Admission Control to Minimize Rejections and Online Set Cover with Repetitions

Admission Contr ol to Minimiz e Rejections and Online Set Co ver with Repetiti ons Noga Alon ∗ Schools of Mathematics and Computer Science T el-A viv Univ ers ity T el-A viv , 69978, Israel noga@math .t au.ac.il Y ossi Azar † School of Computer Science T el-A viv Univ ers ity T el-A viv , 69978, Israel azar@tau.ac.il Shai Gutner ‡ School of Computer Science T el-A viv University T el-A viv , 69978, Israel gutner@tau.a c. il ABSTRA CT W e study the admission control problem in general n et- w orks. Comm unication requests a rrive o ver time, and the online algorithm accepts or re jects each request w hile main- taining the capacity limitations of the netw ork. The admis- sion con trol problem has b een usually analyzed as a b ene- fit problem, where t he goal is to devise an online algorithm that accepts the maxim um number of requests p ossible. The problem w ith this ob jectiv e function is that even algorithms with optimal comp etitive ratios may reject almost all of th e requests, when it w ould hav e b een possible to reject only a few. This could b e inappropriate for settings in which rejections are intended to b e rare even ts. In this pap er, w e consider preemptive online algorithms whose goal is to minimize the num b er of rejected requests. Eac h requ est arrives together with the path it should b e routed on. W e sho w an O (log 2 ( mc ))-comp etitive random- ized algorithm for the weigh ted case, where m is the num b er of edges in the graph and c is th e maximum edge capac- it y . F or th e unw eighted case, w e give an O (log m log c )- compet itive rand omized algo rithm. This settles an open question of Blum, K alai and Klein b erg raised in [10]. W e note that allo wing preemption and handling requests with giv en p aths are essential for a voiding t rivial low er b ounds. The admission control problem is a generalization of t h e online set co ver with rep etitions problem, whose inp ut is a family of m sub sets of a ground set of n elements . Elements of the groun d set are giv en to t he online algorithm one by ∗ Researc h supp orted in part by a gran t from t he Israel Sci- ence F ou n dation, and by the Hermann Minko wski Minerva Cen ter for Geometry at T el Aviv U niv ersity . † Researc h supp orted in part by the Israel Science F ound a- tion and by th e German-Israeli F oundation. ‡ This pap er forms p art of a Ph.D. th esis written by the author un der the sup ervision of Prof. N . Alon and Prof. Y. Azar in T el Av iv U niver sity . Permission to make digital or hard copies of all or part of this work for personal or classroom use is granted without fee provided that copies are not made or distrib uted for profit or commercial adv antage and that copies bear this notic e and the full cita tion on the first page . T o c opy otherwise, to republi sh, to post on serv ers or to redistrib ute to lists, re quires prior specific permission and/or a fee. SP AA’05, July 18 –20, 2005, L as V egas, Ne vada , USA. Copyri ght 2005 A CM 1-58113-986-1/05/ 0007 ... $ 5.00. one, p ossibly requ esting each elemen t a multiple num b er of times. (If eac h element arrives at most once, this cor- respond s to the online set cov er p roblem.) The algorithm must co ver each element by different subsets, according t o the number of times it has b een req u ested. W e give an O (log m log n )-comp etitive ran d omized algo- rithm for the online set cov er with rep etitions problem. This matc h es a recen t lo wer b ound of Ω(log m log n ) giv en by F eige and Korman for the competitive ratio of an y ran- domized p olynomial time algorithm, un der th e B P P 6 = N P assumption. Give n any constant ǫ > 0, an O (log m log n )- compet itive deterministic bicriteria algorithm is shown that co vers eac h element by at least (1 − ǫ ) k sets, where k is the num b er of times the element is co vered by the op t imal solution. Categories and Subject Descriptors C.2.2 [ Computer-Communication Netw orks ]: Netw ork Protocols— R outing pr oto c ols ; F.2.2 [ Anal ysis of Algo- rithms and Problem Com plexity ]: N on numerical Al- gorithms and Problems General T erms Algorithms, Theory . Keyw ords On-line, Comp etitive, A dmission control, Set Cov er. 1. INTR ODUCTION W e study the admission control problem in general graphs