Better Bounds for Online Load Balancing on Unrelated Machines

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1 Beer Bound for Online Load Balancing on Unrelaed Machine Ioanni Caragianni Abrac We udy he roblem of cheduling ermanen ob on unrelaed machine when he obecive i o minimize he L norm of he machine load. he roblem i known a load balancing under he L norm. We reen an imroved uer bound for he greedy algorihm hrough imle analyi; hi bound i alo hown o be be oible wihin he cla of deerminiic online algorihm for he roblem. We alo addre he queion wheher randomizaion hel online load balancing under L norm on unrelaed machine; hi i a challenging queion which i oen for more han a decade even for he L 2 norm. We rovide a oiive anwer o hi queion by reening he fir randomized online algorihm which ouerform deerminiic one under any inegral L norm for 2,..., 37. Our algorihm eenially comue in an online manner a fracional oluion o he roblem and ue he fracional value o make random choice. he local oimizaion crierion ued a each e i novel and raher counerinuiive: he value of he fracional variable for each ob correond o flow a an aroximae Wardro equilibrium for an aroriaely defined non-aomic congeion game. A corollarie of our analyi and by exloiing he relaion beween he L norm and he makean of machine load, we obain new comeiive algorihm for online makean minimizaion, making rogre in anoher longanding oen roblem. Inroducion We udy he following claical cheduling roblem. We have m arallel machine and n indeenden ob, where ob i induce a oibly infinie oiive inegral load w i when roceed by machine. he load of a machine i he oal load of he ob aigned o i. he co of an aignmen or chedule of a e of ob i defined a he L norm of he load vecor, / i.e., l m l for finie and l max m { l } for he load vecor l l,..., l m. he goal of a load balancing algorihm i o aign all ob Reearch Academic Comuer echnology Iniue & Dearmen of Comuer Engineering and Informaic, Univeriy of Para, Rio, Greece. E mail: hi work wa arially uored by he Euroean Union under IS FE Inegraed Proec FP AEOLUS and by a Caraheodory gran from he Univeriy of Para. o machine o ha he co i minimized. hi i he mo general verion of he roblem known a load balancing or cheduling on unrelaed arallel machine. Imoran ecial cae are hoe wih relaed machine and rericed aignmen. In he relaed machine model, each machine ha a oiive eed and each ob i ha a weigh w i and he load of ob i when i i aigned o machine i w i /. he rericed aignmen model i imilar; he main difference i ha each ob ha a ube of ermiible machine where i can be cheduled. he load vecor of ob i in he rericed aignmen model can be hough of a w i / for hoe machine which are ermiible for ob i, and for all oher machine. Secial cae where all machine have he ame eed idenical machine or all ob have he ame weigh are alo inereing. Almo all verion of he roblem are known o be NP-hard [22]; hence, mo of he work ha focued on efficien aroximaion algorihm. A cenario which beer reflec racical iuaion i he one when he informaion abou he ob i no known in advance and i only revealed gradually. Comuaion i hen erformed in e; when a ob aear ogeher wih i load vecor, an online algorihm ha o make an irrevocable deciion and aign i o a machine. A naural online algorihm i he greedy algorihm for he L norm which aign each ob o ha machine ha minimize he increae of he -h ower of he co. he erformance of online algorihm i aeed hrough he noion of comeiive raio defined a he maximum raio over all equence of ob of he execed co of he algorihm over he oimal co. hi definiion i quie general o caure randomized algorihm and aee he erformance of he online algorihm again obliviou adverarie, i.e., adverarie which may have acce o he robabiliy diribuion ued by he algorihm o make random choice bu no o he random choice hemelve. Scheduling o minimize he L norm of he load vecor alo called he makean ha received much aenion. Lenra e al. [29] and Shmoy and ardo [34] rovided 2-aroximaion algorihm for unrelaed machine while he roblem ha been roved o be inaroximable wihin a raio beer han 3/2 [29].

