thus, he online algorihm anno inlude any of hese veries. On he oher hand, Op knows he enire graph, and so i rejes v and insead akes all forhoming veries. In fa, one an easily show ha, A preliminary version of his paper appeared in he proeedings of he 32nd Inernaional Symposium on Theoreial Aspes of Compuer Siene STACS 2015, Leibniz Inernaional Proeedings in Informais 30: , 2015 This work was parially suppored by he Villum Foundaion and he Danish Counil for Independen Researh, Naural Sienes.

2 even if we allow randomizaion, no online algorihm for his problem an obain a ompeiive raio beer han Ωn, where n is he number of veries in he graph. A naural quesion for online problems, whih is no answered by ompeiive analysis, is he following: Is here some small amoun of informaion suh ha, if he online algorihm knew his, hen i would be possible o ahieve a signifianly beer ompeiive raio? Our main resul is a negaive answer o his quesion for an enire lass of hard online problems, inluding independen se. We prove our main resul in he reenly inrodued advie omplexiy model. In his model, he online algorihm is provided wih b bis of advie abou he inpu. No resriions are plaed on he advie. This means ha he advie ould poenially enode some knowledge whih we would never expe o be in possession of in praie, or he advie ould be impossible o ompue in any reasonable amoun of ime. Lower bounds obained in he advie omplexiy model are herefore very robus, sine hey do no rely on any assumpions abou he advie. If we know ha b bis of advie are neessary o be -ompeiive, hen we know ha any piee of informaion whih an be enoded using less han b bis will no allow an online algorihm o be -ompeiive. In his paper, we use advie omplexiy o inrodue he firs omplexiy lass for online problems. The omplee problems for his lass, one of whih is independen se, are very hard in he online seing. We essenially show ha for he omplee problems in he lass, a -ompeiive online algorihm needs as muh advie as is required o expliily enode a soluion of he desired qualiy. One imporan feaure of our framework is ha we inrodue an absra online problem whih is omplee for he lass and well-suied o use as he saring poin for reduions. This makes i easy o prove ha a large number of online problems are omplee for he lass and hereby obain igh bounds on heir advie omplexiy. Advie Complexiy Advie omplexiy [7, 14, 15, 22] is a quaniaive and sandardized, i.e., problem independen, way of relaxing he online onsrain by providing he algorihm wih parial knowledge of he fuure. The main idea of advie omplexiy is o provide an online algorihm, Alg, wih some advie bis. These bis are provided by a rused orale, O, whih has unlimied ompuaional power and knows he enire reques sequene. In he firs model proposed [14], he advie bis were given as answers of varying lenghs o quesions posed by Alg. One diffiuly wih his model is ha using a mos 1 bi, hree differen opions an be enoded giving no bis, a 0, or a 1. This problem was addressed by he model proposed in [15], where he orale is required o send a fixed number of advie bis per reques. However, for he problems we onsider, one bi per reques is enough o guaranee an opimal soluion, and so his model is no appliable. Insead, we will use he advie-on-ape model [7], whih allows for a sublinear number of advie bis while avoiding he problem of enoding informaion in he lengh of eah answer. Before he firs reques arrives, he orale prepares an advie ape, an infinie binary sring. The algorihm Alg may, a any poin, read some bis from he advie ape. The advie omplexiy of Alg is he maximum number of bis read by Alg for any inpu sequene of a mos a given lengh. When advie omplexiy is ombined wih ompeiive analysis, he enral quesion is: How many bis of advie are neessary and suffiien o ahieve a given ompeiive raio? Definiion 1 Compeiive raio [23, 32] and advie omplexiy [7, 22]. The inpu o an online problem, P, is a reques sequene σ = r 1,..., r n. An online algorihm wih advie, Alg, 2

3 ompues he oupu y = y 1,..., y n, under he onsrain ha y i is ompued from ϕ, r 1,..., r i, where ϕ is he onen of he advie ape. Eah possible oupu for P is assoiaed wih a sore. For a reques sequene σ, Algσ Opσ denoes he sore of he oupu ompued by Alg Op when serving σ. If P is a maximizaion problem, hen Alg is n-ompeiive if here exiss a onsan, α, suh ha, for all n N, Opσ n Algσ + α, for all reques sequenes, σ, of lengh a mos n. If P is a minimizaion problem, hen Alg is n-ompeiive if here exiss a onsan, α, suh ha, for all n N, Algσ n Opσ + α, for all reques sequenes, σ, of lengh a mos n. In boh ases, if he inequaliy holds wih α = 0, we say ha Alg is srily n-ompeiive. The advie omplexiy, bn, of an algorihm, Alg, is he larges number of bis of ϕ read by Alg over all possible inpus of lengh a mos n. The advie omplexiy of a problem, P, is a funion, fn,, 1, suh ha he smalles possible advie omplexiy of a srily -ompeiive online algorihm for P is fn,. In his paper, we only onsider deerminisi online algorihms wih advie. Noe ha boh he advie read and he ompeiive raio may depend on n, bu, for ease of noaion, we ofen wrie b and insead of bn and n. Also, by his definiion, 1, for boh minimizaion and maximizaion problems. For minimizaion problems, he sore is also alled he os, and for maximizaion problems, he sore is also alled he profi. Furhermore, we use oupu and soluion inerhangeably. Lower and upper bounds on he advie omplexiy have been obained for many problems, see e.g. [2, 4 10, 13 15, 17, 18, 22, 24, 26, 28, 30, 31]. Sring guessing In [5, 15], he advie omplexiy of he following sring guessing problem, SG, is sudied: For eah reques, whih is simply empy and onains no informaion, he algorihm ries o guess a single bi or more generally, a haraer from some finie alphabe. The orre answer is eiher revealed as soon as he algorihm has made is guess known hisory, or all of he orre answers are revealed ogeher a he very end of he reques sequene unknown hisory. The goal is o guess orrely as many bis as possible. The problem was firs inrodued under he name generalized mahing pennies in [15], where a lower bound for randomized algorihms wih advie was given. In [5], he lower bound was improved for he ase of deerminisi algorihms. In fa, he lower bound given in [5] is igh up o lower-order addiive erms. While SG is raher unineresing in he view of radiional ompeiive analysis, i is very useful in an advie omplexiy seing. Indeed, i has been shown ha he sring guessing problem an be redued o many lassial online problems, hereby giving lower bounds on he advie omplexiy for hese problems. This inludes bin paking [10], he k-server problem [18], lis updae [9], merial ask sysem [15], se over [5] and a erain version of maximum lique [5]. 3

