Approximating the Online Set Multicover Problems Via Randomized Winnowing

Appro ximating the Online Set Multico v er Problems Via Randomized Wi nno wing ∗ Piotr Berman † Departmen t of Computer Science and Engineering P ennsylv ania State Univ ersit y Univ ersit y Park, P A 16802 Email: berman@cs e.psu.edu Bhask ar DasGupta ‡ Departmen t of Computer Science Univ ersit y of Illinois at Chicago Chicago, IL 60607 Email: dasgupta@cs.uic. edu August 27, 20 18 Abstract In this pap er, we consider the w eighted online set k -m ulticov er problem. In this problem, we hav e a universe V of ele ments, a family S of subsets o f V with a p o s itive real cost for every S ∈ S , a nd a “cov er a ge factor” (pos itive integer) k . A subset { i 0 , i 1 , . . . } ⊆ V of elements are presented online in an arbitrar y or de r . When each elemen t i p is presented, w e are a lso told the collection of all (at least k ) sets S i p ⊆ S and their co s ts to which i p belo ngs and w e need to select additiona l sets from S i p if necessar y s uch that our c o llection of selected sets contains at le ast k sets that con tain the elemen t i p . The g oal is to minimize the total c ost o f the s e le cted sets 1 . In this pap er, w e describ e a new randomized a lg orithm for the o nline multico ver problem based on a randomized version of the winnowing approa ch of [15]. This a lgorithm g eneralizes and improv es some earlier results in [1, 2]. W e also discuss low er b ounds on comp etitive ra tios for deterministic algorithms for general k based on the a pproaches in [2]. 1 In tro d uction In this pap er, we consider the W eigh ted Online Set k -multic o v er problem (abbreviated as W OSC k ) defined as follo ws. W e hav e an universe V = { 1, 2, . . . , n } o f elements, a family S of subsets of U with a cost (p ositive real n umb er) c S for ev ery S ∈ S , and a “co v erage factor” (p ositiv e in teger) k . A su bset { i 0 , i 1 , . . . } ⊆ V of elements are presente d in an arbitrary order. When eac h element i p is present ed , w e are also told the collectio n of all (at least k ) sets S i p ⊆ S in wh ic h i p b elongs and w e need to select additional sets from S i p , if necessary , suc h that our collection of sets con tains at le ast k sets that con tain the elemen t i p . T he goal is to minimize the total cost of the selected sets. The sp ecia l c ase of k = 1 will b e sim p ly d enoted by WOSC (W eigh ted Online Set Co ver). The unw eigh ted v ersions of these problems, when the cost an y s et is one, will b e denoted by OSC k or OSC . ∗ A p reliminary v ersion of these results app eared in 9 th W orkshop on Algorithms and D ata S t ructures, F. Dehne, A. L´ opez- Ortiz and J. R. Sac k (editors), LNCS 3608, pp. 110-121, 2005. † Supp orted by NSF grant CCR-0208821. ‡ Supp orted in part by N SF grants DBI- 054336 5, I IS- 0612044 and I IS-0346973. 1 Our algorithm and competitive ratio bound s can be extended to the case when a set can b e selected at most a presp ecified num b er of times instead of just once; we do not rep ort th ese extensions for simplici ty and also b ecause they ha ve n o relev ance to the biological applications that motiv ated our work. 1 The p erformance of an y online algorithm can b e measur ed b y the c omp etitive r atio , i.e. , the ratio of the total cost of the online algorithm to that of an optimal offline algorithm that kno ws the en tire inp ut in adv ance; fo r r andomized algorithms, we measure the p erf orm ance by the exp e cte d comp etitiv e r atio, i.e. , the ratio of the exp ected cost of the solution found by our algorithm to the optim um cost computed b y an adversary that kno w s the entire input s equence and has no limits on computational p o wer, but w h o is not familiar with our random c h oices. The follo wing n otations w ill b e used uniformly throughout the rest of the paper un less otherwise stated explicitly: • V is the univ erse of elemen ts; • m = m ax i ∈ V |{ S ∈ S | i ∈ S }| is th e ma x imum fr e quency , i.e. , the ma x imum num b er of sets in whic h an y elemen t of V b elongs; • d = max S ∈S | S | is th e maxim um set size; • k is the co verag e factor; • e is the base of natural logarithm. None of m , d or | V | is known to the online algorithm i n advanc e. 1.1 Motiv a tions and ap plications One of our m ain motiv ation for in vestig ating these problems, esp ecia lly f or large v a lu es of the “co v erage factor”, is their applications to rev erse engineering p roblems in systems biology . Ho wev er, other applications ha v e also b een noted in previous literat ures and b elo w w e mention one s uc h application in addition to the biological motiv ations. 1.1.1 Clien t /serv er proto cols [2 ] Suc h a situation is mo deled by the problem W OSC in whic h there is a n et wo r k of serv ers , clien ts arriv e one-b y-one in arbitrary ord er, and eac h clien t can b e serv ed b y a subset of the servers based on their geographical distance from the clien t. The extension to W OSC k handles the scenario in whic h a clien t m ust b e attended to b y at least a m inim u m num b er of serv ers for, sa y , r eliabilit y , robustness and impro ved resp onse time. In add ition, in our motiv a tion, we wa nt a distributed algorithm for the v arious s erv ers, namely an algorithm in wh ic h eac h serv er lo cally decide ab out the requests without communicat ing with the other serv ers or kn o wing their act ions (and, thus for example, not allo wed to maint ain a p oten tial fu nction based on a su bset of the servers suc h as in [2]). 1.1.2 Rev erse engineering of gene/protein net w orks [4, 7 , 8, 10, 13, 14, 18, 20] W e briefly explain this motiv ation here du e to lac k of s p ace; the r eader m ay consu lt the references for more details. This motiv ation concerns unr a ve ling (or “rev erse engineering”) th e w eb of interact ions among the comp onen ts o f co mplex p r otein and genetic regulatory net works b y ob s erving global c hanges to deriv e in teractions b etw een individ u al no des. In this application our atten tion is fo cused solely on one su c h approac h, originally describ ed in [13, 14], fur ther elab orated u p on in [4, 18], 2 and reviewe d in [10, 20]. Here o n e assumes t h at th e time evolutio n of a v ector of state v ariables x ( t ) = ( x 1 ( t ) , . . . , x n ( t )) is d escrib ed by a system of differenti al equations: ∂ ~ x ∂t = f ( ~ x, ~ p ) ≡          ∂x 1 ∂t = f 1 ( x 1 , . . . , x n , p 1 , . . . , p m ) ∂x 2 ∂t = f 2 ( x 1 , . . . , x n , p 1 , . . . , p m ) . . . ∂x n ∂t = f n ( x 1 , . . . , x n , p 1 , . . . , p m ) where ~ p = ( p 1 , . . . , p m ) is a v ector of parameters, suc h as lev els of h ormones or of enzymes, whose half-liv es are long compared to the rate at which the v ariables evol ve and wh ic h can b e manipu- lated but remain constant dur in g an y giv en exp eriment. The comp onen ts x i ( t ) of the state vecto r represent quan tities that can b e in principle measured, such as lev els of activit y of selected proteins or transcrip tion rates of certain genes. T here is a reference v alue ¯ p of ~ p , which repr esen ts “wild t yp e” (that is, normal) conditions, and a corresp onding steady state ¯ x of ~ x , suc h that f ( ¯ x, ¯ p ) = 0 . W e are in terested in obtaining inf ormation ab out the Jacobian of the v ector field f ev aluated at ( ¯ x, ¯ p ) , or at le ast ab out the signs o f the d eriv ativ es ∂f i /∂x j ( ¯ x, ¯ p ) . F or example, if ∂f i /∂x j > 0 , this means that x j has a p ositiv e (catalytic ) effect up on the r ate of formation of x i . T o b e more precise, the goal is to find as m u c h information as p ossible ab ou t an u nkno