2 Horowiz and Sahni [26] reened a olynomial-ime aroximaion cheme PAS when he number of machine i conan. For relaed machine, Hochbaum and Shmoy [25] ee alo [24] reened a PAS. In he online cae, Azar e al. [9] howed ha no randomized online algorihm can be beer han ln m-comeiive again obliviou adverarie even on he rericed aignmen model, where m i he number of machine. heir lower bound for deerminiic algorihm i log m. hey alo howed ha he greedy algorihm ha aign each ob o ha machine o ha he increae in he makean i minimized ha oimal comeiivene in he ame model. Ane e al. [3] oberved ha hi greedy algorihm i only Ωm-comeiive on m unrelaed machine. hey obained a deerminiic 4 log m-comeiive algorihm by combining everal ohiicaed echnique uch a exonenial weighing and doubling of eimae of he oimal makean. A randomized e log m-comeiive algorihm wa obained by inroducing randomized doubling o he algorihm of Ane e al.; hi reul i aribued o Indyk in [2]. he inereed reader may look in andard exbook on online comuaion uch a [3] for a coverage of hee reul and he relaed echnique. Conan comeiivene i oible for he relaed machine model [3, 2]. An imoran ecial cae i cheduling o minimize he makean on idenical machine. Graham [23] howed ha he greedy algorihm i 2 m comeiive in hi cae. he greedy algorihm i oimal only for m 3; for any m > 3, beer algorihm exi [2, 33]. Baral e al. [0] were he fir o how an algorihm whoe comeiive raio i below 2 δ for ome conan δ > 0 and arbirary m. See [] for he be known bound for he roblem. he L norm obecive ha been mainly inroduced ince, in many alicaion, he makean i no a uiable way o meaure how well he ob are balanced. Chandra and Wong [5] and Cody and Coffman [8] were he fir o conider he obecive of minimizing he um of quare of he machine load i.e., he quare of he L 2 norm. For unrelaed machine, Awerbuch e al. [6] howed ha he greedy algorihm ha comeiivene a mo + 2 under he L 2 norm and a mo c + Olog for higher norm, where c.7632 i he roo of he equaion c ln c. Furhermore, hey roved a lower bound of aroximaely for any deerminiic online algorihm. Among oher reul, Caragianni e al. [4] howed ha he uer bound of + 2 for he greedy algorihm under he L 2 norm i igh even for rericed aignmen. Beer comeiivene bound under he L 2 norm are known for rericed aignmen and ob wih equal weigh [4, 7, 35]. For he idenical machine cae, Avidor e al. [4] roved comeiivene bound of 4/3 under he L 2 norm and 2 O for higher norm. ln We oin ou ha he greedy algorihm rovided he be known aroximaion guaranee for he offline cae a well unil he recen work of Azar and Eein [7] who ued convex rogramming o comue fracional oluion o he roblem and rounding cheme o obain efficien inegral oluion. hee echnique yielded a randomized 2-aroximaion algorihm for he L 2 norm and deerminiic 2-aroximaion algorihm for higher norm. Beer aroximaion guaranee eecially for mall norm were reened in [28]. Prior o [7] and [28], 2-aroximaion algorihm for all norm imulaneouly had been reened in [8] for rericed aignmen while, in he ame aer, he roblem wa roved o be APX-hard for any L norm wih 2. Alon e al. [2] roved ha PASe are oible for rericed aignmen on idenical machine or for a conan number of unrelaed machine. Eein and Sgall [9] reened a PAS for relaed machine. Before reening our reul, we oen a arenhei o menion ha he recenly emerging field of Algorihmic Game heory [3] ha exenively conidered load balancing from a differen erecive. An aumion made here i ha he aignmen of ob i no cenrally conrolled bu, inead, each ob i owned by a elfih agen. Each agen favor aignmen of her ob o a machine o ha he laency he exerience i minimum given he aignmen of he ob conrolled by he oher agen. Following he eminal work of Kououia and Paadimiriou [27], a va amoun of he recen relaed lieraure udie how much he overall erformance of load balancing under everal obecive deeriorae a Nah equilibria, i.e., a aignmen where no agen ha an incenive o change he aignmen of her ob. Load balancing game are u ecial cae of aomic congeion game [5, 6] which model elfih behavior in nework rouing. Acually, load balancing can be hough of a rouing demand beween wo node hrough m arallel link. A lighly differen model which ha received much aenion recenly and whoe udy ha ared a lea half a cenury ago in he Economic and ranoraion lieraure ee [32] and he reference herein i ha of non-aomic congeion game in nework. We decribe he imle cae of uch game here; hee are he only game-heoreic definiion which we acually ue in he curren aer. In a non-aomic congeion game on m arallel link connecing a ource node o a deinaion node, here are infiniely many agen, each conrolling a negligible amoun of raffic flow from he ource o he deinaion o ha he oal amoun of raffic flow i uniy. Each link ha a non-negaive laency funcion f which indi-