4 Asymmeri sring guessing. In his paper, we inrodue a new sring guessing problem alled asymmeri sring guessing, ASG, formally defined in Seion 2. The rules are similar o hose of he original sring guessing problem wih an alphabe of size wo, bu he sore funion is asymmeri: If he algorihm answers 1 and he orre answer is 0, hen his ouns as a single wrong answer as in he original problem. On he oher hand, if he algorihm answers 0 and he orre answer is 1, he soluion is deemed infeasible and he algorihm ges an infinie penaly. This asymmery in he sore funion fores he algorihm o be very auious when making is guesses. As wih he original sring guessing problem, ASG is no very ineresing in he radiional framework of ompeiive analysis. However, i urns ou ha ASG apures, in a very preise way, he hardness of problems suh as online independen se and online verex over. Problems Many of he problems ha we onsider are graph problems, and mos of hem are sudied in he verex-arrival model. In his model, he veries of an unknown graph are revealed one by one. Tha is, in eah round, a verex is revealed ogeher wih all edges onneing i o previously revealed veries. For he problems we sudy in he verex-arrival model, whenever a verex, v, is revealed, an online algorihm Alg mus irrevoably deide if v should be inluded in is soluion or no. Denoe by V Alg he veries inluded by Alg in is soluion afer all veries of he inpu graph have been revealed. The individual graph problems are defined by speifying he se of feasible soluions. The os profi of an infeasible soluion is. The problems we onsider in he verex-arrival model are: Online Verex Cover. A soluion is feasible if i is a verex over in he inpu graph. The problem is a minimizaion problem. Online Cyle Finding. A soluion is feasible if he subgraph indued by he veries in he soluion onains a yle. We assume ha he presened graph always onains a yle. The problem is a minimizaion problem Online Dominaing Se. A soluion is feasible if i is a dominaing se in he inpu graph. The problem is a minimizaion problem. Online Independen Se. A soluion is feasible if i is an independen se in he inpu graph. The problem is a maximizaion problem. We emphasize ha he lassial 2-approximaion algorihm for offline verex over anno be used in our online seing, even hough he algorihm is greedy. Tha algorihm greedily overs he edges by seleing boh endpoins one by one, bu his is no possible in he verex-arrival model. Apar from he graph problems in he verex-arrival model menioned above, we also onsider he following online problems. Again, he os profi of an infeasible soluion is. Online Disjoin Pah Alloaion. A pah wih L + 1 veries {v 0,..., v L } is given. Eah reques v i, v j is a subpah speified by he wo endpoins v i and v j. A reques v i, v j mus immediaely be eiher aeped or rejeed. This deision is irrevoable. A soluion is feasible if he subpahs ha have been aeped do no share any edges. The profi of a feasible soluion is he number of aeped pahs. The problem is a maximizaion problem. Online Se Cover se-arrival version. A finie se U known as he universe is given. The inpu is a sequene of n finie subses of U, A 1,..., A n, suh ha 1 i n A i = U. A subse an be eiher aeped or rejeed. Denoe by S he se of indies of he subses aeped in some 4

5 soluion. The soluion is feasible if i S A i = U. The os of a feasible soluion is he number of aeped subses. The problem is a minimizaion problem. Preliminaries Throughou he paper, we le n denoe he number of requess in he inpu. We le log denoe he binary logarihm log 2 and ln he naural logarihm log e. By a sring we always mean a bi sring. For a sring x {0, 1} n, we denoe by x 1 he Hamming weigh of x ha is, he number of 1s in x and we define x 0 = n x 1. Also, we denoe he i h bi of x by x i, so ha x = x 1 x 2... x n. For n N, define [n] = {1, 2,..., n}. For a subse Y [n], he haraerisi veor of Y is he sring y = y 1... y n {0, 1} n suh ha, for all i [n], y i = 1 if and only if i Y. For x, y {0, 1} n, we wrie x y if x i = 1 y i = 1 for all 1 i n. If he orale needs o ommuniae some ineger m o he algorihm, and if he algorihm does no know of any upper bound on m, he orale needs o use a self-delimiing enoding. For insane, he orale an wrie logm + 1 in unary a sring of 1 s followed by a 0 before wriing m iself in binary. In oal, his enoding uses 2 logm = Olog m bis. Slighly more effiien enodings exis, see e.g. [6]. Our onribuion In Seion 3, we give lower and upper bounds on he advie omplexiy of he new asymmeri sring guessing problem, ASG. The bounds are igh up o an addiive erm of Olog n. Boh upper and lower bounds hold for he ompeiive raio as well as he sri ompeiive raio. More preisely, if b is he number of advie bis neessary and suffiien o ahieve a sri ompeiive raio > 1, hen we show ha 1 1 b = log 1 + n ± Θlog n, 1 where 1 n 1 e ln 2 log n n. This holds for all varians of he asymmeri sring guessing problem minimizaion/maximizaion and known/unknown hisory. See Figure 1 on page 12 for a graphial plo. For he lower bound, he onsan hidden in Θlog n depends on he addiive onsan α of he -ompeiive algorihm. We only onsider > 1, sine in order o be srily 1-ompeiive, an algorihm needs o orrely guess every single bi. I is easy o show ha his requires n bis of advie see e.g. [5]. By Remark 1 in seion 3, his also gives a lower bound for being 1-ompeiive. In Seion 4, we inrodue a lass, AOC, of online problems. The lass AOC essenially onsiss of hose problems whih an be redued o ASG. In pariular, for any problem in AOC, our upper bound on he advie omplexiy for ASG applies. This is one of he few known examples of a general ehnique for onsruing online algorihms wih advie, whih works for an enire lass of problems. On he hardness side, we show ha several online problems, inluding Online Verex Cover, Online Cyle Finding, Online Dominaing Se, Online Independen Se, Online Se 5