wn matrix A ∈ R n × n whic h is the Jacobian matrix ∂f/∂x . Th e critical assumption is that, while w e m a y n ot kno w the form of f , w e often do kno w that c ertain p ar ameters p j do not dir e ctly affe ct c ertain v ariables x i . This amounts to a priori biological kno w ledge of sp ecificit y of enzymes and similar data. Suc h a kno wledge can b e summarized b y a bin ary matrix C = ( c ij ) ∈ { 0, 1 } n × m , wh er e “ c ij = 1 ” means that p j do es not ap p ear in the equation for ˙ x i , that is, ∂ f i /∂p j ≡ 0 . In ou r curr ent con text, eac h ro w of C corresp ond to an elemen t, eac h column of C corresp ond to a set, and 0 - 1 entries indicate the mem b ership s of elemen ts in s ets. A cru cial con trib ution of the ab ov e-men tioned references in this con text is as follo ws. Supp ose that we solv e this set-m ultico v er instance in whic h eac h elemen t is co v ered at least some β times. Then with β = n − 1 w e can reco ver the elemen ts of A uniquely up to a sc a lar multiple (and , thus can know the signs of the deriv ati ves ∂f i /∂x j ( ¯ x, ¯ p ) pr ecisely) and with β = n − k for s ome small k w e can reco ve r the elemen ts of A up to a mo dest am biguit y that can b e tolerat ed in pr actice. I f the corresp onding exp erimen tal pr otocols are carried out using measuremen ts via a suitable b iologic al r ep orting mechanisms su c h as fluorescent pr oteins in an online fash ion, one arrives at the online set multico v er prob lems d iscu ssed in this pap er. 1.2 Summary of prior w or k Offline versions WSC k and SC k of the problems W O SC k and O SC k , in wh ich all the | V | elemen ts are p resen ted at the same time, ha ve b een w ell stu died in the literature. F ollo wing a b rief su mmary of some of th e results only ab out these problems. Assuming NP 6⊆ DTIME ( n log log n ) , the SC 1 problem in general cann ot b e app ro ximated to within a factor of ( 1 − ε ) ln | V | for any constant 0 < ε < 1 in p olynomial time [11] and cann ot b e app ro ximated to within a factor of ln d − O ( ln ln d ) in p olynomial time when restricted to set-co v er instances with maxim um set size d for all sufficien tly large d unless P = NP . On the other hand , an instance of the SC k problem can b e ( 1 + ln d ) - appro ximated in O ( | V | · | S | · k ) time by a simple greedy heur istic that, at ev ery step, selects a new s et that co ve r s the maxim u m num b er of those elements that has not b een co vered at least k times y et [12, 22]; these results we r e recen tly imp r o ve d up on in [7] w ho p ro vided a randomized appro ximation algorithm with an exp ected p erformance ratio of ( 1 + o ( 1 )) ln  d k  when d/k is at least ab out e 2 ≈ 7.39 , and for s m aller v alues of d/k th e exp ected p erformance ratio w as 1 + 2 p d/k . Regarding previous r esults for th e online versions, the authors in [1, 2] considered the W O SC problem and pro vid ed b oth deterministic and simple randomized algorithms with a comp etitiv e 3 ratio or exp ected comp etit ive ratio of O ( lo g m log | V | ) a n d an almo s t matc hing lo wer b oun d of Ω  log | S | log | V | log log | S | + log log | V |  on the comp etit ive r atio for any deterministic algorithm for almost all v alues 2 of | V | and | S | . The authors of [5] pro vided an efficien t randomized on lin e appro ximation algorithm and a corresp ond ing matc hing lo we r b ound (for an y r andomized algorithm) for a d ifferen t v ersion of the online set-co v er p roblem in which one is allo w ed to pick at most k sets for a giv en k and the goal is to m aximize the num b er of presented elemen ts for wh ic h at least one s et contai n ing them w as selected on or b efore the element wa s presente d . Concurrent to our conference publication, Alon, Azar and Gu tner [3] considered the weig hted online set-co v er pr oblem with rep etitions which is studied in a bigger con text of admissions con trol problem in general netw orks. Here, an element can b e pr esen ted m ultiple times and, if the elemen t is p resen ted k times, our goal is to co v er it by at least k d ifferen t sets. F or this problem [3] con tains a randomized O ( log m log | V | ) -comp etitiv e algo- rithm as w ell as a determin istic bi-criteria app ro ximation algo rithm, i.e. , a deterministic algo r ithm that has a comp etit ive ratio of O ( log m log | V | ) and co v ers an elemen t by at least ( 1 − ε ) k differen t sets for an y fixed ε > 0 ; it is easy to see that these b ounds carry o v er to the problem WOSC k . Con- v ersely , it is not d ifficult to see th at our algorithm A- Universal and analysis can easily b e adap ted to this problem to ac hiev e an exp ecte d comp etitiv e ratio of log 2 m ln d + O ( log 2 m + ln d ) w ith ar- bitrary set w eights; one wo u ld need to mo dify app r opriate places of Section 3.4. F or unw eigh ted sets, via Corollary 2(b), Algorithm A-Univ ersal pro vid es an improv ed exp ecte d comp etitiv e r atio of “roughly” (neglecting small constan ts) max  5 log 2 m, log 2 m ln  d k log 2 m   and the co n stan ts in volv ed in this b ound are further impro ved in Theorem 10. 1.3 Summary of our results and techniqu es Let r ( m, d , k ) denote the comp etitiv e ratio of an y online algorithm for W O SC k as a fu n ction of m , d and k . In this pap er, w e describ e a new randomized algorithm for the online multic ov er p r oblem based on a randomized version of the winnowing approac h of [15]. Ou r main contributions are then as follo ws : • W e fi rst pro vid e a uniform analysis of our algorithm for all cases of th e online set multic ov er problems. As a corollary of our analysis, we observ e the follo wing. – F or OSC , WOSC and WOSC k our randomized algorithm has E [ r ( m, d, k )] equ al to log 2 m ln d plus small lo w er ord er term s . While the authors in [1, 2] did pro vide a deter- ministic algorithm and a simple randomized algorithm for W OSC with a comp etitiv e ratio and an exp ected comp etitiv e r atio of O ( log m log | V | ) , resp ectiv ely , the im p ro ve- men ts of our approac h and analysis are as follo ws: ∗ W e pro vid e b etter constan t factors and lo w er-order terms. Note that tight analysis of the appr o ximabilit y or inap p ro ximability b ounds for set-co v er t yp e problems in- v olving tigh t estimates of the constan ts and lo wer-order terms is not a new idea; for example, s ee [6, 7, 17, 19, 21]. ∗ W e use the maxim u m set size d rather th an the larger universe size | V | in the comp etitiv e ratio b ound. ∗ F or large co verage factor k (the case of utmost imp ortance in our app lications to s ys- tems b iology in Section 1.1.2), ou r uniform an alysis via the quantit y κ (see Section 3) 2 T o b e precise, when log 2 | V | ≤ | S | ≤ e | V | 1 2 − δ for any fixed δ > 0 ; we will refer to similar b ound s as “almost all v alues” of t hese parameters in the sequel. 