with edge capacities. An online algorithm can receive a se- quence of communications requests on a v irtual p ath, that ma y b e accepted or rejected, while staying within the ca- pacit y limitations. This problem has typically been studied as a b enefit prob- lem. This means that the online algorithm has to b e com- p etitiv e with resp ect t o the num b er of accepted requests. A problem with this ob jective fun ct ion is th at in some cases an online algorithm with a go od comp etitive ratio may re- ject the va st ma jority of the requ ests, whereas th e optimal solution rejects only a small fraction of them. In this pap er we consider the goal of minimizing th e num- b er of rejected requests, which wa s first studied in [10]. This approac h is suitable for applications in which rejections are intended to b e rare even ts. A situation in which a signifi- cant fraction of the requests is rejected even by the optimal solution means that th e netw ork needs to b e upgraded. W e consider preemptive online algorithms for the admis- sion control problem. A llowing preemption is necessary for ac hieving reasonable b ounds for the comp etitive ratio. Each request arriv es together with the path it should b e routed on. The admission control algorithm decides whether to ac- cept or reject it. An online algorithm for b oth admission contro l and routing easily admits a triv ial lo wer b ound [10]. The admission contro l to minim ize rejections prob- lem. W e now formally d efine th e admission con trol p roblem. The in put consist of the follo wing: • A directed graph G = ( V , E ), where | E | = m . Each edge e has an integer capacity c e > 0. W e denote c = m ax e ∈ E c e . • A sequence of requests r 1 , r 2 , . . . , eac h of whic h is a simple path in th e graph. Every request r i has a cost p i > 0 asso ciated with it. A feas ible solution for the problem m ust assure that for every edge e , the num b er of accepted requ ests whose paths conta in e is at most its capacit y c e . The goal is to find a feasible solution of minimum cost of the rejected requests. The online algorithm is given req uests on e at a time, and must d ecide whether to accept or reject each request. It is also allo w ed t o preemp t a request, i.e. to reject it after already accepting it, but it cann ot accept a request after rejecting it. Let O P T b e a feasible solution having minimum cost C O P T . An algorithm is β -comp etitiv e if the total cost of the requests rejected by this algorithm is at most β C O P T . Previous results for admissi on contro l . Tigh t b ound s w ere achiev ed for the admission control problem, where the goal is t o maximize th e number of accepted requests. A w er- buch, Azar and Plotkin [6] p ro vide an O (log n )-comp etitive algorithm for general graphs. F or the admission con trol problem on a tree, O (log d )-comp etitiv e randomized algo- rithms app ear in [7, 8], where d is the diameter of the tree. Adler and A zar presen ted a constan t- competitive preemp- tive algorithm for admission control on th e line [1]. The admission control to minimize rejections p roblem was studied by Blum, Kalai and Kleinberg in [10], where tw o deterministic algorithms with comp etitive ratios of O ( √ m ) and c + 1 are given ( m is the num b er of edges in th e graph and c is the maximum capacity). They raised the question of whether an online algorithm with p olylogari th mic com- p etitiv e ratio can b e obtained. W e note th at one can combine an algorithm for maximiz- ing throughput of accepted requests and an algori th m for minimizing rejections and get one algorithm which ac h iev es b oth sim ultaneously with sligh tly degrading the comp etitive ratio [9, 11]. In this pap er we sh o w that the admission control to mini- mize rejections problem is a generalization of t he online set co ver with rep etitions problem describ ed b elow: The online s et cov e r wi th rep etitions problem. The online set cov er problem is defined as follow s: Let X b e a ground set of n elements, and let S b e a family of subsets of X , |S | = m . Eac h S ∈ S has a n on-negative cost associated with it. An adversary gives elements to the algorithm from X one by one. Eac h element of X can b e given an arbitrary num b er of times, not necessarily consecutively . An element should b e co vered by a num b er of sets which