3 cae ha he laency exerienced by all agen rouing heir raffic flow hrough i f x, where x i he oal amoun of flow roued hrough. Again, agen favor aignmen of heir flow o ha he laency exerienced i minimum given he aignmen of he oher agen. Aignmen from where no agen ha an incenive o deviae are called Wardro equilibria [36] and aify he following condiion: a flow vecor x i a a Wardro equilibrium if for any wo link, wih x > 0, i hold ha f x f x. Beckmann e al. [] have reened a convex oenial funcion whoe local minima correond o Wardro equilibria; hence, Wardro equilibria for non-aomic congeion game in nework can be arbirarily aroximaed uing convex rogramming echnique e.g., [30]. See alo [20] and he reference herein for recen diribued aroximaion of uch equilibria. Much eaier and faer aroximaion of Wardro equilibria are oible for game on arallel link rovided ha he laency funcion have ome reaonable form e.g., hey are olynomial. In hi aer, we udy he online oimizaion verion of load balancing in a coordinaed way. Alhough everal recen aer have conidered imoran ecial cae of online load balancing, no rogre on he general roblem wih he L norm obecive ha been made ince he work of Awerbuch e al. [6] in 995. We make uch a rogre and imrove mo of he reul of [6]. We renghen and ignificanly imlify he analyi of he greedy algorihm and how ha i i a mo 2 / - comeiive under he L norm. We alo how ha hi bound i igh for any, i.e., we how ha no deerminiic online algorihm can have beer comeiive- ne han greedy. Our bound aroache ln from below a increae. A a corollary of our analyi and by exloiing he relaion of he L norm and he makean of machine load, we obain ha he greedy algorihm for he L ln m norm i e log m-comeiive for makean minimizaion on m unrelaed machine. hi i a raher urriing reul ince i mean ha he ame comeiivene wih he randomized verion of he algorihm of Ane e al. [3] can be obained by a much imler deerminiic greedy-like algorihm. hee reul are reened in Secion 3. In Secion 4, we reen he mo inereing and echnically involved reul of he aer. We demonrae ha we can bea he lower bound on he comeiivene of deerminiic algorihm for load balancing under L norm by uing randomizaion. We reen a general algorihm called Balance which i arameerized by and a arameer vecor α. he algorihm i waching he momen of he random variable denoing he load on each machine and, a each e, i ue hi informaion ogeher wih he ob load in order o comue he robabiliy diribuion according o which i will make he deciion for he ob conidered. hi can be hough of a comuing a fracional oluion o he roblem which i rounded in a randomized way in order o obain an inegral aignmen. A each e, he crierion ued in order o comue he robabiliy diribuion of he random choice ugge a nice inerlay wih game. he algorihm conider a non-aomic congeion game on arallel link correonding o he machine wih aroriaely defined laency funcion, comue an aroximae Wardro equilibrium, and e he robabiliie of he random choice equal o he flow in hi equilibrium. Noe ha, unlike he aroach in Algorihmic Game heory menioned above, he game i ued a a ool by our algorihm in order o coordinae he aignmen of ob efficienly. Our analyi lead o ufficien condiion for he elecion of he arameer vecor α. Uing hee condiion, we are able o comue aroriae arameer o ha algorihm Balance i a mo.222-comeiive again obliviou adverarie for inegral value of u o 37. For he L 2 norm, our algorihm i 5-comeiive. By exloiing he relaion of L norm o he makean, we alo obain he fir comeiivene uer bound for makean minimizaion which bea he reul of Ane e al. [3] when he number of machine i u o an aronomically large conan i.e., e 37. We begin wih ome reliminary definiion in Secion 2 and conclude wih exenion and oen roblem in Secion 5. Due o lack of ace, ome roof have been omied. 2 Preliminarie We briefly reen he noaion ued in he analyi of our algorihm for he L norm. For an ineger k, we uually ue [k] o denoe he e {, 2,..., k}. When we conider an inance of he roblem, we denoe by l he load of machine in he oimal oluion. We ue he binary variable y i {0, } o denoe wheher ob i i aigned o machine in he oimal oluion y i or no y i 0. Clearly, i y iw i l and i y iwi i y iw i l for any. We alo denoe by Λ i he load of machine afer he aignmen of ob i. Hence, in order o rove uer bound on he comeiive [ raio, i uffice o bound he raio ] / / of IE Λ n or imly Λ n when he /. greedy algorihm i concerned over l In he analyi we ue he Minkowki inequaliy or he riangle inequaliy for he L norm aing ha k / k / + a + b / k a b