4 pro vid es an exp ected comp etitiv e ratio of roughly max    5 log 2 m, log 2 m ln   d max  1,  k log 2 m c        where c ≥ 1 is the ratio of the largest to the smallest weigh t among the sets in an optimal solution. This p ro vides a sm o oth transition of the exp ected comp etitiv e ratio b et w een “roughly” log 2 m ln d plus smal l lower or der terms for W O SC k when the w eights are arbitrary p ositive num b ers to max  5 log 2 m, log 2 m ln  d k log 2 m   for OSC k when all the w eights are the same. ∗ As a corollary of the ab ov e, for (the unw eigh ted ve rsion) O SC k for general k the exp ected comp etit ive ratio E [ r ( m, d, k )] d ecreases logarithmically w ith decreasing d/k w ith a v alue of roughly 5 log 2 m in the limit 3 for all sufficien tly large k . • W e next provide an improv ed analysis of E [ r ( m, d, 1 )] for OSC with b ette r constant s . • W e next pro vide an improv ed analysis of E [ r ( m, d, k )] f or O SC k with b ette r constants and asymptotic limit for large k . The case of large k is imp ortan t for its application in r everse engineering of biological net works as outlined in S ection 1.1. More precisely , we s h o w th at E [ r ( m, d , k )] is at most  1 2 + log 2 m  ·  2 ln d k + 3.4  + 1 + 2 log 2 m if k ≤ ( 2 e ) · d and at most 1 + 2 log 2 m otherwise. • Finally , w e discuss lo wer b oun ds on comp etiti v e ratios for deterministic algorithms for OSC k and WOSC k for general k using the approac hes in [2]. The lo we r b ounds obtained are Ω  log | S | k log | V | k log log | S | k + log log | V | k  for OSC k and Ω  log | S | log | V | log log | S | + log log | V |  for WOSC k for man y v alues of the p arameters. 1.4 Comparison With Previous W ork The structure of our algorithm is similar to and the analysis metho d of our algorithm is motiv ated b y the imp licit randomized algorithm (wh ic h was subsequently derand omized) in the pap er The online set c over pr oblem by Alon et al. [2]. F or ev ery set we maint ain a n umb er that will guide the p ro cess of selection; w e us e αp [ S ] , Alon et al. use w S . When a new element is receiv ed, and it is n ot co v ered (or sufficien tly co v ered) y et, in b oth pap ers this n umb er is m ultiplied by a constan t — if th e new elemen t b elongs to S (in the w eight ed case, this num b er is incremented by a constant d ivided b y c S ). T he p ro cess of set selection is a b it different : w e simply select set S with probabilit y that equals the incremen t of αp [ S ] , while Alon et al. the pro cedure is ac hieving a similar effect rather indirectly — it v ery m uch looks lik e a de-randomization of our ap p roac h (we knew their approac h when w e wo r k ed on our s, so ours w as a de-de-rand omization). The analysis of Alon et al. uses an ingenious p ote ntial function, w hile we use three classes of accoun ts. In either case, this is a form of amortized analysis. The tw o approac hes offer distinct adv an tages. Alon et al. had a m u c h shorter pro of and could obtain a de-rand omized version. As our choi ces w ere more explicitly related to P oisson trial, w e applied our own v ersions of C hernoff b ound to tighten the analysis considerably . 3 Notice the similarit y of th is dep endence of the ex p ected comp etitive ratio on d/k to th at in our results in [7] for the offline version of the problem where w e pro vided an approximatio n algorithm with an exp ected p erformance ratio of ab out max { ( 1 + o ( 1 )) ln  d k  , 1 + 2 p d/k } . 5 A fractional solution to th e set co v er p roblem is implicit in these solutions, as the “guiding num- b ers” can b e in terpreted as fractional c hoices, and making th e “guided c h oices” can b e in terp r eted as roun ding. Ho wev er, neither our analysis, nor th at of Alon et al. u se that fact explicitly . 2 A G eneric Randomized Winno wing Algorithm W e first d escrib e a generic rand omized winnowing algorithm A-Univ ersal b elo w in Fig. 1. The winno w ing algorithm has tw o scaling factors: a multi plicativ e scaling factor µ c S that dep ends on the particular set S co ntaining i and another ad d itiv e scaling factor | S i | − 1 that d ep ends on the n u m b er of sets that con tain i . These scaling f actors quan tify the appropriate lev el of “promotion” in the winno w ing approac h. In the next few sections, we will analyze th e ab o ve algo r ithm for the v arious online set-m ultico ve r pr ob lems. T he f ollo w ing notations will b e used u niformly th r oughout the analysis: • J ⊆ V b e the set of elemen ts receiv ed in a run of the algorithm. • T ∗ b e an optim um s olution. 2.1 A Guided T our — Rough Sk et c h of the Analysis of A-Univ ersal for the Un we igh ted Case W e first sk etc h the o v er all analysis of A-Univ ersal for the case when ev ery set has co s t 1 to pro vid e the r eader an in tuition b ehind the o verall analysis of the algorithm. Be ar i n mind that this analysis is neither the most pr e cise nor the simplest, b u t it c an b e extende d to the gener al c ase . In particular we ma y ov erestimate or underestimate the constan ts slightly in the d escription to omit tedious d etails in fav or of p ro viding a b etter in tuition. Since the fun ction Stat alw ays returns 1 , w e can remov e line A4 an d simplify line A6 to p [ S ] ← min ( αp [ S ] + | S i | − 1 , 1 ) . The cost of h andling an elemen t i by A-Universal is the n u mb er of sets that are selected. Th e analysis is conditional on quantit y s = ξ ( i ) , where ξ ( i ) is the sum of αp [ S ] ’s o ve r S ∈ S i − T ∗ at the time wh en i is r eceiv ed , and we tak e the worst case o ver all p ossible v alues s . W e define even t E ( b ) that exactly b sets from S i − T ∗ w ere already s elected b efore elemen t i wa s receiv ed. Note that these selecti ons were su ccesses in P oisson trials that ha ve su m of probabilities s , so th e pr obabilit y of E ( b ) can b e expressed as some p ( s, b ) , e.g. usin g Lemma 13. The cost is s p lit in to three comp onents: (i) s electio n s of sets from T ∗ , (ii) selections fr om S i − T ∗ made in lines A8-9, and (iii) selection from S i − T ∗ made in lines A11-12. Selections of t yp e (i) are charged to ac c ount ( T ∗ ) , ob viously the fi n al v alue of this account con tributes at most 1 to the comp etitiv e ratio. Rather than payi ng for the actual cost of selections of typ e (ii) and (iii), w e pay the exp e cte d cost of these selectio ns, and on av erage w e w ill b e p a ying enough. W e estimate this cost as s + deficit , and we pa y it as f ollo w s: w e charge a fixed amount 1 + ψ to ev ery ac c ount ( S ) suc h that S ∈ S i ∩T ∗ − T , and the left-o ver cost is c harged to ac c ount ( i ) . The con trib ution of ac c ount ( S ) to the comp etitiv e ratio is the ratio of the exp ected fin al v alue of ac c ount ( S ) to the p ortion of c ( T ∗ ) attributed to S , and the latter h app ens to b e 1 (in the unw eight ed case!). Thus this contribution is ( 1 + ψ ) β w here β is th e exp ect ed n u m b er of times we can c h arge ac c ount ( S ) . W e in tro d uce fu n ction Λ ( S ) to estimate β . The initial v alue of Λ ( S ) = log 2 1 = 0 . When w e charge ac c ount ( S ) after receiving elemen t i , the v alue of ξ ( S ) increases fr om some x to at least x + x + m − 1 , so mx + 1 increases to at least 2mx + 2 , so Λ ( S ) increases by at least 1 — except 6 F1 function Stat ( B , j ) F2 A ← ∅ F3 while ( | A | < j ) do // select j least cost sets fr om B // F4 S ← least cost set from B − A ; insert S to A F5 return c S // r eturn the cost of the last selected set // // definition // D1 for ( i ∈ V ) D2 S i ← { s ∈ S : i ∈ S } // initialization // I1 T ← ∅ // T is our collection of selected sets // I2 for ( S ∈ S ) I3 αp [ S ] ← 0 // accum u lated p robabilit y of eac h set // // after r eceiving an elemen t i // A1 deficit ← k − | S i ∩ T | // k is the co verage factor // A2 if deficit ≤ 0 // we need deficit more sets for i // A3 fin ish the pro cessing of i A4 µ ← Stat ( S i − T , deficit ) A5 for ( S ∈ S i − T ) A6 p [ S ] ← µ c S  αp [ S ] + | S i | − 1  // probability for this step // A7 αp [ S ] ← α p [ S ] + p [ S ] // accumulate d pr obabilit y // A8 wit h p robabilit y min { p [ S ] , 1 } A9 insert S to T // r andomized selection // A10 deficit ← k − | S i ∩ T | A11 rep eat deficit t imes // greedy selecti on // A12 insert a least cost set from S i − T to T Figure 1: Algo rithm A-Univ ersal when ξ [ S ] increases to x + 1 and S is deterministically