is eq ual to th e num b er of times it arrived. W e assume that the elements of X and the mem b ers of S are known in adv ance to the algorithm, how ever, t he elements given by the adversary are not known in adva n ce. The ob jectiv e is to minimize the cost of the sets chosen by the algorithm. Previous results for onl ine s et cov er. The offline versi on of the set cov er problem is a classic N P-hard prob- lem that w as stu d ied extensively , and the b est appro xima- tion factor achiev able for it in p olynomial time (assuming P 6 = N P ) is Θ(log n ) [12, 13]. The b asic online set co ver problem, where repetitions are not allo wed, was studied in [2, 14]. A different v ariant of t h e p roblem, d ealing with maximum b enefit, is presented in [5]. An O ( log m log n )- compet itive deterministic algorithm for the online set cover problem w as giv en by [2] where n is the number of ele- ments and m is the num b er of sets. A lo wer b ound of Ω( log m log n log log m +log log n ) was also show n for any deterministic on- line algori th m. A recent result of F eige and Korman [14] establishes a low er b ound of Ω(log m log n ) for th e comp eti- tive ratio of any randomized p olynomi al time algorithm for the online set cov er p roblem, under th e B P P 6 = N P as- sumption. They also prove the same low er b ound for any deterministic p olynomi al time algorithm, und er the P 6 = N P assumption. Our results. The main result we give in th is pap er is an O (log 2 ( mc ))-comp etitive randomized algorithm for the admission control to minimize rejections p roblem. This set- tles the op en question raised b y Blum et al. [10]. F or the unw eigh ted case, where all costs are equal to 1, we slightly impro ve th is b ound and give an O (log m log c )-comp etitive randomized algorithm, W e present a simple reduction b etw een online set cov er with rep etitions and the admission control to minimize re- jections problem. This implies an O (log 2 ( mn ))-comp etitive randomized algorithm for the online set cov er with rep eti- tions problem. F or the unw eighted case (all costs are equal to 1), we get an O (log m log n )-comp etitiv e rand omized al- gorithm. This matches th e low er b ound of Ω(log m log n ) giv en by F eige and Korman. Their results also imply a low er b ound of Ω(log m log c ) for the comp etitive ratio of any ran- domized p olynomial time algorithm for the admission control to minimize rejections problem (assuming B P P 6 = N P ). The derandomization techniques used in [2] for th e online set cov er problem do not seem to apply here. This is why w e also consider th e b icriteria version of the on line set cov er with rep etition p rob lem. F or a given constant ǫ > 0, the on- line algorithm is req u ired to cove r eac h elemen t by a fraction of 1 − ǫ times the num b er of its app earances. S pecifically , at any p oint of time, if an element has b een requested k times so far, then the optimal solution cov ers it by k different sets, whereas the online algorithm cov ers it by (1 − ǫ ) k different sets. W e giv e an O (log m log n )-comp etitive deterministic bicriteria algorithm for this problem. T ech ni ques. The techniques w e use follo w those of [2, 3] t ogether with some new ideas. W e start with an online fractional solution which is monotone increasing during the algorithm. Then, the fractional solution is conve rted into a randomized algorithm. Interestingly , to get a deterministic bicriteria algorithm we use a differen t fractional algo rithm than t h e one used for t he rand omized algorithm. 2. FRA CTIONAL ALGORITHM FOR ADMISSION CONTR OL In this section we describ e a fractional algorithm for the problem. A fractional algorithm is allo wed to reject a frac- tion of a request r i . W e use a weig ht f i for this fraction. Sp ecifically , if 0 ≤ f i < 1, we reject with p ercen tage of p re- cisely f i . If f i ≥ 1, t h en t he request is completely rejected. At any stage of the fractional algorithm w e will u se the fol- lo wing notation: • RE Q e will denote the set of requests that arrived so far whose path s contain the edge e . • RE Q will d enote S e ∈ E RE Q e . • ALI V E e will denote t he requ ests from RE Q e that hav e not b een fully rejected (requests r i for which f i < 1). • n e will denote the excess of edge e caused by t he re- quests in A LI V E