6 Cover and Online Disjoin Pah Alloaion are AOC-omplee, ha is, hey have he same advie omplexiy as ASG. We prove his by providing reduions from ASG o eah of hese problems. The reduions preserve he ompeiive raio and only inrease he number of advie bis by an addiive erm of Olog n. Thus, we obain bounds on he advie omplexiy of eah of hese problems whih are essenially igh. Finally, we give a few examples of problems whih belong o AOC, bu are provably no AOC-omplee. This firs omplexiy lass wih is many omplee problems ould be he beginning of a omplexiy heory for online algorihms. As a key sep in obaining our resuls, we esablish a onneion beween he advie omplexiy of ASG and he size of overing designs a well-sudied obje from he field of ombinaorial designs. Disussion of resuls. Noe ha he offline versions of he AOC-omplee problems have very differen properies. Finding he shores yle in a graph an be done in polynomial ime. There is a greedy 2-approximaion algorihm for finding a minimum verex over. No olog n-approximaion algorihm exiss for finding a minimum se over or a minimum dominaing se, unless P = NP [29]. For any ε > 0, no n 1 ε -approximaion algorihm exiss for finding a maximum independen se, unless ZPP = NP [21]. Ye hese AOC-omplee problems all have essenially he same high advie omplexiy. Remarkably, he algorihm presened in his paper for problems in AOC is oblivious o he inpu: i ignores he inpu and uses only he advie o ompue he oupu. Our lower bound proves ha for AOC-omplee problems, his oblivious algorihm is opimal. This shows ha for AOC-omplee problems, an adversary an reveal he inpu in suh a way ha an online algorihm simply anno dedue any useful informaion from he previously revealed requess when i has o answer he urren reques. Thus, even hough he AOC-omplee problems are very differen in he offline seing wih respe o approximaion, in he online seing, hey beome equally hard sine an adversary an preven an online algorihm from using any non-rivial sruure of hese problems. Finally, we remark ha he bounds 1 are under he assumpion ha he number of 1s in he inpu sring ha is, he size of he opimal soluion is hosen adversarially. In fa, if denoes he number of 1s in he inpu sring, we give igh lower and upper bounds on he advie omplexiy as a funion of boh n,, and. We hen obain 1 by alulaing he value of whih maximizes he advie needed i urns ou ha his value is somewhere beween n/e and n/2. If is smaller or larger han his value, hen our algorihm will use less advie han saed in 1. Comparison wih previous resuls. The original sring guessing problem, SG, an be viewed as a maximizaion problem, he goal being o orrely guess as many of he n bis as possible. Clearly, Op always obains a profi of n. Wih a single bi of advie, an algorihm an ahieve a sri ompeiive raio of 2: The advie bi simply indiaes wheher he algorihm should always guess 0 or always guess 1. This is in sark onras o ASG, where linear advie is needed o ahieve any onsan ompeiive raio. On he oher hand, for boh SG and ASG, ahieving a onsan ompeiive raio < 2 requires linear advie. However, he exa amoun of advie required o ahieve suh a ompeiive raio is larger for ASG han for SG. See Figure 1 for a graphial omparison. The problems Online Independen Se and Online Disjoin Pah Alloaion, whih we show o be AOC-omplee, have previously been sudied in he onex of advie omplexiy or 6

7 similar models. Sine hese problems are AOC-omplee, we presen a deailed omparison of our work o hese previous resuls. In [7], among oher problems, he advie omplexiy of Online Disjoin Pah Alloaion is onsidered. I is shown ha a srily -ompeiive algorihm mus read a leas n bis of advie. Comparing wih our resuls, we see ha his lower bound is asympoially igh. On he oher hand, he auhors show ha for any 2, here exiss a srily -ompeiive online algorihm reading a mos b bis of advie, where b = min { n log 1 1/, n log n } + 3 log n + O1. We remark ha n log / 1 1/ n log /, for 2. Thus, his upper bound is a faor of 2 log away from he lower bound. In [19], he problem Online Independen Se is onsidered in a muli-soluion model. In his model, an online algorihm is allowed o mainain muliple soluions. The algorihm knows a priori he number n of veries in he inpu graph. The model is parameerized by a funion rn. Whenever a verex v is revealed, he algorihm an inlude v in a mos rn differen soluions some of whih migh be new soluions wih v as he firs verex. A he end, he algorihm oupus he soluion whih onains he mos veries. The muli-soluion model is losely relaed o he advie omplexiy model. Afer proessing he enire inpu, an algorihm in he muli-soluion model has reaed a mos n rn differen soluions sine a mos rn new soluions an be reaed in eah round. Thus, one an onver a muli-soluion algorihm o an algorihm wih advie by leing he orale provide logn rn bis of advie indiaing whih soluion o oupu. In addiion, he orale needs o provide Olog n bis of advie in order o le he algorihm learn n whih was given o he muli-soluion algorihm for free. On he oher hand, an algorihm using bn bis of advie an be onvered o 2 bn deerminisi algorihms. One an hen run hem in parallel o obain a muli-soluion algorihm wih rn = 2 bn. These simple onversions allow one o ranslae boh upper and lower bounds beween he wo models almos exaly up o a lower-order addiive erm of Olog n. I is shown in [19] ha for any 1, here is a srily -ompeiive algorihm in he mulisoluion model if log rn 1 n/. This gives a srily -ompeiive algorihm reading n + Olog n bis of advie. On he oher hand, i is shown ha for any srily -ompeiive algorihm in he muli-soluion model, i mus hold ha n/2 logn rn. This implies ha any srily -ompeiive algorihm wih advie mus read a leas n 2 log n bis of advie. Thus, he upper and lower bounds obained in [19] are asympoially igh. Comparing our resuls o hose of [19] and [7], we see ha we improve on boh he lower and upper bounds on he advie omplexiy of he problems under onsideraion by giving igh resuls. For he upper bound on Online Disjoin Pah Alloaion, he improvemen is a faor of log /2. The resuls of [19] are already asympoially igh. Our improvemen onsiss of deermining he exa oeffiien of he higher-order erm. Perhaps even more imporan, obaining hese igh lower and upper bounds on he advie omplexiy for Online Independen Se and Online Disjoin Pah Alloaion beomes very easy when using our sring guessing problem ASG. We remark ha he reduions we use o show he hardness of hese problems redues insanes of ASG o insanes of Online Independen Se resp. Online Disjoin Pah Alloaion ha are idenial o he hard insanes used in [19] resp. [7]. Wha enables us o improve he previous bounds, even hough we use he same hard insanes, is ha we have a deailed analysis of he advie omplexiy of ASG a our disposal. 7