selected. s maller. Th e av erage fi nal v alue of Λ ( S ) is at most log 2 m (cf. Lemma 4). Th u s ac c ount ( S ) ’s cont r ibute r oughly ( 1 + ψ ) log 2 m to the comp etitiv e ratio. Note that th er e m u st b e at least deficit man y sets in S i ∩ T ∗ − T , so 1 term in 1 + ψ surely co v ers the cost of selections of type (iii). If there are b su c h sets and s > bψ , we charge s − bψ to ac c ount ( i ) . T o find the con trib ution of ac c ount ( i ) to the comp etitiv e ratio w e m ust ascrib e p art of c ( T ∗ ) to i and to estimate the final v alue of ac c ount ( i ) . If w e ha ve receiv ed b elemen ts s o far, c ( T ∗ ) ≥ kb/d , so w e can ascrib e k/d to i . Note th at w e m ak e only on e c h arge to ac c ount ( i ) . Ho w can we estimate this charge u n der condition E ( j − 1 ) ? First, b ecause j − 1 “incorrect” sets were already selected, deficit would b e 0 if only j − 1 “correct” sets r emained un selected, so the charges are 0 unless we hav e at least j unselected “correct” sets. T hus u nder condition E ( j − 1 ) , if we mak e an y charge s at all, at least jψ wa s c harged to a c c ount ( S ) ’s to co ver the a verag e cost of selections of type (ii). Th us un der condition E ( b ) we c harge at most s − ( b + 1 ) ψ to ac c ount ( i ) . As w e estimate the probabilit y of E ( b ) with p ( s, b ) , we can estimate the a v erage fin al v alue of ac c ount ( i ) as P ⌊ s/ψ ⌋ j = 1 p ( s, j − 1 )[ s − jψ ] . 7 Using Lemma 13, one can sho w that ψ = max { 2, ln ( k/ d ) } assur es that ac c ount ( i ) do n ot con- tribute more than a log 2 m factor to the comp etitiv e ratio. 3 An Uniform A nalysis of Algorithm A-Univ ersal In this section, we pr esen t a un iform analysis of Algorithm A-Univ ersal that applies to all v ers ions of the online s et multico ve r problems, i.e. , OSC , OSC k , W OSC and WOSC k . Abu s ing notations sligh tly , define c ( S ′ ) = P S ∈S ′ c S for any sub col lection of sets S ′ ⊆ S . Our b ound on the comp etiti v e ratio w ill b e in fl uenced by the parameter κ defined as: κ = min i ∈J & S ∈S i ∩T ∗  c ( S i ∩ T ∗ ) c S  . It is easy to c h ec k that κ =    1 for OSC k for OSC k ≥ 1 f or W O SC and WOSC k . The main r esult pr o v ed in this section is the follo w ing theorem. Theorem 1 The exp e cte d c omp etitive r atio E [ r ( m, d, k )] of Algorith m A-Univ ersal is at most 1 + log 2 m × max  5, 2 + ln d κ log 2 m  Corollary 2 (a) F or OSC , W OSC and WOSC k , setting κ = 1 we obtain E [ r ( m, d, k )] to b e at most log 2 m ln d plus lower or der terms. (b) F or OSC k , setting κ = k , we obtain E [ r ( m, d, k )] to b e at most 1 + log 2 m × max  5,  2 + ln d k log 2 m   ≈ log 2 m × max  5, ln d k log 2 m  (c) L et c ≥ 1 is the r atio of the lar ge st to the smal lest weight among the sets in an optimal solution. Then, setting κ = max  1, k c  , we obtain E [ r ( m, d, k )] to b e at most 1 + log 2 m × max  5,  2 + ln d max { 1, k c } log 2 m   ≈ log 2 m × max  5, ln d max  1,  k log 2 m c   !  In the n ext few subsections we pro ve the ab ov e theorem. 3.1 The ov erall sc heme W e first roughly describ e the o v erall sc heme of our analysis. The a verage cost of a run of A- Univ ersal is the sum of a verag e costs that are in curred w h en elemen ts i ∈ J are receiv ed. W e will account for these costs b y dividing these costs in to three p arts cost 1 + P i ∈J cost i 2 + P i ∈J cost i 3 where: cost 1 ≤ c ( T ∗ ) upp er b ounds the total cost incurred by the algorithm f or selecting s ets in T ∩ T ∗ . cost i 2 is the cost of selecting sets from S i − T ∗ in line A9 for eac h i ∈ J . cost i 3 is the cost of selecting sets from S i − T ∗ in line A12 for eac h i ∈ J . 8 W e will use the accoun ting sc h eme to count these costs by cr eating the follo win g th ree t yp es of accoun ts: ac c ount ( T ∗ ) ; ac c ount ( S ) for eac h set S ∈ T ∗ − T ; ac c ount ( i ) for eac h receiv ed element i ∈ J . cost 1 ob viously add s at most 1 to the a v er age competitiv e ratio; we will charge this cost to ac c ount ( T ∗ ) . T h e other t w o kinds of costs, namely cost i 2 + cost i 3 for eac h i , will b e distributed to th e remaining tw o accounts. Let D = d κ log 2 m . The distribution of charges to these t w o accoun ts will satisfy the follo wing: • P i ∈J ac c ount ( i ) ≤ log 2 m · c ( T ∗ ) . T h is claim in turn will b e satisfied by: – dividing the optimal cost c ( T ∗ ) int o pieces c i ( T ∗ ) for eac h i ∈ J such that P i ∈J c i ( T ∗ ) ≤ c ( T ∗ ) ; and – sho wing that, for eac h i ∈ J , ac c ount ( i ) ≤ log 2 m · c i ( T ∗ ) . • P S ∈T ∗ ac c ount ( S ) ≤ log 2 m · max { 4, ln D + 1 } · c ( T ∗ ) . This will ob viously prov e an exp ected comp etitiv e ratio of at most the m axim um of 1 + 5 ( 1 + log 2 m ) and 1 + ( 1 + log 2 m )( 2 + ln D ) , as p romised. W e will p erform our analysis from the p oin t of view of eac h receiv ed element i ∈ J . T o defi ne and analyze the c harges we w ill defin e sev eral quan tities: µ ( i ) the v alue of µ calculat ed in line A4 after receiving i ξ ( i ) the sum of αp [ S ] ’s o ver S ∈ S i − T ∗ at the time when i is receiv ed a ( i ) | T ∩ S i − T ∗ | at the time when i is receiv ed Λ ( S ) log 2 ( m · αp [ S ] + 1 ) f or eac h S ∈ S ; it c hanges d uring the execution of A-Universal Finally , let ∆ ( X ) denote the amoun t of c hange (increase or decrease) of a quantit y X when an elemen t i is p ro cessed. 3.2 The role of Λ ( S ) W e will ensure the invariant ac c ount ( S ) ≤ max { 4, ln D + 1 } · Λ ( S ) · c S for ev ery S ∈ T ∗ . W e will simp ly not accept larger c harges to the acco u n ts of sets than this inv arian t allo ws. This in v arian t is useful b ecause w e will p r o ve a u niv ersal upp er b ound U on the exp ected final v alue of Λ ( S ) , and th us the con tribution of the accoun ts of sets to the exp ecte d comp etitiv e ratio will b e max { 4, ln D + 1 } · U . Definition 3 When we determine th e char ge s to ac c ounts made when element i i s r e c eive d, we classify sets fr om S i ∩ T ∗ − T as he avy if c S ≥ µ ( i ) and light otherwise. When i is receiv ed w e c harge accoun ts of S ∈ T ∗ ∩ S i − T in the follo w ing manner: • for a ligh t set, ∆ ( ac c ount ( S )) = c S while we can show that ∆ ( Λ ( S )) > 1 and • for a hea vy set ∆ ( ac c ount ( S )) = max { 4, ln D + 1 } µ ( i ) w hile ∆ ( Λ ( S )) ≥ µ ( i ) /c S . 9 The ab ov e estimates of ∆ ( Λ ( S )) are easy to s h o w: in lines A6-7 w e incremen t αp [ S ] + m − 1 with µ ( i ) c S ( αp [ S ] + | S i | − 1 ) ≥ µ ( i ) c S ( αp [ S ] + m − 1 ) , whic h increments Λ ( S ) = log 2 ( αp [ S ] + | S i | − 1 ) − log 2 m by at least log 2 ( 1 + µ ( i ) /c S ) ; for a light set this increment is at least log 2 2 = 1 , and for a h eavy set we hav e µ ( i ) /c S ≤ 1 , and w e u se the follo wing fact: log 2 ( 1 + x ) ≥ x for x ≤ 1. Of cour se, suc h an approac h mak es sense only if w e can p ro ve an upp er b ound on E [ Λ ( S )] . Note that in step A6 w e ma y calculat e a v alue of p [ S ] that is larger than 1 . W e analyze E [ Λ ( S )] from the follo win g p oin t of view: consider a fixed sequence of p [ S ] o ver the execution of the algorithm; eac h time p [ S ] > 0 there is a c h an ce that S gets selected and th is is the last step when Λ ( S ) in creases. Ou r b ound will hold true f or ev ery p ossib le sequence. Lemma 4 E [ Λ ( S )] ≤ log 2 m for m ≥ 7 . Pro of. W e w ant to find the exp ected fi nal v alue of Λ ( S ) = log 2 ( m · αp [ S ]+ 1 ) = log 2 m + log 2 ( αp [ S ]+ m − 1 ) . It is a function of the sequence of probabilities, say p 1 , p 2 , . . . , th at p [ S ] computed when elemen ts of S were receiv ed. W e will b e w orking with sequ en ces formed from p