e . n e = | ALI V E e | − c e The requirement from a fractional algorithm is that for every edge e , X i ∈ ALI V E e f i ≥ n e The cost associated with a fractional algorithm is d efi ned to b e P i ∈ RE Q min { f i , 1 } p i . W e will now describ e an O (log ( mc ))-comp etitive algo - rithm for the problem, even versus a fractional optimum. The cost of the op t imal fractional solution, C O P T is denoted by α . W e ma y assume, by d oubling, that the v alue of α is known up to a factor of 2. T o determine th e initial v alue of α we look for the fi rst time in which we must reject a request from an edge e . W e can start guessing α = min i ∈ RE Q e p i , and th en run t he algorithm with th is b ound on the optimal solution. If it turns out that the v alue of the optimal so- lution is larger than our current guess for it, (that is, the cost exceeds Θ( α log( mc )) ) , then we ”forget” ab out all the request fractions rejected so far, up date the v alue of α by doubling it, and contin ue. W e note that t h e cost of the re- quest fractions that we have ”forgotten” about can increase the cost of our solution by at most a factor of 2, since the v alue of α was doub led in each step. W e thus assume that α is known. D enote by R big the requests with cost exceedin g 2 α . The opt imal fractional solution can reject a total fraction of at most 1 / 2 out of th e requests of R big . Hence, when an edge is requested more than its capacit y , the fractional optimum must reject a total fraction of at least 1 / 2 out of the requests not in R big whose paths conta in the edge. By doubling the fraction of rejection for all the requ ests not in R big (keeping fractions to be at most 1) and completely accepting all the requests in R big , w e get a feasible fractional solution whose cost is at most tw ice the opt imum. Hence, the online algorithm can alwa y s completely accept requ ests of cost exceedin g 2 α (an d adjust the edge capacities c e accordingly). Denote by R small the requests with cost at most α/ ( mc ). W e claim that w e can completely reject all the requests from R small . F or each edge e , t he optimal solution can accept a total fraction of at most c out of th e requests whose p aths contain th e edge e , and therefore it can ac- cept a total fraction of at most mc requests. The frac- tions of requests accepted out of R small hav e t otal cost at most mc · α/ ( mc ) = α . It follo ws that the optimal solu- tion pays at least cost ( R small ) − α for the fractions of re- quests out of R small that it rejected. Therefore, the on- line algorithm can reject all th e requests in R small and pay cost ( R small ). If cost ( R small ) < 2 α , then this adds only O ( α ) to the cost of the online algori th m. If cost ( R small ) ≥ 2 α , then cost ( R small ) ≤ 2( cost ( R small ) − α ), so the online algo- rithm is 2-comp etitive with resp ect t o the requests in R small . By the ab o ve arguments, all the req uests of cost smaller than α/ ( mc ) or greater than 2 α are rejected immed iately or accepted p ermanently (edge capacities are decreased in this case), resp ectiv ely . An algorithm n eeds to h andle only requests of cost b et ween α/ ( mc ) and 2 α . W e normalize the costs so that the minimum cost is 1 and the maxim um cost is g ≤ 2 mc , and fix α appropriately . The algorithm maintains a w eight f i for each req u est r i . The w eights can only increase during the run of t h e algo- rithm. In itially f i = 0 for all the requests. Assume now that the algorithm receives a req uest r i for a path of cost p i . F or eac h edge e , we up date RE Q e , ALI V E e and n e according to the definitions given ab o ve. The follo wing is p erformed for all the edges e of the path of r i , in an arbitrary order. 1. If P i ∈ ALI V E e f i ≥ n e , t h en do nothing. 