8 Relaed work The advie omplexiy of Online Disjoin Pah Alloaion has also been sudied as a funion of he lengh of he pah as opposed o he number of requess, see [3, 7]. The advie omplexiy of Online Independen Se on biparie graphs and on sparse graphs has been deermined in [13]. I urns ou ha for hese graph lasses, even a small amoun of advie an be very helpful. For insane, i is shown ha a single bi of advie is enough o be 4-ompeiive on rees reall ha wihou advie, i is no possible o be beer han Ωn-ompeiive, even on rees. I is lear ha online maximum lique in he verex arrival model is essenially equivalen o Online Independen Se. In [5], he advie omplexiy of a differen version of online maximum lique is sudied: The veries of a graph are revealed as in he verex-arrival model. Le V Alg be he se of veries seleed by Alg and le C be a maximum lique in he subgraph indued by he veries V Alg. The profi of he soluion V Alg is C 2 / V Alg. In pariular, he algorihm is no required o oupu a lique, bu is insead punished for inluding oo many addiional veries in is oupu. The Online Verex Cover problem and some variaions hereof are sudied in [11]. The advie omplexiy of an online se over problem [1] has been sudied in [24]. However, he version of online se over ha we onsider is differen and so our resuls and hose of [24] are inomparable. 2 Asymmeri Sring Guessing In his seion, we formally define he asymmeri sring guessing problem and give simple algorihms for he problem. There are four varians of he problem, one for eah ombinaion of minimizaion/maximizaion and known/unknown hisory. Colleively, hese four problems will be referred o as ASG. We have deliberaely ried o mimi he definiion of he sring guessing problem SG from [5]. However, for ASG, he number, n, of requess is no revealed o he online algorihm as opposed o in [5]. This is only a minor ehnial deail sine i hanges he advie omplexiy by a mos Olog n bis. 2.1 The Minimizaion Version We begin by defining he wo minimizaion varians of ASG: One in whih he oupu of he algorihm anno depend on he orreness of previous answers unknown hisory, and one in whih he algorihm, afer eah guess, learns he orre answer known hisory 1. We olleively refer o he wo minimizaion problems as minasg. Definiion 2. The minimum asymmeri sring guessing problem wih unknown hisory, minasgu, has inpu? 1,...,? n, x, where x {0, 1} n, for some n N. For 1 i n, round i proeeds as follows: 1. The algorihm reeives reques? i whih onains no informaion. 2. The algorihm answers y i, where y i {0, 1}. 1 The onep of known hisory for online problems also appears in [19, 20] where i is denoed ranspareny. 8

9 The oupu y = y 1... y n ompued by he algorihm is feasible, if x y. Oherwise, y is infeasible. The os of a feasible oupu is y 1, and he os of an infeasible oupu is. The goal is o minimize he os. Thus, eah reques arries no informaion. While his may seem arifiial, i does apure he hardness of some online problems see for example Lemma 7. Definiion 3. The minimum asymmeri sring guessing problem wih known hisory, minasgk, has inpu?, x 1,..., x n, where x = x 1... x n {0, 1} n, for some n N. For 1 i n, round i proeeds as follows: 1. If i > 1, he algorihm learns he orre answer, x i 1, o he reques in he previous round. 2. The algorihm answers y i = fx 1,..., x i 1 {0, 1}, where f is a funion defined by he algorihm. The oupu y = y 1... y n ompued by he algorihm is feasible, if x y. Oherwise, y is infeasible. The os of a feasible oupu is y 1, and he os of an infeasible oupu is. The goal is o minimize he os. The sring x in eiher version of minasg will be referred o as he inpu sring or he orre sring. Noe ha he number of requess in boh versions of minasg is n + 1, sine here is a final reques ha does no require any response from he algorihm. This final reques ensures ha he enire sring x is evenually known. For simpliiy, we will measure he advie omplexiy of minasg as a funion of n his hoie is no imporan as i hanges he advie omplexiy by a mos one bi. Clearly, for any deerminisi minasg algorihm whih someimes answers 0, here exiss an inpu sring on whih he algorihm ges a os of. However, if an algorihm always answers 1, he inpu sring ould onsis solely of 0s. Thus, no deerminisi algorihm an ahieve any ompeiive raio bounded by a funion of n. One an easily show ha he same holds for any randomized algorihm. We now give a simple algorihm for minasg whih reads On/ bis of advie and ahieves a sri ompeiive raio of. Theorem 1. For any 1, here is a srily -ompeiive algorihm for minasg whih reads n + Ologn/ bis of advie. Proof. We will prove he resul for minasgu. Clearly, i hen also holds for minasgk. Le x = x 1... x n be he inpu sring. The orale enodes p = n/ in a self-delimiing way, whih requires Ologn/ bis of advie. For 0 j < p, define C j = {x i : i j mod p}. These p ses pariion he inpu sring, and he size of eah C j is a mos n/p. The orale wries one bi, b j, for eah se C j. If C j onains only 0s, b j is se o 0. Oherwise, b j is se o 1. Thus, in oal, he orale wries n/ + Ologn/ bis of advie o he advie ape. The algorihm, Alg, learns p and he bis b 0,..., b p 1 from he advie ape. In round i, Alg answers wih he bi b i mod p. We laim ha his algorihm is srily -ompeiive. I is lear ha he algorihm produes a feasible oupu. Furhermore, if Alg answers 1 in round i, i mus be he ase ha a leas one inpu bi in C i mod p is 1. Sine he size of eah C j is a mos n/p, his implies ha Alg is srily -ompeiive. 9