ossib le sequences of probabilities by deleting an in itial part; let the s u m of this initial p art and m − 1 is z . W e d efine βp i = z + P i − 1 j = 1 p j whic h stands for the v alue of αp [ S ] + m − 1 in lin e A6 wh en we compute p i . W e say that ~ p = ( p 1 , p 2 , . . . ) is z -legal if for i ≥ 1 we ha ve 0 ≤ p i ≤ βp i , and if p i ≥ 1 then p i is the last term of ~ p . Let tail ( ~ p ) = ( p 2 , . . . ) . W e define F ( z, ~ p ) as follo ws. If ~ p is an empt y sequence then F ( z, ~ p ) = 0 , otherwise F ( z, ~ p ) = p 1 log 2 ( p 1 + z ) + ( 1 − p 1 ) F ( z + p 1 , tail ( ~ p )) ( ∗ ) In turn, F ( z ) is the s u premum v alue of F ( z, ~ p ) o v er all z -legal sequences. Our goal is to sho w that F ( 1/m ) < 0 for m ≥ 7 . W e fi rst show that if the supremum defin ing F ( z ) is limited to in finite sequences, then it is fin ite. By rep etitiv ely applying f orm ula (*) we get F ( z, p ) = ∞ X i = 1 i − 1 Y j = 1 ( 1 − p j ) p i log 2 ( βp i + p i ) < e z Z ∞ z e − x log 2 ( x + 1 ) dx where the summation can b e con verted to an in tegral as f ollo w s: p i can b e a sum of dx ’s o v er an in terv al of length p i , say from βp i to βp i + 1 , the pro du ct can b e the probabilistic d ensit y function that can b e b oun ded from ab ov e with e z − x and log 2 can b e the fun ction that we compute exp ectation of, and it can b e estimated f rom ab o ve with log 2 ( x + 1 ) ; this ju stifies the estimate w ith of F ( z, ~ p ) with a con v ergent integral. Next we sho w that for z ≥ log 2 e we ha ve F ( z ) = F ( z, ( z )) = 1 + log 2 z . Su pp ose that F ( z ) > 1 + log 2 z . Then for some fin ite ~ p and for some z ≥ log 2 e w e h a ve F ( z, ~ p ) > F ( z, ( z )) = 1 + log 2 z . Consider a shortest suc h sequence. Beca u se of ( ∗ ) we can conclude that ~ p has length 2 , since otherwise F ( z + p 1 , tail ( ~ p )) ≤ F ( z + p 1 , ( z + p 1 )) , b ut in that case w e can replace tail ( ~ p ) w ith the 10 single term z + p 1 . S o w e can assu me that ~ p = ( x, z + x ) for some x > 0 . Then w e ha ve F ( z, ~ p ) = x log 2 ( z + x ) + ( 1 − x ) log 2 ( z + x + z + x ) > 1 + log 2 z whic h implies x log 2 ( z + x ) + ( 1 − x )( 1 + log 2 ( z + x )) > 1 + log 2 z whic h implies x  log 2 z + log 2 z + x z  + ( 1 − x )  1 + log 2 z + log 2 z + x z  > 1 + log 2 z whic h implies log 2 z + x z > x The latter is not p ossible, b ecause f or x ≥ z ≥ log 2 e th e deriv ativ e of the left-hand-side is log 2 e z + x ≤ 1 , while the deriv ativ e of the righ t-han d -side is 1 . In a z -legal sequence ~ p we hav e p 1 ≤ min { 1, z } . As the third observ ation we can sho w th at if βp has more than one term, th en p 1 + p 2 > m in { 1, z } , otherwise we increase F ( z, ~ p ) when w e coalesce the fi r st t wo terms of ~ p into one. Let p 1 = x, p 2 = y, p 1 + p 2 = p , w e ha ve x log 2 ( z + x ) + ( 1 − x ) y log 2 ( z + p ) + ( 1 − x )( 1 − y ) F ( z + p ) < p log 2 ( z + p ) + ( 1 − p ) F ( z + p ) whic h implies x  log 2 z + x z + p + log 2 ( z + p )  + ( 1 − x ) y log 2 ( z + p ) + xyF ( z + p ) < p log 2 ( z + p ) whic h implies x log 2 z + x z + p − xy log 2 ( z + p ) + xyF ( z + p ) < 0 whic h implies F ( z + p ) < log 2 ( z + p ) + 1 y log 2  1 + y z + p − y  Because w e alw ays ha ve F ( z ) ≤ log 2 ( z ) + 1 , it suffices to s h o w that 1 y log 2 ( 1 + y z + p − y ) > 1 . This follo ws from the fact that for x < log 2 e the deriv ativ e of log 2 x is larger than 1 . The metho d s used to sho w the last tw o fact allo w to c haracterize the optimal (or worst case) sequences: if z ≥ log 2 e , use 1 -term sequence consisting of z , otherwise start from m in { z, 1, log e − z } . As a consequence, if 1 2 log 2 e ≤ z ≤ log 2 e then F ( z ) = F ( z, ( log 2 e − z, log 2 e ) = log 2 log 2 e + 1 − log 2 e + z , and for z ≤ 1 2 log 2 e we kno w th at F ( z ) = z log 2 ( 2z ) + ( 1 − z ) F ( 2z ) . It is easy to see that for F ( z/2 ) < F ( z ) , and we can compute the v alues of F ( 1/m ) for m = 2, 3, . . . , 7 : m 1 2 3 4 5 6 7 8 F ( 1/m ) 1.086 0.543 0.397 0.157 0.120 0.067 − 0.016 − 0.112 ❑ Observe that it is ve r y easy to show the comp etitiv e ratio of m , so for m = 1 it m akes no sense to d iscuss the comp etitiv e ratio, while for 1 < m ≤ 16 , s ince 4 log 2 m ≥ m , th e upp er b ound we are pr o ving is trivial. 3.3 Charges due to the c osts of line A12 When we make greedy selections in line A12, there are at least d eficit man y sets in S i ∩ T ∗ − T ; w e can order them according to their costs, say S 1 , S 2 , . . . ; and let c S i = a i . Because we could mak e greedy selectio n s of these sets, the costs of actual selections cannot b e larger, so if th ese costs are ordered b 1 ≤ . . . ≤ b deficit , w e hav e b i ≤ a i for i = 1, . . . , deficit . Therefore we can charge b i to ac c ount ( S i ) a n d the exp ected sum of suc h c harges made to eac h ac c ount ( S ) is at most c S · log 2 m . T herefore these c harges con tribu te log 2 m to the exp ected comp etitiv e ratio. 11 3.4 Charges due to the c osts of line A9 The exp ected sum of charges d u e to th e costs of line A9 equals µ ( i ) ξ ( i ) + µ ( i ) : ev ery set fr om S i − T − T ∗ con tributes, regardless of its weig h t, µ ( i )( αp [ S ] + | S i | − 1 ) , αp [ S ] terms add to ξ ( i ) , wh ile | S i | − 1 terms add to 1 . W e will r efer to these t wo terms as A9a c h arges and A9b c h arges. A9b c harges will b e giv en to an arbitrary accoun t of a hea v y set (in the worst case, there is only one). A9a c harges are d istributed among the accoun ts of heavy sets and ac c ount ( i ) . The idea is the follo wing: we will fi x the A9a charge to eac h h ea vy set accoun t to some ψ su c h that the con tribution of these charge s to the comp etiti ve ratio will b e exactly µ ( i ) ψ . W e estimate the num b er of the hea vy sets as follo w s. Lemma 5 Ther e ar e at le ast a ( i ) + 1 he avy se ts. Pro of. Our assumption is that at the time i is receiv ed, a ( i ) sets from S i − T ∗ are already selected to T . Thus w hen w e compute µ ( i ) in a call to Stat ( S i − T ) in line A4 we can form set A from S i ∩ T ∗ after excluding a ( i ) sets w ith the largest cost. W ould w e do that, µ ( i ) w ould b ecome the largest cost in S i ∩ T ∗ − T , after excluding a ( i ) costs that are y et larger, so we indeed ha v e at least a ( i ) sets of cost µ ( i ) or more — hence hea vy . When we include other sets in A as well, the v alue of µ ( i ) can only d ecrease, and then the num b er of hea vy sets can only increase. ❑ Therefore at most µ ( i )( ξ ( i ) − ( a ( i ) + 1 ) ψ ) will b e c harged to ac c ount ( i ) . Th us w e need to sho w that E [ ξ ( i ) − ( a ( i ) + 1 ) ψ ] is sufficiently small. The in tuition is that when ξ ( i ) is small, th e charge s cannot b e made, and when ξ ( i ) is large, the av erage v alue of a ( i ) is equally large an d th u s th e probabilit y of making c h arges is su fficien tly small to assure a very small a verage v alue. In the next sub section we analyze these pr obabilities, bu t it is easy to see that the higher ψ , the smaller E [ ξ ( i ) − ( a ( i ) + 1 ) ψ ] . W e w ant to set th e a ve r age c harge to ac c ount ( i ) in s uc h a wa y that the exp ected con trib ution of these accoun ts to the comp etitiv e ratio is at most log 2 m . S o the question is: ho w large p ortion of c ( T ∗ ) can w e attribute to elemen t i ? T o simplify our calculations, we rescale the costs of sets so µ ( i ) = 1 and thus c S ≥ 1 for eac h hea vy set S and the sum of c harges du e to line A9 is simply ξ ( i ) . W e associate with i a piece c i ( T ∗ ) of th e optim u m cost c ( T ∗ ) : c i ( T ∗ ) = X S ∈S i ∩T ∗ c S / | S | ≥ 1 d c ( S i ∩ T ∗ ) ≥ κ d µ ( i ) = κ/d. It is then easy to v erify that X i ∈J c i ( T ∗ ) ≤ X i ∈J 1 d c ( S i ∩ T ∗ ) ≤ c ( T ∩ T ∗ ) ≤ c ( T ∗ ) Th us w e can charge ac c ount ( i ) in suc h a wa y that on av erage it receiv es ( κ/d ) log 2 m , and let D − 1 = ( κ/d ) log 2 m . I n the n ext su bsection, we find a sufficient ly h igh v alue of ψ to mak e it s o. F or no w observ e that the comp etitiv e ratio will b e 1 + ( 3 + ψ ) log 2 m : 1 for the c h arges to ac c ount ( T ∗ ) , log 2 m for the c h arges due to line A12, log 2 m for the c h arges to ac c ount ( i ) ’s, log 2 m for A9b c h arges and ψ log 2 m for A9a c h arges. 