2. Else, while P i ∈ ALI V E e f i < n e , p erform a wei ght aug- mentation : (a) F or eac h i ∈ ALI V E e , if f i = 0, then set f i = 1 / ( g c ). (b) F or each i ∈ ALI V E e , f i ← f i (1 + 1 n e p i ). (c) Up date ALI V E e and n e . Note that the fractional algorithm starts with all weig hts equal t o zero. This is necessary , since the online algorithm must reject 0 requests in case the optimal solution rejects 0 requ ests. Hence, the algorithm is comp etitive for α = 0, and from n o w on w e assume without loss of generality that α > 0. In the follo wing w e analyze th e p erformance of the algorithm. Lemma 1. The total numb er of weight augmentations p er- forme d during the al gorithm is at m ost O ( α log ( g c )) . Pr oof. Consider the follo wing p otentia l function: Φ = Y i ∈ RE Q max { f i , 1 / ( g c ) } f ∗ i p i where f ∗ i is the weigh t of the req uest r i in the optimal frac- tional solution. W e now show t hree prop erties of Φ: • The initial v alue of th e p otenti al function is: ( g c ) − α . • The p otenti al function never exceeds 2 α . • In eac h weigh t augmentation step, the p otential func- tion is multiplied by at least 2. The first tw o prop erties follow directly from th e initial v alue and from the fact that no request gets a weigh t of more than 1 + 1 /p i ≤ 2. Consider an iteration in whic h the adversary giv es a request r i with cost p i . No w su ppose that a wei ght augmenta tion is p erformed for an edge e . W e m ust have P i ∈ ALI V E e f ∗ i ≥ n e since the optimal solution m ust sat- isfy the capacity constrain t. Th us, the p oten tial function is multipli ed by at least: Y i ∈ ALI V E e „ 1 + 1 n e p i « f ∗ i p i ≥ Y i ∈ ALI V E e „ 1 + 1 n e « f ∗ i ≥ 2 The first inequality foll ows since for all x ≥ 1 and z ≥ 0, (1 + z /x ) x ≥ 1 + z and the last inequalit y follow s since P i ∈ ALI V E e f ∗ i ≥ n e . It follow s that th e total number of w eight augmentatio n steps is at most: log 2 (2 g c ) α = O ( α log g c ) Theorem 2. F or the weighte d c ase, t he fr actional algo- rithm is O (log( mc )) -c omp etitive. In c ase al l the c osts ar e e qual to 1 , the algorithm is O (log c ) -c omp etitive. Pr oof. The cost associated with the online algorithm is P i ∈ RE Q min { f i , 1 } p i , which we will denote by C O N . Con- sider a weigh t augmen tation step p erformed for an edge e . In step 2a of the algorithm, th e wei ghts of at most c + 1 requests change from 0 to 1 / ( g c ). This is b ecause b efore the current requ est arrived, there could have b een at most c requests containing the edge e and ha vin g f i = 0 (the maximum capacity is c ). Since the max im um cost is g , th e total increase of C O N in step 2a of the algorithm is at most ( c + 1) 1 gc g = 1 + 1 /c . If follow s that in step 2a, the quantity P i ∈ ALI V E e f i can increase b y at most 1 + 1 /c . A w eight augmenta tion is p erformed as long as P i ∈ ALI V E e f i < n e . Before step 2b we h a ve that P i ∈ ALI V E e f i < n e + 1 + 1 /c . Thus, the total increase of C O N in step 2b of the algorithm does not ex ceed X i ∈ ALI V E e f i p i 1 n e p i = X i ∈ ALI V E e f i n e < 2 + 1 /c It follo ws that the total increase of C O N in a weigh t augmen- tation step is at most 3 + 2 /c . Using lemma 1 which b ounds the num b er of augmentation steps, w e conclude t hat the algorithm is O (log ( g c ))-comp etitive. F or the weigh ted case, we saw that the input can b e trans- formed so that g ≤ 2 mc , which implies that the algorithm is O (log ( mc ))-comp etitive. In case all the costs are equ al to 1, g is also eq u al to 1 and the algorithm is O (log c )-comp etitive. 3. RANDOMIZED ALGORITHM FOR ADMISSION CONTR OL W e describ e in this section an O ( log 2 ( mc ))-comp etitive randomized algorithm for t h e weig hted case and a sligh tly b etter O (log m log c )-competitive rand omized algorithm for the unw eighted case. The algorithm main tains a wei ght f i for each req u est r i , exactly like the fractional algorithm. Assume now that the algorithm receives a request r i with cost p i . The follo wing is p erformed in this case. 1. Perf orm all the wei ght augmentations according to the fractional algorithm. 2. Reject all requ ests whose weig ht is at least 1 12 l og( mc ) . 3. F or every request r , if its w eight f increased by δ , t hen reject th e request r with p robab ility 12 δ log ( mc ). 