10 2.2 The Maximizaion Version We also onsider ASG in a maximizaion version. One an view his as a dual version of minasg. Definiion 4. The maximum asymmeri sring guessing problem wih unknown hisory, max- ASGu, is idenial o minasgu, exep ha he sore funion is differen: The sore of a feasible oupu y is y 0, and he sore of an infeasible oupu is. The goal is o maximize he sore. The maximum asymmeri sring guessing problem wih known hisory is defined similarly: Definiion 5. The maximum asymmeri sring guessing problem wih known hisory, max- ASGk, is idenial o minasgk, exep ha he sore funion is differen: The sore of a feasible oupu y is y 0, and he sore of an infeasible oupu is. The goal is o maximize he sore. We olleively refer o he wo problems as maxasg. Similarly, minasgu and maxasgu are olleively alled ASGu, and minasgk and maxasgk are olleively alled ASGk. An algorihm for maxasg wihou advie anno aain any ompeiive raio bounded by a funion of n. If suh an algorihm would ever answer 0 in some round, an adversary would le he orre answer be 1 and he algorihm s oupu would be infeasible. On he oher hand, answering 1 in every round gives an oupu wih a profi of zero. Consider insanes of minasg and maxasg wih he same orre sring x. I is lear ha he opimal soluion is he same for boh insanes. However, as is usual wih dual versions of a problem, hey differ wih respe o approximaion. For example, if half of he bis in x are 1s, hen we ge a 2-ompeiive soluion y for he minasg insane by answering 1 in eah round. However, in maxasg, he profi of he same soluion y is zero. Despie his, here is a similar resul o Theorem 1 for maxasg. Theorem 2. For any 1, here is a srily -ompeiive algorihm for maxasg whih reads n/ + Olog n bis of advie. Proof. We will prove he resul for maxasgu. Clearly, i hen also holds for maxasgk. The orale pariions he inpu sring x = x 1... x n ino disjoin bloks, eah onaining a mos n onseuive bis. Noe ha here mus exis a blok where he number of 0s is a leas x 0 /. The orale uses Olog n bis o enode he index i in whih his blok sars and he index i in whih i ends. Furhermore, he orale wries he sring x i... x i ono he advie ape, whih requires a mos n bis, sine his is he larges possible size of a blok. The algorihm learns he sring x i... x i and answers aordingly in rounds i o i. In all oher rounds, he algorihm answers 1. Sine he profi of his oupu is a leas x 0 /, i follows ha Alg is srily -ompeiive. In he following seion, we deermine he amoun of advie an algorihm needs o ahieve some ompeiive raio > 1. I urns ou ha he algorihms from Theorems 1 and 2 use he asympoially smalles possible number of advie bis, bu he oeffiien in fron of he erm n/ an be improved. 3 Advie Complexiy of ASG In his seion we give upper and lower bounds on he number of advie bis neessary o obain -ompeiive ASG algorihms, for some > 1. The bounds are igh up o Olog n bis. For 10

11 ASGu, he gap beween he upper and lower bounds sems only from he fa ha he advie used for he upper bound inludes he number, n, of requess and he number,, of 1-bis in he inpu. Sine he lower bound is shown o hold even if he algorihm knows n and, his sligh gap is o be expeed. The following wo observaions will be used exensively in he analysis. Remark 1. Suppose ha a minasg algorihm, Alg, is -ompeiive. By definiion, here exiss a onsan, α, suh ha Algσ Opσ + α. Then, one an onsru a new algorihm, Alg, whih is srily -ompeiive and uses Olog n addiional advie bis as follows: Use Olog n bis of advie o enode he lengh n of he inpu and use α log n = Olog n bis of advie o enode he index of a mos α rounds in whih Alg guesses 1 bu where he orre answer is 0. Clearly, Alg an use his addiional advie o ahieve a sri ompeiive raio of. This also means ha a lower bound of b on he number of advie bis required o be srily -ompeiive implies a lower bound of b Olog n advie bis for being -ompeiive where he onsan hidden in Olog n depends on he addiive onsan α of he -ompeiive algorihm. The same ehnique an be used for maxasg. Remark 2. For a minimizaion problem, an algorihm, Alg, using b bis of advie an be onvered ino 2 b algorihms, Alg 1,..., Alg 2 b, wihou advie, one for eah possible advie sring, suh ha Algσ = min i Alg i σ for any inpu sequene σ. The same holds for maximizaion problems, exep ha in his ase, Algσ = max i Alg i σ. For ASG wih unknown hisory, he oupu of a deerminisi algorihm an depend only on he advie, sine no informaion is revealed o he algorihm hrough he inpu. Thus, for minasgu and maxasgu, a deerminisi algorihm using b advie bis an produe only 2 b differen oupus, one for eah possible advie sring. 3.1 Using Covering Designs In order o deermine he advie omplexiy of ASG, we will use some basi resuls from he heory of ombinaorial designs. We sar wih he definiion of a overing design. For any k N, a k-se is a se of ardinaliy k. Le v k be posiive inegers. A v,k,- overing design is a family of k-subses alled bloks of a v-se, S, suh ha any -subse of S is onained in a leas one blok. The size of a overing design, D, is he number of bloks in D. The overing number, Cv, k,, is he smalles possible size of a v, k, -overing design. Many papers have been devoed o he sudy of hese numbers. See [12] for a survey. The onneion o ASG is ha for inpus o minasg where he number of 1s is, an n,, -overing design an be used o obain a srily -ompeiive algorihm. I is lear ha a v, k, -overing design always exiss. Sine a single blok has exaly k -subses, and sine he oal number of -subses of a se of size v is v, i follows ha v / k Cv, k,. We will make use of he following upper bound on he size of a overing design: Lemma 1 Erdős, Spener [16]. For all naural numbers v k, v v k Cv, k, k k 1 + ln 11

12 We use Lemma 1 o express boh he upper and lower bound in erms of a quoien of binomial oeffiiens. This inrodues an addiional differene of log n beween he saed lower and upper bounds. Lemma 17 in Appendix A shows how he bounds we obain an be approximaed by a losed formula, avoiding binomial oeffiiens. This approximaion oss an addiional addiive differene of Olog n beween he lower and upper bounds. The approximaion is in erms of he following funion: Bn, = log n For > 1, we show ha Bn, ± Olog n bis of advie are neessary and suffiien o ahieve a sri ompeiive raio of, for any version of ASG. See Figure 1 for a graphial view. I an be shown Lemma 15 ha 1 n e ln2 Bn, n. In pariular, if = on/ log n, we see ha Olog n beomes a lower-order addiive erm. Thus, for his range of, we deermine exaly he higher-order erm in he advie omplexiy of ASG. Sine his is he main fous of our paper, we will ofen refer o Olog n as a lower-order addiive erm. The ase where = Ωn/ log n is reaed separaely in Seion Advie bis per reques Compeiive raio ASG SG 1 1 e ln2 Fig. 1. The upper solid line green shows he number of advie bis per reques whih are neessary and suffiien for obaining a sri ompeiive raio of for ASG ignoring lower-order erms. The lower solid line brown shows he same number for he original binary sring guessing problem SG [5]. The dashed lines are he funions 1/ and 1/e ln2. 12