12 3.5 Split of A9a c harges b etw een i and the hea vy sets In this section w e pro ve that for ψ = max { 2, ln D − 1 } w e ha ve E [ ξ ( i ) − ( a ( i ) + 1 ) ψ ] ≤ D − 1 . Define E ( i, b ) =  1 if a ( i ) ≤ b 0 otherwise Let charge ( i, ψ, ℓ, x ) b e the formula for the c h arge to ac c ount ( i ) assuming we use ψ w ith ℓψ ≤ x = ξ ( i ) ≤ ( ℓ + 1 ) ψ . W e can estimate charge ( i, ψ, ℓ, x ) in the follo wing manner: • If E ( i, ℓ − 1 ) = 1 , then a ( i ) + 1 = ℓ , the total c harge to all the heavy sets is ℓψ and thus we ha ve to charge ac c ount ( i ) with x − ℓψ . • if E ( i, ℓ − 2 ) = 1 then we also ha ve E ( i, ℓ − 1 ) = 1 , so we c harged ac c ount ( i ) with x − ℓψ already , but we n eed to c harge ac c ount ( i ) with an additional amount of ψ . • Cont inuing in a similar man n er, it follo ws that for eac h b ≤ ℓ − 2 , if E ( i, b ) = 1 we charge ac c ount ( i ) with an additional amount of ψ . Th us w e get th e follo wing estimate: E [ ch arge ( i, ψ, ℓ, x )] = Pr [ E ( i, ℓ − 1 ) = 1 ] · ( x − ℓψ ) + ψ ℓ − 2 X j = 0 Pr [ E ( i, j ) = 1 ] . Since ψ ( a ( i ) + 1 ) < ξ ( i ) and ψ ≥ 2 , a ( i ) + 1 is less than 1 2 ξ ( i ) . Thus, w e can use Lemma 13 with X = x = ξ ( i ) and a = j to obtain Pr [ E ( i, j ) = 1 ] < e − x x j j ! for j = ℓ − 1, ℓ − 2, . . . , 0 . Let C ( ψ, ℓ, x ) b e the estimate of of E [ charge ( i, ψ, ℓ, x )] thus obtained: C ( ψ, ℓ, x ) = e − x   x ℓ − 1 ( ℓ − 1 )! ( x − ℓψ ) + ψ ℓ − 2 X j = 0 x j j !   . Lemma 6 If ψ ≥ 2 , x ≥ 1 and ℓ = ⌊ x/ψ ⌋ ≥ 1 then C ( ψ, ℓ, x ) ≤ e −( ψ + 1 ) . Pro of. W e firs t consider the case of ℓ = 1 . Because E ( i, − 1 ) is not p ossible, charge ( i, ψ , 1, x ) = E ( i, 0 )( x − ψ ) and C ( ψ, 1, x ) = e − x ( x − ψ ) . No w since ∂ ∂x C ( ψ, 1, x ) = e − x (− x + ψ + 1 ) , C ( ψ, 1, x ) is maximized f or x = ψ + 1 with a maxim um v alue of e −( ψ + 1 ) . F or ℓ ≥ 2 the summation part of the formula for C ( ψ, ℓ, x ) is non-trivial; in that case one can calculate that ∂ ∂x C ( ψ, ℓ, x ) = e − x x ℓ − 2 ( ℓ − 1 )! (− x 2 + ℓ ( ψ + 1 ) x − ( ℓ 2 − 1 ) ψ ) . As w e see, this deriv ativ e is a pr o duct of a p ositiv e fu nction with a trinomial. T h is trinomial has the maxim um for x = ℓ ( ψ + 1 ) /2 , so in our range, ℓψ ≤ x ≤ ( ℓ + 1 ) ψ , it is decreasing. F or x = ℓψ the v alue of the trin omial is ψ > 0 , and for x = ℓψ + 2/ℓ th e v alue of the trinomial is 2 − ψ − 4ℓ − 2 < 0 . Therefore the m axim um m u st o ccur in the int erv al b et w een ℓψ and ℓψ + 2/ ℓ and it will suffice to pro ve our claim in this range. F or x = ℓψ + z with 0 < z < 2/ℓ the inequalit y we w ant to prov e is equiv alen t to LHS = ( ℓψ + z ) ℓ − 1 ( ℓ − 1 )! z + ψ ℓ − 2 X j = 0 ( ℓψ + z ) j j ! ≤ e ( ℓ − 1 ) ψ − 1 + z = RHS (1) 13 Supp ose that (1) is tr u e for some ψ ; then for ψ ′ = ψ + ε RHS increases by a factor of e ( ℓ − 1 ) ε , wh ile eac h monomial ( ℓψ + z ) j j ! , for j = 0, 1, . . . , ℓ − 1 , increases by a factor of  1 + ε ψ + z ℓ  j ≤  1 + ε ψ  ℓ − 1 < e ( ℓ − 1 ) ε ψ and th us the entire LHS increases b y a factor of at most ψ e ( ℓ − 1 ) ε ψ < e ( ℓ − 1 ) ε . Because LHS increases less that RHS, the inequalit y for ψ implies that for ψ + ε and th us for ev ery higher v alue. F or this r eason it suffi ces to pro ve the inequalit y f or ψ = 2 and for ℓψ < x < ℓψ + 2/ℓ (thus, f or 0 < z < 2/ℓ ). F or ψ = 2 , our claim is redu ces to LHS = ( 2ℓ + z ) ℓ − 1 ( ℓ − 1 )! z + 2 ℓ − 2 X j = 0 ( 2ℓ + z ) j j ! ≤ e 2ℓ − 3 + z = RHS F or conv en ience, let y = 2ℓ + z . Th us , w e need to pr o v e LHS = y ℓ − 1 ( ℓ − 1 )! ( y − 2ℓ ) + 2 ℓ − 2 X j = 0 y j j ! ≤ e y − 3 = RHS sub ject to 2ℓ < y < 2ℓ + 2 ℓ . Since ℓ ≥ 2 , y < 2ℓ + 2 ℓ < 2 ( ℓ + 1 ) and thus y − 2ℓ < 2 . Thus LHS < 2 P ℓ − 1 j = 0 y j j ! , and since, by the wel l-kno wn series expansion, e y = P ∞ j = 0 y j j ! it suffi ces to show that 2 e 3 ℓ − 1 X j = 0 T j ≤ ∞ X j = 0 T j for ℓ ≥ 2 , 2ℓ < y < 2ℓ + 2 ℓ and T j = y j j ! . First, w e verify b y induction that T j ≥ P j − 1 i = 0 T i for 1 ≤ j ≤ ℓ . Note that for 1 ≤ j ≤ ℓ , T j /T j − 1 = y/j > 2 . F or the basis case of j = 1 , it is therefore obvio u s. Otherwise, T j > 2T j − 1 > T j − 1 + P j − 2 i = 0 T i = P j − 1 i = 0 T i b y inductive h yp othesis. Thus, it suffices to sho w that 2 e 3 T ℓ ≤ ∞ X j = 0 T j F or ℓ + 1 ≤ j ≤ 2ℓ , T j /T j − 1 = y/j > 1 . Thus, P ∞ j = 0 T j ≥ ℓT ℓ , an d th us it suffi ces to show that 2 e 3 T ℓ ≤ ℓ · T ℓ whic h holds pr o vided ℓ ≥ 2 e 3 ≈ 40.17 . Thus, the claim holds for ℓ > 40 . F or 2 ≤ ℓ ≤ 40 and ψ = 2 , w e can v erify our claim by easy numerical calculation. Notice that we ju st need to v er if y C ( 2, ℓ, x 0 ) ≤ e − 3 where x 0 is the real ro ot of the qu adratic fu nction f ( x ) = − x 2 + 3ℓx − 2 ( ℓ 2 − 1 ) that lies in the range 2ℓ < x < 2ℓ + 2/ℓ . By n u m erical calculation, one can tabulate the results as sho wn in T able 1 and ve rify that C ( 2, ℓ, x 0 ) < 0.049 < e − 3 . ❑ No w, since ψ = max { 2, ln D − 1 } ≥ 2 we conclude using Lemma 6 th at the a ve r age c harge to ac c ount ( i ) is at most e − ln D = D − 1 . 4 Impro v ed A nalysis of Algorithm A-Univ ersal for Un we igh ted Cases In this section, we pro vide improv ed analysis of the exp ected comp etitiv e ratios of Algorithm A- Univ ersal or its minor v ariation for the unw eigh ted cases of the online set multico ver problems. These impro vemen ts p ertain to provi ding impro ve d constan ts in the b ound for E [ r ( m, d, k )] . Th e follo wing notations will b e used in this s ection: 14 ℓ x 0 C ( 2, ℓ, x 0 ) 40 80.049 938 0.00000 0267802482750 39 78.051 215 0.00000 0367770130466 38 76.052 559 0.00000 0505162841918 37 74.053 975 0.00000 0694037963620 36 72.055 470 0.00000 0953753092710 35 70.057 050 0.00000 1310973313578 34 68.058 722 0.00000 1802442476141 33 66.060 495 0.00000 2478811076980 32 64.062 378 0.00000 3409926108503 31 62.064 382 0.00000 4692144890365 30 60.066 519 0.00000 6458452590756 29 58.068 802 0.00000 8892465898008 28 56.071 247 0.00001 2247826675415 27 54.073 872 0.00001 6875076361489 26 52.076 697 0.00002 3258920058581 25 50.079 746 0.00003 2069930688629 24 48.083 046 0.00004 4236337186173 23 46.086 630 0.00006 1043767052413 22 44.090 537 0.00008 4273925651732 21 42.094 810 0.00011 6397546202183 20 40.099 505 0.00016 0843029165595 19 38.104 686 0.00022 2370693445282 18 36.110 434 0.00030 7594429791974 17 34.116 844 0.00042 5709065373619 16 32.124 038 0.00058 9504628397967 15 30.132 169 0.00081 6780125566277 14 28.141 428 0.00113 2311971151022 13 26.152 067 0.00157 0588251431389 12 24.164 414 0.00217 9590204991318 11 22.178 908 0.00302 5980931596380 10 20.196 152 0.00420 2124182703906 9 18.216 991 0.00583 5328094363729 8 16.242 641 0.00809 9376451161879 7 14.274 917 0.01122 7174827357965 6 12.316 625 0.01551 9482245119539 5 10.372 281 0.02133 3034990024608 4 8.4494 90 0.02899 5023101223379 3 6.5615 53 0.03846 8799615120751 2 4.7320 51 0.04812 9928161242959 T able 1: V erification of C ( 2, ℓ, x 0 ) < e − 3 for 2 ≤ ℓ ≤ 40 . 15 σp [ i ] = P S ∈S i p [ S ] ; σαp [ i ] = P S ∈S i αp [ S ] ; e T is the set of elemen ts of T for whic h line A5 was executed. 