4. If the current req uest r i cannot b e accepted (some ed ge w ould b e ov er capacity), t h en reject the request. Else, accept the requ est r i . W e can assume that | R E Q e | , t h e total number of requests whose paths contain a sp ecific edge e , is less t h an 4 mc 2 . T o see this, note th at t h e fractional algorithm normalizes the costs so that the minimum cost is 1 and the maxim um cost is at most 2 mc . If | RE Q e | ≥ 4 mc 2 , then since the optimal solution can accept at most c requests from RE Q e , it must pay a cost of at least t − 2 mc 2 for requests rejected out of RE Q e , where t is the total cost of these requests. The online algorithm can reject all the requests in RE Q e , p a y t and it will still be 2-comp etitive with resp ect to th e requests in RE Q e , since t ≥ 4 mc 2 . Theorem 3. F or the weighte d c ase, the r andomize d algo- rithm i s O (log 2 ( mc )) -c omp etitive. Pr oof. Denote by C f r ac the cost of the fractional algo- rithm. The exp ected cost of requests rejected in step 3 of the algorithm is at most 12 C f r ac log( mc ). The cost of requests rejected in step 2 has the same u pp er b ound. W e now calculate the probabilit y for a request r to b e rejected in step 4. This can happ en only if the path of request r con tains an edge e for which P i ∈ ALI V E e f i ≥ n e but the randomized algorithm rejected less than n e requests whose paths con tain the edge e . A ll the requests with w eight at least 1 12 l og( mc ) are rejected for sure, so we can assume that f i < 1 12 log( mc ) for all i ∈ ALI V E e . Supp ose that i ∈ ALI V E e and that du ring all runs of step 3 of the algorithm the request r i has b een rejected with probabilities q 1 , . . . , q n , where P n k =1 q k = 12 f i log( mc ). The probabilit y that r i will b e rejected is at least 1 − n Y k =1 (1 − q k ) ≥ 1 − e − P n i = k q k = 1 − e − 12 f i log mc ≥ 6 f i log mc The last ineq ualit y follow s since for all 0 ≤ x ≤ 1, 1 − e − x ≥ x/ 2. The number of requ ests in ALI V E e whic h were rejected by the algorithm is a random v ariable whose v alue is the sum of mutually ind ep endent { 0 , 1 } -v alued random vari ables and its ex pectation is at least µ = 6 n e log mc . By Chernoff b ound (c.f., e.g., [4]), the probability for this rand om vari - able to get a v alue less than (1 − a ) µ is at most e − a 2 µ/ 2 for every a > 0. Therefore, the probability to b e less than n e is at most e − (1 − 1 6 log mc ) 2 (6 n e log mc ) / 2 ≤ 3 m 3 c 3 The request costs w ere normalized, so that the maximum cost is at most 2 mc . Each edge is contained in the paths of at most 4 mc 2 requests. Therefore, the exp ected cost of requests which are rejected in step 4 because of th is edge is at most (4 mc 2 )(2 mc )3 / ( m 3 c 3 ) = 24 /m . Thus, the total exp ected cost of requests rejected in step 4 is 24. The result now follow s from Theorem 2. F or the unw eighted case we sligh tly change the algorithm as follow s. In step 3 of the algori th m we reject a request with probabilit y 4 δ log m , and in step 2 we reject all th e requests whose weigh t is at least 1 / (4 log m ). Theorem 4. F or the unweighte d c ase, the r andomize d al- gorithm is O (log m log c ) -c omp etitive. Pr oof. F ollo wing the pro of of Theorem 3, we get that the p robab ility for an edge to cause a sp ecific request to b e rejected in step 4 of the randomized algorithm is at most e − (1 − 1 2 log m ) 2 (2 n e log m ) / 2 ≤ 3 m Denote by Q the q uan tity m a x e ∈ E ( | RE Q e | − c e ). Hence, Q is the maximum excess capacity in the netw ork. The total exp ected cost of requests rejected in step 4 is at most Q (3 /m ) m = 3 Q . It is obvious th at the optimal solution must reject at least Q requests. The result now follo ws from Theorem 2. 4. REDUCING ONLINE SET CO VER TO ADMISSION CONTR OL W e n o w describe the reduction b et ween online set co ver and admission control. Sup pose we are give n th e follo wing input to the online set cove r with repetitions problem: X is a ground set of n elements and S is a family of m subsets of X , with a p ositiv e cost c S associated with eac h S ∈ S . The instance of the admission control to minimize rejections problem is constructed as follow s: The graph G = ( V , E ) has an edge e j for eac h element j ∈ X . The capacity of the edge e j is d efi ned to b e the num b er of sets that contain the elemen t j . The maxim u m capacity is th erefore at most m . The requests are given to th e admission control algorithm in tw o phases. In th e first phase, b efore any elemen t is give n to the on line set cov er algorithm, we generate m requests t o the admission con trol online algorithm. F or every S ∈ S , the req u est consists of all the edges e j such t hat j ∈ S . The online algorithm can accept all the req uests and this will cause the edges to reach their full capacity . In