13 3.2 Advie Complexiy of minasg We firs onsider minasg wih unknown hisory. Clearly, an upper bound for minasgu is also valid for minasgk. We will show ha he overing number Cv, k, is very losely relaed o he advie omplexiy of minasgu. Theorem 3. For any > 1, here exiss a srily -ompeiive algorihm for minasg reading b bis of advie, where b Bn, + Olog n. Proof. We will define an algorihm Alg and an orale O for minasgu suh ha Alg is srily -ompeiive and reads a mos b bis of advie. Clearly, he same algorihm an be used for minasgk. Le x = x 1... x n be an inpu sring o minasgu and se = x 1. The orale O wries he value of n o he advie ape using a self-delimiing enoding. Furhermore, he orale wries he value of o he advie ape using log n bis his is possible sine n. Thus, his par of he advie uses a mos 3 log n + 1 bis in oal. If n, hen Alg will answer 1 in eah round. If = 0, Alg will answer 0 in eah round. If 0 < < n, hen Alg ompues an opimal n,, -overing design as follows: Alg ries in lexiographi order, say all possible ses of -bloks, saring wih ses onsising of one blok, hen wo bloks, and so on. For eah suh se, Alg an hek if i is indeed an n,, - overing design. As soon as a valid overing design, D, is found, he algorihm an sop, sine D will be a smalles possible n,, -overing design. Now, O piks a -blok, S y, from D, suh ha he haraerisi veor y of S y saisfies ha x y. Noe ha, sine Alg is deerminisi, he orale knows whih overing design Alg ompues and he ordering of he bloks in ha design. The orale hen wries he index of S y on he advie ape. This requires a mos log Cn,, bis of advie. Alg reads he index of he -blok S y from he advie ape and answers 1 in round i if and only if he elemen i belongs o S y. Clearly, his will resul in Alg answering 1 exaly imes and produing a feasible oupu. I follows ha Alg is srily -ompeiive. Furhermore, he number of bis read by Alg is b log max Cn,, : <n + 3 log n + 1. The heorem now follows from Lemma 17, Inequaliy 8. We now give an almos mahing lower bound. Theorem 4. For any > 1, a -ompeiive algorihm Alg for minasgu mus read b bis of advie, where b Bn, Olog n. Proof. By Remark 1, i suffies o prove he lower bound for srily -ompeiive algorihms. Suppose ha Alg is srily -ompeiive. Le b be he number of advie bis read by Alg on 13

14 inpus of lengh n. For 0 n, le I n, be he se of inpu srings of lengh n wih Hamming weigh, and le Y n, be he orresponding se of oupu srings produed by Alg. We will argue ha, for eah, 0 n, Y n, an be onvered o an n,, -overing design of size a mos 2 b. By Remark 2, Alg an produe a mos 2 b differen oupu srings, one for eah possible advie sring. Now, for eah inpu sring, x I n,, here mus exis some advie whih makes Alg oupu a sring y, where y 1 and x y. If no, hen Alg is no srily -ompeiive. For eah possible oupu y {0, 1} n ompued by Alg, we onver i o he se S y [n] whih has y as is haraerisi veor. If y 1 <, we add some arbirary elemens o S y so ha S y onains exaly elemens. Sine Alg is srily -ompeiive, his onversion gives he bloks of an n,, -overing design. The size of his overing design is a mos 2 b, sine Alg an produe a mos 2 b differen oupus. I follows ha Cn,, 2 b, for all, 0 n. Thus, b log max Cn,,. : <n The heorem now follows from Lemma 17, Inequaliy 6. Noe ha he proof of Theorem 4 relies heavily on he unknown hisory in order o bound he oal number of possible oupus. However, Theorem 5 below saes ha he lower bound of Bn, Olog n also holds for minasgk. In order o prove his, we show how an adversary an ensure ha revealing he orre answers for previous requess does no give he algorihm oo muh exra informaion. The way o ensure his depends on he speifi sraegy used by he algorihm and orale a hand, and so he proof is more ompliaed han ha of Theorem 4. Theorem 5. For any > 1, a -ompeiive algorihm for minasgk mus read b bis of advie, where b Bn, Olog n. Proof. By Remark 1, i suffies o prove he lower bound for srily -ompeiive algorihms. Consider he se, I n,, of inpu srings of lengh n and Hamming weigh, for some suh ha n. Resriing he inpu se o srings wih one pariular Hamming weigh an only weaken he adversary. Le Alg be a srily -ompeiive algorihm for minasgk whih reads a mos b bis of advie for any inpu of lengh n. For an advie sring ϕ, denoe by I ϕ I n, he se of inpu srings for whih Alg reads he advie ϕ. Sine we are onsidering minasgk, in any round, Alg may use boh he advie sring and he informaion abou he orre answer for previous rounds when deiding on an answer for he urren round. We will prove he lower bound by onsidering he ompuaion of Alg, when reading he advie ϕ, as a game beween Alg and an adversary. This game proeeds aording o he rules speified in Definiion 3. In pariular, a he beginning of round i, he adversary reveals he orre answer x i 1 for round i 1 o Alg. Thus, a he beginning of round i, he algorihm knows he firs i 1 bis, x 1,..., x i 1, of he inpu sring. We say ha a sring s I ϕ is alive in round i if s j = x j for all j < i, and we denoe by I i ϕ I ϕ he se of srings whih are alive in round i. The adversary mus reveal he orre answers in a way ha is onsisen wih ϕ. Tha is, in eah round, here mus exis a leas one sring in I ϕ whih is alive. We firs make wo simple observaions: 14