4.1 Impro ved p erformance b ounds for OSC Theorem 7 E [ r ( m, d, 1 )] ≤ log 2 m ln d, if m > 15  1 2 + log 2 m  ( 1 + ln d ) , otherwise In the rest of the sectio n , w e p ro ve the ab o ve th eorem via a series of claims. Note that for OSC we substitute µ = c S = k = 1 in the psuedo co de of Algorithm A-Univ ersal and th at deficit ∈ { 0, 1 } . Lemma 8 F or any T ∈ T ∗ , E h | e T | i ≤ 1 2 + log 2 m, if m ≤ 7 log 2 m, otherwise Pro of. W e can us e the p ro of of Lemma 4 w ith s m all exceptions. The sequence of probabilities that are compu ted are alw ays doubling th e previous one, so for z ≥ 1 we alw ays use p robabilit y 1 and as the r esult, F ( z ) = log 2 z + 1 , and thus F ( 1 ) = 1 . Similarly , for 1 2 ≤ z ≤ 1 we h a ve F ( z ) = z ( log 2 z + 1 ) + ( 1 − z )( log 2 z + 2 ) = log 2 z + 2 − z , and th u s F ( z ) = 1 2 . In turn, E h | e T | i = log 2 m + F ( 1 / m ) , so for m ≥ 2 we ha ve E h | e T | i ≤ log 2 m + 1 2 and for m ≥ 7 w e hav e E h | e T | i ≤ log 2 m . ❑ Ob viously E [ | T | ] is equal to the su m of p robabilities used in line A12 plus the num b er of times w e execute line A12. Let ξ ( i ) b e the v alue of σαp [ i ] at the time the algo r ithm receiv es elemen t i as the inpu t. If the test of line A2 is false, the su m of pr ob ab ilities u sed in line A6 is ξ ( i ) + 1 , w hile by Lemma 13 with α = 0 lin e A12 is executed with pr obabilit y at most 1 e < 0.37 , so the cont r ibution of i to the exp ected cost is smaller than ξ ( i ) + 1.37 . Lemma 9 F or T ∈ T ∗ , if | e T | > 0 then E h P i ∈ e T ξ ( i ) i < E h | e T | i  ln | T | − ln E h | e T | i . Pro of. Before the co n dition in line A2 is ev aluated for el emen t i th e algorithm p erform s in- dep end ent random selectio n s of sets from S i with the sum of pr obabilities of su ccess equal to ξ ( i ) . By Lemma 13 with α = 0 the pr ob ab ility that all th ese selections fail, and th u s the test in line A2 is false, is Pr h i ∈ e T i < e − ξ ( i ) . Let Γ b e a parameter to b e established la ter, and let ζ ( i ) = max { 0, ξ ( i ) − ln | T | + Γ } . Clearly , E   X i ∈ e T ξ ( i )   ≤ E h | e T | i ( ln | T | − Γ ) + X i ∈ T Pr h i ∈ e T i ζ ( i ) Let T ′ = { i ∈ T : ζ ( i ) > 0 } . T h en X i ∈ T Pr h i ∈ e T i ζ ( i ) ≤ X i ∈ T ′ e − ζ ( i )− ln | T | + Γ ζ ( i ) = | T | − 1 e Γ X i ∈ T ′ e − ζ ( i ) ζ ( i ) < e Γ − 1 . where the last inequalit y follo ws from F act 2 and T ′ ⊆ T . Thus, E   X i ∈ e T ξ ( i )   ≤ E h | e T | i   ln | T | − Γ + e Γ − 1 E h | e T | i   16 W e can use Γ = 1 + ln E h | e T | i to get the desired estimate. ❑ No w, w e are ready to finish the pro of of th e claim on E [ r ( m, d , 1 )] in the theorem. E [ r ( m, d, 1 )] = E [ | T | ] | T ∗ | < P T ∈T ∗ E [ P i ∈ e T ξ ( i )+ 1.37 ] | T ∗ | < P T ∈T ∗ E [ | e T | ]( ln | T | − ln E [ | e T | ] + 1.37 ) | T ∗ | (b y Lemma 9) = E h | e T | i  ln | T | − ln E h | e T | i + 1.37  The last qu an tit y is an in creasing function of E h | e T | i , so w e can r eplace it with its o verestimat e. F or ev ery m ≥ 2 we can u se estimate E h | e T | i ≤ 0.5 + log 2 m and the fact that ln ( 0.5 + log 2 2 ) > 0.37 . F or m ≥ 16 we can use estimate E h | e T | i ≤ log 2 m and the fact that ln log 2 16 > 1.37 . 4.2 Impro ved p erformance b ounds for OSC k Note that for OSC k w e s ubstitute µ = c S = 1 in the psu edo co de of Algorithm A-Univ ersal and that deficit ∈ { 0, 1, 2, . . . , k } . F o r impro ved analysis, w e change Algorithm A-Universal slightly , namely , line A6 (with µ = c S = 1 ) A6 p [ S ] ← min   αp [ S ] + | S i | − 1  , 1  // probability for this step // is c h an ged to A6’ p [ S ] ← min   αp [ S ] + deficit · | S i | − 1  , 1  // probability for this step // Theorem 10 With the ab ove mo dific ation of Algorithm A-Univ ersal , E [ r ( m, d, k )] ≤   1 2 + log 2 m  ·  2 ln d k + 3.4  + 1 + 2 log 2 m if k ≤ ( 2 e ) · d 1 + 2 log 2 m otherwise W e no w p ro ceed with the pro of of the ab o v e theorem. As b efore, T ∗ is an optimal solution and for T ∈ T ∗ w e define e T as the set of element s of T f or which line S3 w as executed. Since Lemma 8 is still true with th e same pr o of, we ha v e E h e T i ≤ log 2 m + 1 2 for all m . W e will distribu te the av erage cost of th e obtained solution as f ollo w s. Eac h elemen t of e T giv es a c harge to T and a charge to its element s . I f the algorithm hav e receiv ed the set of element X ⊆ U , then clearly | T ∗ | ≥ | X | · k d ; our goal is to give charge s to the element s so that their exp ected sum equals xk /d ≤ | T ∗ | . W e will again p erform an analysis of th e a verage cost of receiving an elemen t i for whic h th e test in lin e A2 is false. W e define or redefine the follo wing notatio n s: σαp [ i ] = P S ∈S i − T ∗ αp [ S ] ; ξ ( i ) is the v alue of σαp [ i ] w hen line A1 is executed for i ; β ( i ) = | ( S i ∩ T ∗ ) − ( S i ∩ T ) | ; ψ ( i ) = | ( S i ∩ T ) − ( S i ∩ T ∗ ) | ; 17 The v alue of deficit in lin e A1 is at most β ( i ) − ψ ( i ) . Elemen t i will b elong to some e T only if ψ ( i ) < β ( i ) . W e will view ξ ( i ) and β ( i ) as fixed parameters of the ev ent when i is receiv ed. T he quan tity ψ ( i ) is the num b er of successes in indep end en t trials with su ccess p robabilities that add to ξ ( i ) . Let p ( i ) = Pr [ ψ ( i ) < β ( i )] . W e charge elemen t i with a v alue of π e ( i ) = k dp ( i ) . T he in tu ition is that, b ecause we m ake this c h arge with pr obabilit y p ( i ) , on an a v erage it equals p ( i ) π e ( i ) = k/d and the s u m of these c harges therefore cannot b e larger that | T ∗ | . W e th en d istribute the remaining cost equally among ψ ( i ) < β ( i ) many elemen ts of ( S i ∩ T ∗ ) − ( S i ∩ T ) . Clearly , eac h of the v alue of deficit computed in line A1 and computed in line A10 cann ot exceed β ( i ) . Th e term deficit · | S i | − 1 in lin e A6’ adds at most deficit to the sum of pr obabilities computed in line A6’, th u s the cost attribu table to this term, as well as the cost due to line A12 add to at most 2 p er T ∈ T ∗ . I t remains to estimate the cost due to the terms αp [ S ] . W e decrease this cost by the c harge made to i , so eac h set T ∈ T ∗ suc h that i ∈ e T recei ves a charge of at most π s ( i ) = max  0, ξ ( i )− π e ( i ) β ( i )  = max  0, ξ ( i )− k dp ( i ) β ( i )  . The exp ect ed num b er of s ets selected b y us is therefore at most P T ∈T ∗ P i ∈ e T ( π s ( i ) + 2 ) + P i ∈ X p ( i ) π e ( i ) ≤ | T ∗ | · P i ∈ e T π s ( i ) + 2 · | e T | · | T ∗ | + | X | · k d ≤  1 2 + log 2 m  π s ( i ) + 2 log 2 m + 1  · | T ∗ | whic h means we need to estimate the quan tity π s ( i ) . F or this, w e fir st need to calculate a b ound for p ( i ) . Remem b er that ψ ( i ) is the num b er of successes of a set of ind ep endent trials with success probabilities that add up to ξ ( i ) . The s tand ard Chernoff b oun d theorem [9, 16] states that if we ha ve a set of ind ep enden t trials w ith th e sum of success probabilities µ , the pr obabilit y that the n u m b er of su ccesses is b elo w ( 1 − δ ) µ is b elo w e − δ 2 µ/2 . In our case, µ = ξ ( i ) and ( 1 − δ ) µ is β ( i ) . W e introdu ce the follo wing notat ions for simplicit y: β = β ( i ) , φ = ξ ( i ) /β and κ = d/ k . No w µ = φβ and δ = ( φ − 1 ) / φ ; thus via Chernoff b ound we hav e p ( i ) < e − ( φ − 1 ) 2 2φ 2 φβ = e − ( φ − 1 ) 2 2φ β . Hence π s ( i ) < max  0, φ − 1 κβ e ( φ − 1 ) 2 2φ β  < max  0, φ − 1 κβ e ( φ 2 − 1 ) β  By us in g simple calculus and the fact that β ≥ 1 , it can b e sh o wn that the maximum v alue of the function f ( φ ) = φ − 1 κβ e ( φ 2 − 1 ) β is at most 2 ln κ + 2 ln ( 2 e ) < 2 ln κ + 3.4 . This sho w s that π s ( i ) <  2 ln κ + 3.4 if k < ( 2 e ) · d 0 otherwise 4.3 Lo wer b ounds on comp etitive ratios for OSC k and W OSC k Lemma 11 4 Ther e exists an instanc e with m = | S | sets over n = | V | elements such that for any fixe d δ > 0 any deterministic algorithm must have a c omp etitive r atio of (i) Ω  log m k log n k log log m k + log log n k  for OSC k pr ovide d k log 2 n k + 1 ≤ m ≤ ( k + 1 ) e ( n k ) 1 2 − δ and k < min { m, n } ; (ii) Ω  log m log n log log m + log log n  for WOSC k pr ovide d k + log 2 ( n − 1 − ⌈ log 2 ( k + 1 ) ⌉ ) ≤ m ≤ k + e ( n − 1 − ⌈ log 2 ( k + 1 ) ⌉ ) 1 2 − δ and k < 1 2 · min { m, 2 n − 1 } . 