the second ph ase, eac h time the adversa ry giv es an elemen t j to the online set co ver algorithm, we generate a request which consists of the one edge e j and give it to t he admission control algorithm. In case the req u est caused the edge e j to b e ov er capacity , the algorithm will h a ve to reject one request in order to keep the capacit y constraint. In case there is a feasible cov er for the input given t o the online set cov er p roblem, th ere is no reason for the admission contro l algorithm to reject requests that were given in the second phase. This is b ecause requests in the second phase consist of only one edge. Th us, w e can assume th at the admission con trol algorithm rejects only requests given in the first p hase, which corresp ond to subsets of X . It is easy to see that t he requ ests rejected by the admis- sion control algorithm correspond to a legal set cove r. W e reduced an online set cov er problem with n elements and m sets to an admission con trol problem with n edges and maximum capacit y at most m . The fact that the requests w e generated are n ot simple paths in the graph can b e easily fixed by adding ex t ra edges. 5. DETERMINISTIC BICRITERIA ALGO- RITHM FOR ONLINE SET CO VER In this section we describe, giv en any constant ǫ > 0, an O (log m log n )-comp etitive deterministic bicriteria algo- rithm that cove rs eac h element by at least (1 − ǫ ) k sets, where k is th e num b er of times th e elemen t has b een requ ested, whereas the optimum cove rs it k times. W e assume for sim- plicit y that all the sets hav e cost eq ual to 1. The result can b e easily generalized for the weigh ted case using techniques from [2]. The algorithm maintai n s a w eight w S > 0 for each S ∈ S . Initially w S = 1 / (2 m ) for each S ∈ S . The weigh t of each elemen t j ∈ X is defined as w j = P S ∈S j w S , where S j denotes th e collection of sets containing element j . Initially , the algorithm starts with the empty co ver C = ∅ . F or each j ∈ X , we define cov er j = |S j T C | , which is the num b er of times elemen t j is cov ered so far. The follo wing p oten tial function is u sed throughout the algorithm: Φ = X j ∈ X n 2( w j − cov er j ) W e give a high level d escription of a single iteration of the algorithm in whic h the adversary gives an element j and the algorithm chooses sets that co ver it. W e denote by k the num b er of times that the element j has b een requested so far. 1. If cov er j ≥ (1 − ǫ ) k , then do nothing. 2. Else, while cov er j < ( 1 − ǫ ) k , p erform a weight aug- mentation : (a) F or eac h S ∈ S j − C , w S ← w S (1 + 1 2 k ). (b) Add to C all the subsets for which w S ≥ 1. (c) Choose from S j at most 2 log n sets to C so that the va lue of the potential function Φ do es not exceed its v alue b efore the weigh t augmentation. In th e follo wing w e analyze the p erformance of the algo- rithm and explain whic h sets to add to the cov er C in step 2c of t h e algorithm. The cost of the optimal solution C O P T is d en oted by α . Lemma 5. The total numb er of weight augmentations p er- forme d during the al gorithm is at m ost O ( α log m ) . Pr oof. Consider the follo wing p otentia l function: Ψ = Y S ∈C OP T w S W e now show th ree prop erties of Ψ: • The initial val ue of the p otenti al function is: (2 m ) − α . • The p otenti al function never exceeds 1 . 5 α . • In eac h weigh t augmentation step, the p otential func- tion is multiplied by at least 2 ǫ/ 2 . The first tw o prop erties follow directly from th e initial v alue and from the fact that no request gets a weigh t of more than 1 . 5. Consider an iteration in which the adversary giv es an element j for th e k th time. Now supp ose th at a weigh t augmenta tion is p erformed for element j . W e must ha ve that cov er j < ( 1 − ǫ ) k , which means that the online algorithm has co vered element j less than (1 − ǫ ) k times. The optimal solution OP T co vers element j at least k times, which means that t here are at least ǫk subsets of OP T containing j whic h w ere n ot c h osen yet. Th u s, in step 2a of the algorithm t he p oten tial function is multiplied by at least: (1 + 1 2 k ) ǫk ≥ 2 ǫ/ 2 It follo ws that for fixed ǫ > 0 the total number of wei ght augmenta tion steps is at most: log(3 m ) α log 2 ǫ/ 2 = O ( α log m ) Lemma 6. Consider an iter ation in which a weight