15 Suppose ha, in some round i, here exiss a sring s I i ϕ suh ha s i = 1. Then, Alg mus answer 1, or else he adversary an hoose s as he inpu sring and hereby fore Alg o inur a os of. Thus, we will assume ha Alg always answers 1 in suh rounds. On he oher hand, if, in round i, all s I i ϕ have s i = 0, hen Alg is free o answer 0. We will assume ha Alg always answers 0 in suh rounds. Assume ha, a some poin during he ompuaion, I ϕ onains exaly m srings and exaly h 1s are sill o be revealed. We le L 1 m, h be he larges number suh ha for every se of m differen srings of equal lengh, eah wih Hamming weigh h, he adversary an fore Alg o inur a os of a leas L 1 m, h when saring for his siuaion. In oher words, L 1 m, h is he minimum number of rounds in whih he adversary an fore Alg o answer 1. Claim: For any m, h 1, { L 1 m, h min d: m } d. 2 h Before proving he laim, we will show how i implies he heorem. For any, 0 < n, here are n possible inpu srings of lengh n and Hamming weigh. By he pigeonhole priniple, here mus exis an advie sring ϕ suh ha I ϕ n /2 b. Now, if m = I ϕ >, hen L 1 m, + 1. This onradis he fa ha Alg is srily -ompeiive. Thus, i mus hold ha I ϕ. Combining he wo inequaliies involving I ϕ, we ge n n Iϕ 2 b 2 b Sine his holds for all values of, we obain he lower bound n b log max : <n The heorem hen follows from Lemma 17 and Inequaliies 7 and 6. Proof of laim: Fix 1 i n and assume ha, a he beginning of round i, here are m srings alive, all of whih sill have exaly h 1 s o be revealed. The res of he proof is by induion on m and h. For he base ase, suppose firs ha h = 1. Then, for eah of he m srings, s 1,..., s m I i ϕ, here is exaly one index, i 1,..., i m, suh ha s 1 i 1 = = s m i m = 1. Sine all srings in I i ϕ mus be differen, i follows ha i j i k for j k. Wihou loss of generaliy, assume ha i 1 < i 2 < < i m. In rounds i 1,..., i m 1, he adversary hooses he orre answer o be 0, while Alg is fored o answer 1 in eah of hese rounds. Finally, in round i m, he adversary reveals he orre answer o be 1 and hene he inpu sring mus be s m. In oal, Alg inurs a os of m, whih shows ha L 1 m, 1 = m for all m 1. Assume now ha m = 1. I is lear ha L 1 m, h h for all values of h. In pariular, L 1 1, h = h. This finishes he base ase. For he induive sep, fix inegers m, h 2. Assume ha he formula is rue for all i, j suh ha j h 1 or suh ha j = h and i m 1. We will show ha he formula is also rue for m, h.. 15

16 Consider he srings s 1,..., s m I i ϕ alive a he beginning of round i. We pariion I i ϕ ino wo ses, S 0 = {s j : s j i = 0} and S 1 = {s j : s j i = 1}, and le m 0 = S 0 and m 1 = S 1. Reall ha if all sequenes s Iϕ i have s i = 0, we assume ha Alg answers 0, leaving m and h unhanged. Thus, we may safely ignore suh rounds and assume ha m 0 < m. We le { } d d = min d : m, h { } d d 0 = min d : m 0, and h { } d d 1 = min d : m 1. h 1 If d d, hen he adversary hooses 1 as he orre answer in round i. By he induion hypohesis, L 1 m 1, h 1 d 1. Togeher wih he fa ha Alg is fored o answer 1 in round i, his shows ha he adversary an fore Alg o inur a os of a leas L 1 m 1, h 1+1 d 1 +1 d. On he oher hand, if d < d, he adversary hooses 0 as he orre answer in round i. Noe ha his implies ha eah sring alive in round i + 1 sill has exaly h 1 s o be revealed. We mus have d 1 d 2 sine d 1 and d are boh inegers. Moreover, by definiion of d, i holds ha m > d 1 h. Thus, we ge he following lower bound on m0 : m 0 = m m 1 d 1 > h d 1 h d 2 = h d1 h 1 d 2, sine h 1, by Pasal s Ideniy. a is inreasing in a b This lower bound on m 0 shows ha d 0 > d 2, and hene d 0 d 1. Combining his wih he induion hypohesis gives L 1 m 0, h d 0 d 1. Sine m 1 1, Alg is fored o answer 1 in round i, so he adversary an make Alg inur a os of a leas L 1 m 0, h + 1 d. 3.3 Advie Complexiy of maxasg In his seion, we will show ha he advie omplexiy of maxasg is he same as ha of minasg, up o a lower-order addiive erm of Olog n. We use he same ehniques as in Seion 3.2. As noed before, he diffiuly of ompuing a -ompeiive soluion for a speifi inpu sring is no he same for minasg and maxasg. The key poin is ha ompuing a -ompeiive soluion for maxasg, on inpu srings wih u 0 s, is roughly as diffiul as ompuing a -ompeiive soluion for minasg, on inpu srings wih u/ 1 s. We show ha he proofs of Theorems 3 5 an easily be modified o give upper and lower bounds on he advie omplexiy of maxasg. These bounds wihin he proofs look slighly differen from he ones obained for minasg, bu we show in Lemmas 19 and 20 ha hey differ from Bn, by a mos an addiive erm of Olog n. Theorem 6. For any > 1, here exiss a srily -ompeiive online algorihm for maxasg reading b bis of advie, where b Bn, + Olog n. 16

17 Proof. We will define an algorihm Alg and an orale O for maxasgu suh ha Alg is srily -ompeiive and reads a mos b bis of advie. Clearly, he same algorihm an be used for maxasgk. As in he proof of Theorem 3, we noe ha, for any inegers n, u where 0 < u < n, he algorihm Alg an ompue an opimal n, n u/, n u-overing design deerminisially. Le x = x 1... x n be an inpu sring o maxasgu and se u = x 0. The orale O wries he values of n and u o he advie ape using a mos 3 log n + 1 bis in oal. If 0 < u < n, hen O piks an n u/ -blok, S y, from he opimal n, n u/, n u-overing design, as ompued by Alg, suh ha he haraerisi veor y of S y saisfies ha x y. The orale wries he index of S y on he advie ape. This requires a mos log Cn, n u/, n u bis of advie. The algorihm, Alg, firs reads he values of n and u from he advie ape. If u = 0, hen Alg will answer 1 in eah round, and if u = n, hen Alg will answer 0 in eah round. If 0 < u < n, hen Alg will read he index of he n u/ -blok S y from he advie ape. Alg will answer 1 in round i if and only if he elemen i belongs o he given blok. Clearly, his will resul in Alg answering 0 exaly n n u/ = u/ imes and produing a feasible oupu. I follows ha Alg will be srily -ompeiive. Furhermore, he number of bis read by Alg is b log max C n, n u: 0<u<n u, n u + 3 log n + 1. The heorem now follows from Lemma 20. Theorem 7. For any > 1, a -ompeiive algorihm Alg for maxasgu mus read b bis of advie, where b Bn, Olog n. Proof. By Remark 1, i suffies o prove he lower bound for srily -ompeiive algorihms. Suppose ha Alg is srily -ompeiive. Le b be he number of advie bis read by Alg on inpus of lengh n. For 0 u n, le I n,u be he se of inpu srings x of lengh n wih x 0 = u, and le Y n,u be he orresponding se of oupu srings produed by Alg. We will argue ha, for eah u, Y n,u an be onvered o an n, n u/, n u-overing design of size a mos 2 b. By Remark 2, Alg an produe a mos 2 b differen oupu srings, one for eah possible advie sring. Now, for eah inpu sring, x = x 1... x n wih x 0 = u and, hene, x 1 = n u, here mus exis some advie whih makes Alg oupu a sring y = y 1... y n where y 0 u/ and, hene, y 1 n u/ and x y. If no, hen Alg is no srily -ompeiive. For eah possible oupu y {0, 1} n ompued by Alg, we onver i o he se S y [n] whih has y as is haraerisi veor. If y 1 < n u/, we add some arbirary elemens o S y so ha S y onains exaly n u/ elemens. Sine Alg is srily -ompeiive, his onversion gives he bloks of an n, n u/, n u-overing design. The size of his overing design is a mos 2 b, sine Alg an produe a mos 2 b differen oupus. I follows ha Cn, n u/, n u 2 b, for all u. Thus, b log max C n, n u: 0<u<n u, n u. The heorem now follows from Lemma 20. As was he ase for minasg, he lower bound for maxasgu also holds for maxasgk. 17