4 The relationships b et ween m , n and k were referred to as “for almost all v alues of the parameters” b efore. 18 Pro of. (i) Alon et al. [2] provided an instance of OSC w ith m ′ sets and n ′ elemen ts with su c h that the optimal (offline) co ver con tains just one s et b ut any online cov er must use Ω  log m ′ log n ′ log log m ′ + log log n ′  sets as long as log 2 n ′ ≤ m ′ ≤ e ( n ′ ) 1 2 − δ for any fixed δ > 0 . Consider a giv en k . W e will us e one additional elemen t x and k additional sets suc h that x app ears in all these sets. T o mak e these k sets m utu ally differen t, we will use an additional ⌈ log 2 ( k + 1 ) ⌉ elements (whic h we will n ev er present) and add a d istinct subset of these additional elemen ts to eac h of the k sets. W e will also ha ve k copies of the instances of Alon et al. [2] with elemen ts r enamed to make eac h cop y distinct from the rest; eac h element of eac h copy is also added to exactly k − 1 of the k add itional sets we menti oned at first. The total n um b er of elemen ts n satisfies kn ′ < n = kn ′ + 1 + ⌈ log 2 k ⌉ < ( k + 1 ) n ′ , and the total n umber of sets is m = k + km ′ < ( k + 1 ) m ′ since k < m . W e first pr esen t the elemen t x to force the adv er s ary to select th e k add itional sets; these sets also co ver any elemen t in the k copies of Alon et al. [2] exactly k − 1 times. After this, we presen t the elements in the k copies of Alon et al. [2] f ollo w ing their sc h eme, present in g elemen ts in one cop y completely b efore present in g elemen ts in the n ext cop y . No w the op timal uses at most 2k sets, whereas by a reasoning similar to that in Alon et al. [2] an y online algorithm must use Ω  k + k · log m ′ log n ′ log log m ′ + log log n ′  sets; th us the p erform ance ratio is at least Ω  log m ′ log n ′ log log m ′ + log log n ′  = Ω  log m k log n k log log m k + log log n k  . Moreo ver, the relationship b et w een m and n is giv en b y k · log 2 n k + 1 < k · log 2 n ′ ≤ k m ′ < m < ( k + 1 ) m ′ ≤ ( k + 1 ) · e ( n ′ ) 1 2 − δ < ( k + 1 ) · e ( n k ) 1 2 − δ (ii) W e again use one additional elemen t x plus ⌈ log 2 ( k + 1 ) ⌉ additional elemen ts (that w e w ill nev er p resen t) to create k additional sets suc h that x app ears in all th ese s ets. W e set the cost of eac h of th ese sets to b e arbitrarily close to zero, say ε . This time we jus t use one cop y of the instance of Alon et al. [2] with eac h set of cost 1 and, as b efore, eac h elemen t of this cop y is also added to exactly k − 1 of the k additional s ets w e men tioned at first. Th e total n u m b er of elemen ts n satisfies n ′ < n = n ′ + 1 + ⌈ log 2 k ⌉ , and the total num b er of sets m satisfies m ′ < m = k + m ′ . W e again fi rst present the element x to force the adversary to s elect the k additional sets; these sets also co ver an y elemen t in the cop y of Alon et al. [2] exactly k − 1 times. After this, w e pr esent the elemen ts in the copy of Alon et al. [2] with n ′ elemen ts and m ′ sets follo wing their scheme. Ov erall, the optimal uses sets of total cost 1 + ε wh ereas by a reasoning similar to that in Alon et al. [2] an y online algorithm m u st u se sets of total cost at least ε + Ω  log m ′ log n ′ log log m ′ + log log n ′  ; thus setting ε to b e su fficien tly small we ac hiev e a comp etitiv e ratio of Ω  log m ′ log n ′ log log m ′ + log log n ′  = Ω  log ( m − k ) l og ( n − 1 − ⌈ log 2 ( k + 1 ) ⌉ ) log log ( m − k )+ log log ( n − 1 − ⌈ log 2 ( k + 1 ) ⌉ )  = Ω  log m log n log log m + log log n  where the last equalit y holds since k < 1 2 · min { m, 2 n − 1 } . Moreo ve r , the relationship b et ween m and n is given b y k + log 2 ( n − 1 − ⌈ log 2 ( k + 1 ) ⌉ ) = k + log 2 n ′ ≤ k + m ′ = m ≤ k + e ( n ′ ) 1 2 − δ = k + e ( n − 1 − ⌈ log 2 ( k + 1 ) ⌉ ) 1 2 − δ ❑ 19 Ac kno w ledgmen ts. 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[21] L. T revisan. Non-appr oximability r esults for optimization pr oblems on b ounde d de gr e e instanc es , pro ceedings of the 33rd annual A CM Symp osium on the Theory of Comp uting, p p. 453-46 1, 2001. [22] V. V azirani. A ppr oximation algorithms , Springer-V erlag, July 2001. App endix A Some com b inatorial and probabilistic facts and results F act 1 If f is a non-ne gative i nte ger r andom f unction, then E [ f ] = P ∞ i = 1 Pr [ f ≥ i ] . F act 2 The function f ( x ) = x e − x is maximize d for x = 1 . The subsequent lemmas d eal w ith N indep enden t 0-1 random v ariables τ 1 , . . . , τ N called trials with even t { τ i = 1 } is the suc c e ss of trial num b er i and s = P N i = 1 τ i is the num b er of successful trials. Let x i = Pr [ τ i = 1 ] = E [ τ i ] and X = P N i = 1 x i = E [ s ] . Lemma 12 If 0 < 2α ≤ X + 1 than Pr [ s = α ] > Pr [ s = α − 1 ] . 21 Pro of. Our elementa ry ev ents are 0/1 vect ors τ = ( τ 1 , . . . , τ N ) . Let E α b e the ev en t { s = α } , i.e. the set of elemen tary ev en ts with α 1’s. Giv en τ ∈ E α − 1 w e can form an elemen tary even t from E α b y conv erting some 0 into 1. If w e do it with τ i , call the result τ i ; observe that Pr  τ i  > x i Pr [ τ ] . Therefore the sum of p robabilities of element ary ev ents f ormed fr om τ is at least Pr [ τ ] P i : τ i = 0 x i ≥ ( X − α + 1 ) Pr [ τ ] ≥ αPr [ τ ] . This sho w s that the su m of p robabilities of the multi- s et of elemen tary ev ents f ormed fr om elemen ts of E α − 1 is larger than αPr [ E α − 1 ] ; in tur n, ev ery elemen ts in this multi-set b elongs to E α and it is present in this multi-set exactly α times. Thus Pr [ E α ] ≥ α − 1 αPr [ E α − 1 ] . ❑ Lemma 13 If 0 ≤ α ≤ X/2 then Pr [ s ≤ α ] < e − X X α / α ! . Pro of. T he case of α = 0 is easy since Pr [ s ≤ 0 ] = Π n i = 1 ( 1 − x i ) < Π n i = 1 e − x i = e − X . So, we assu me in the remaining that α > 0 . W e will sh o w how to alte r the p robabilities so that X remains constan t and Pr [ s ≤ α ] do es not decrease. Let x 0 = x 1 + x 2 , s ′ = s − τ 1 − τ 2 and let q α = Pr [ s ′ ≤ α ] . W e assume that x 0 ≤ 1 . Then Pr [ s ≤ α ] = Pr [ τ 1 = τ 2 = 0 & s ′ ≤ α ] + Pr [ τ 1 + τ 2 = 1 & s ′ ≤ α − 1 ] + Pr [ τ 1 = τ 2 = 1 & s ′ ≤ α − 2 ] = ( 1 − x 1 )( 1 − x 1 ) q α + [( 1 − x 1 ) x 2 + x 1 ( 1 − x 2 )] q α − 1 + x 1 x 2 q α − 2 = ( 1 − x 0 + x 1 x 2 ) q α + ( x 0 − 2x 1 x 2 ) q α − 1 + x 1 x 2 q α − 2 = [ P = ( 1 − x 0 ) q α + x 0 q α − 1 ] + x 1 x 2 ( q α − 2q α − 1 + q α − 2 ) = P + x 1 x 2 ( Pr [ s ′ = α ] − Pr [ s ′ = α − 1 ] ) If we keep x 1 + x 2 fixed, P is constant and w e maximize the latter expression when x 1 = x 2 (b ecause 2α ≤ ( X − x 1 − x 2 ) + 1 , by Lemma 12, the difference of probabilities in the paren thesis is p ositiv e). This sho ws that Pr [ s = α ] is maximized wh en all x i ’s are equal. W e can “pad” the ve ctor of x i ’s with zeros, i.e. add tr ials with zero p robabilit y of success. This sho ws that we can ov erestimate our probabilit y when we go to the limit w ith N → ∞ an d all x i ’s equal to X/N . W e can n ow finish the p r o of b y observing the follo win g from standard estimate s in prob ab ility theory: lim N →∞ N ! ( N − α )! α !  1 − X N  N − α  X N  α = X α e X α ! ❑ 22