aug- mentation i s p erforme d. L et Φ s and Φ e b e the values of the p otential function Φ b efor e and after the iter ation, r e- sp e ctively. Then, ther e exist at most 2 log n sets that c an b e adde d to C during the iter ation suc h that Φ e ≤ Φ s . F ur- thermor e, the value of the p otential f unction never exc e e ds n 2 . Pr oof. The proof is by in d uction on th e iterations of the algorithm. I n itially , the va lue of the p otential function Φ is less than n · n = n 2 . Supp ose that in th e iteration the adversa ry gives elemen t j for the k th time. F or eac h set S ∈ S j , let w S and w S + δ S denote the weigh t of S b efore and after th e iteration, resp ectively . Define δ j = P S ∈S j δ S . By th e induction hyp othesis, we know that 2( w j − cov er j ) < 2, b ecause otherwise Φ s w ould hav e been greater th an n 2 . Thus, w j < cov er j + 1 ≤ ⌊ (1 − ǫ ) k ⌋ + 1 ≤ k . This means that δ j ≤ k · 1 / (2 k ) = 1 / 2. W e now explain which sets from S j are add ed to C . Rep eat 2 log n times: choose at most one set from S j such that each set S ∈ S j is chos en with probabilit y 2 δ S . (This can b e implemented by choosing a num b er uniformly at ran- dom in [0,1], since 2 δ j ≤ 1.) Consider an element j ′ ∈ X . Let the wei ght of j ′ b efore the iteration b e w j ′ and let the weigh t after the iteration b e w j ′ + δ j ′ . Element j ′ contri b utes b efore the iteration to the p otential function the val ue n 2 w j ′ . In eac h random choi ce, th e probability that we do not choose a set containing elemen t j ′ is 1 − 2 δ j ′ . The probability th at th is happ ens in all the 2 log n random choice s is therefore (1 − 2 δ j ′ ) 2 log n ≤ n − 4 δ j ′ . Note th at δ j ′ ≤ 1 / 2. In case we c h oose a set containing elemen t j ′ , then cov er j ′ will increase by at least 1 and t h e contri b ution of elemen t j ′ to th e p otential fun ct ion will b e at most n 2( w j ′ + δ j ′ − 1) ≤ n 2 w j ′ − 1 . Therefore, the exp ected contri b ution of element j ′ to the p oten tial function after the iteration is at most n − 4 δj ′ n 2( w j ′ + δ j ′ ) + (1 − n − 4 δj ′ ) n 2 w j ′ − 1 = n 2 w j ′ ( n − 2 δj ′ + n − 1 − n − 4 δj ′ − 1 ) ≤ n 2 w j ′ where to justify the last inequality , we prov e th at f ( x ) = n x + n − 1 − n 2 x − 1 ≤ 1 for every x ≤ 0. T o show this we note th at f (0) = 1 and f ′ ( x ) = n x log n (1 − 2 n x − 1 ). This implies that f ′ ( x ) ≥ 0 for every x ≤ 0. W e can conclude that f ( x ) ≤ 1 for every x ≤ 0, as n eeded. By linearity of exp ectation it follow s that Exp [Φ e ] ≤ Φ s . Hence, there exists a choice of at most 2 log n sets such that Φ e ≤ Φ s . The choices of the va rious sets S to b e added to C can b e d one d eterministicall y and efficiently , by the metho d of conditional probabilities, c.f., e.g., [4], chapter 15. After eac h weig ht augmentation, we can greedily add sets to C one by one, making sure that the p otential function will decrease as much as p ossible after eac h su ch choice. Theorem 7. The deterministic algorithm for online set c over i s O (log m log n ) -c omp etitive. Pr oof. It follo ws from Lemma 5 that the num b er of iter- ations is at most O ( α log m ). By Lemma 6, in each iteration w e choose at most 2 log n sets to C in step 2c of the algo- rithm. The sets chosen is step 2b of the algorithm are t h ose whic h h a ve we ight at least 1. The sum of w eights of all th e sets is initially 1 / 2 and it increases by at m ost 1 / 2 in each w eight augmenta tion. This means that at the end of the algorithm, there can b e only O ( α log m ) sets whose w eight is at least 1. Therefore , the total number of sets chosen by the algorithm is as claimed. 6. CONCLUDING REMARKS • An interes ting op en problem is to d ecide if the algo- rithm presented here for th e admission control problem can b e derandomized. • Recently , F eige and Korman established a low er b ound of Ω(log m log n ) for the comp etitive ratio of any ran- domized p olynomial time algorithm for th e online set co ver p roblem, u nder the B P P 6 = N P assumption. I t is interesting to decide wheth er this low er boun d ap- plies for sup erp olynomial time algorithms as well. • The algorithms we ga ve for the admission con trol prob- lem did not use the fact that t he requests are simple paths in th e graph. 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