18 Theorem 8. For any > 1, a -ompeiive algorihm Alg for maxasgk mus read a leas b bis of advie, where b Bn, Olog n. Proof. By Remark 1, i suffies o prove he lower bound for srily -ompeiive algorihms. Consider inpu srings, x, of lengh n and suh ha x 0 = u. Le = x 1 = n u. We reuse he noaion from he proof of Theorem 5 and le I ϕ I n, denoe he se of srings for whih Alg reads he advie sring ϕ. Suppose here exiss some advie sring ϕ suh ha m = I ϕ > n u. Sine Inequaliy 2 from he proof of Theorem 5 holds for maxasg oo, we ge ha L 1 m, n u + 1. Bu his means ha here exiss an inpu x I ϕ, wih x 1 =, suh ha Alg mus answer 1 a leas n u + 1 imes. In oher words, for he oupu y, ompued by Alg on inpu x, i holds ha y 0 n n u + 1 u 1. Sine x 0 = u, his onradis he fa ha Alg is srily -ompeiive. Sine here are n u possible inpu srings x suh ha x 0 = u, and sine he above was shown o hold for all hoies of u, we ge he lower bound n u b log max u: 0<u<n n u n u. The heorem now follows from Lemma Advie Complexiy of ASG when = Ωn/ log n Throughou he paper, we mosly ignore addiive erms of Olog n in he advie omplexiy. However, in his seion, we will onsider he advie omplexiy of ASG when he number of advie bis read is a mos logarihmi. Surprisingly, i urns ou ha he advie omplexiy of minasg and maxasg is differen in his ase. Reall ha, by Theorem 6 or Theorem 2, using Olog n bis of advie, an algorihm for n log n maxasg an ahieve a ompeiive raio of. The following heorem shows ha here is a phase-ransiion in he advie omplexiy, in he sense ha using less han log n bis of advie is no beer han using no advie a all. We remark ha Theorem 9 and is proof are essenially equivalen o a previous resul of Halldórsson e al. [19] on Online Independen Se in he muli-soluion model. Theorem 9 f. [19]. Le Alg be an algorihm for maxasg reading b < log n bis of advie. Then, he ompeiive raio of Alg is no bounded by a funion of n. This is rue even if Alg knows n in advane. Proof. We will prove he resul for maxasgk. Clearly, i hen also holds for maxasgu. By Remark 2, we an onver Alg o m = 2 b online algorihms wihou advie. Denoe he algorihms by Alg 1,..., Alg m. Sine b < log n, i follows ha m n/2. We laim ha he adversary an onsru an inpu sring x = x 1... x n for maxasgk suh ha he following holds: For eah 1 j m, he oupu of Alg j is eiher infeasible or onains only 1s. Furhermore, x an be onsrued suh ha x 0 n 2. 18

19 We now show how he adversary may ahieve his. For 1 i n, he adversary deides he value of x i as follows: If here is some algorihm, Alg j, whih answers 0 in round i and Alg j answers 1 in all rounds before round i, he adversary les x i = 1. In all oher ases, he adversary les x i = 0. I follows ha if an algorihm Alg j ever answers 0, is oupu will be infeasible. Furhermore, he number of 1 s in he inpu sring onsrued by he adversary is a mos n/2, sine m n/2. Thus, he profi of Op on his inpu is a leas n/2, while he profi of Alg is a mos 0. For minasg, he algorihm from Theorem 1 ahieves a ompeiive raio of and uses On/ bis of advie, for any > 1. In pariular, i is possible o ahieve a ompeiive raio of e.g. On/log log n using Olog log n bis of advie, whih we have jus shown is no possible for maxasg. The following heorem shows ha no srily -ompeiive algorihm for minasg an use less han Ωn/ bis of advie, even if n/ = olog n. Theorem 10. For any > 1, on inpus of lengh n, a srily -ompeiive algorihm Alg for minasg mus read a leas b = Ωn/ bis of advie. Proof. We will prove he resul for minasgk. Clearly, i hen also holds for minasgu. Suppose ha Alg is srily -ompeiive. Sine =, i follows from he proof of Theorem 5 ha Alg mus read a leas b bis of advie, where n b log By Lemma 18, his implies ha b = Ωn/. 4 The Complexiy Class AOC max : <n In his seion, we define a lass, AOC, and show ha for eah problem, P, in AOC, he advie omplexiy of P is a mos ha of ASG. Definiion 6. A problem, P, is in AOC Asymmeri Online Covering if i an be defined as follows: The inpu o an insane of P onsiss of a sequene of n requess, σ = r 1,..., r n, and possibly one final dummy reques. An algorihm for P ompues a binary oupu sring, y = y 1... y n {0, 1} n, where y i = fr 1,..., r i for some funion f. For minimizaion maximizaion problems, he sore funion, s, maps a pair, σ, y, of inpu and oupu o a os profi in N { } N { }. For an inpu, σ, and an oupu, y, y is feasible if sσ, y N. Oherwise, y is infeasible. There mus exis a leas one feasible oupu. Le S min σ S max σ be he se of hose oupus ha minimize maximize s for a given inpu σ. If P is a minimizaion problem, hen for every inpu, σ, he following mus hold: 1. For a feasible oupu, y, sσ, y = y An oupu, y, is feasible if here exiss a y S min σ suh ha y y. If here is no suh y, he oupu may or may no be feasible. If P is a maximizaion problem, hen for every inpu, σ, he following mus hold: 1. For a feasible oupu, y, sσ, y = y