4 for any and a, b 0, a well a Hölder inequaliy aing ha IE[Z] IE[Z ] / for any non-negaive random variable Z and ee wikiedia.org for a deailed reenaion of hee inequaliie. Our reul for makean minimizaion follow by he analyi of he algorihm for he L norm and he following obervaion. Lemma 2.. Le A be a c-comeiive online algorihm for load balancing on unrelaed machine under he L norm. hen, algorihm A i cm / -comeiive for makean minimizaion on m unrelaed machine. 3 Deerminiic online algorihm In hi ecion we comue he exac comeiivene bound of he greedy algorihm for he L norm and we how ha hi i he be oible among deerminiic online algorihm. heorem 3.. For any, he greedy algorihm ha comeiive raio a mo for load balancing 2 / on unrelaed machine under he L norm. Proof. A he e aociaed wih ob i, we have Λ i 3. Λ i, Λi, + y i w i Λ i, Λn + y i w i Λ n where he fir inequaliy follow by he greedy naure of he algorihm and he econd inequaliy follow ince Λ n Λ i for any i, and he funcion fz z +a z i non-decreaing in [0, for a 0 and. Nex we will need he following echnical lemma. Lemma 3.. Le, 0 and a i 0, for i,..., k. hen, k k + a i + a i. Uing 3. and umming over all e of he algorihm, we obain ha 3.2 i i Λ n Λ i Λ i, Λn + y i w i Λ n Λn + y i w i Λ n i Λ n + i Λ n + l Λ n y i w i Λ n + Λ n l Λ n he econd inequaliy follow by Lemma 3. while he la inequaliy i derived by alying Minkowki inequaliy. We divide all ide of 3.2 by l and e c n Λ / l. Now 3.2 become 2c c+ which yield he deired reul c 2 /. Uing he inequaliy e z + z, we have ha 2 / ln hi imrove he reviou bound of Olog of [6]. We alo how ha greedy i oimal wihin he cla of deerminiic online algorihm for any L norm; hi reul alo imrove he reviouly be known lower bound of of [6]. heorem 3.2. For any ɛ > 0 and any, no deerminiic online load balancing algorihm on unrelaed machine can be beer han 2 / ɛ-comeiive under he L norm. Proof. We ue a imilar conrucion wih [6] i.e., rericed aignmen on idenical machine bu we alo allow ob have differen load. hi alo generalize a conrucion ued in [4] o how a lower bound of + 2 on he comeiivene of he greedy algorihm under he L 2 norm. Le δ > 0 and ineger k be arameer o be defined laer and le be uch ha +δ /. Conider he execuion of a deerminiic online algorihm on m 2 k machine and an adverary ha reveal ob in k hae. Iniially, all 2 k machine are acive. In each hae, he adverary mache he acive machine ino air. In he hae i i 0,..., k, for each air of machine a, b, he adverary reen one ob wih load 2 i/ on machine a and b, and infinie load on any oher machine. he machine ha are aigned ob by he algorihm remain acive for he nex hae; machine ha are no aigned ob become inacive. Denoe by o and de he -h ower of he co of he oimal aignmen and he co of he aignmen comued by he deerminiic algorihm, reecively. We oberve ha a machine ha become inacive immediaely afer hae i for i 0,..., k i no aigned any ob a hae i and laer hae while i i aigned one ob in each hae before hae i if

5 any. he machine ha i aigned he ob a hae k i alo aigned one ob in each hae before hae k. Since 2 k i machine become inacive immediaely afer hae i, we have ha 3.3 de k 2 / 0 k 2 k i k + 2 k i i 2 / 0 i 2 / 0 k 2 k i 2 i/ 2 / k 2 k 2 i/ 2 / k 2 k 2 i/ 2 / 2 k k 2 i/ 2 / 2 k k + 2 / 2 / 2 k k / he econd inequaliy follow ince a a for any a [0, ] and while he la inequaliy follow ince 2 / i increaing for and, hence, 2 / /2 which yield 2. 2 / In order o bound o from above, conider he aignmen in which each ob i aigned in ooiion o he algorihm. Given a ob wih finie load on a air of machine a, b, he ob i aigned o a if he algorihm aign i o b, and vice vera. hen, each machine ha a mo one ob and uing he relaion beween and δ, we obain k 3.4 o 2 k i 2 i/ i0 k 2 k i 2 i/ + i0 + δk2 k. By comaring 3.3 and 3.4, we obain ha for any ɛ > 0 here are ufficienly large k and ufficienly mall δ o ha he raio of he co of he aignmen comued by he deerminiic algorihm over he oi- mal co i de o / 2 / ɛ. We remark ha no rericion on he number of machine aear in he aemen of heorem 3.2. Our lower bound conrucion eenially how ha when o log m here exi an inance wih m machine on which any deerminiic online algorihm ha comeiive raio arbirarily cloe o 2 /. By ighening he inequaliie ued in order o obain 3.3, we can how ha he ame hold when o. log m log log m he lower bound of heorem 3.2 definiely doe no hold for ωlog m ince, in hi cae, he Olog m- comeiive algorihm for makean minimizaion can be eaily roved o be Olog m-comeiive under he L norm. We conclude hi ecion by reening a reul for makean minimizaion. I follow a a corollary of our analyi of he greedy algorihm heorem 3. and Lemma 2.. heorem 3.3. he greedy algorihm for he L ln m norm i e log m-comeiive for makean minimizaion, where m i he number of unrelaed machine. he greedy algorihm for he L ln m norm ha everal advanage over he e log m-comeiive randomized algorihm obained by he echnique of Ane e al. [3] exended wih randomized doubling. Fir, i i deerminiic. Second, i doe no ue exonenial weighing and, hence, olynomial ace i alway ufficien in order o erform comuaion. Acually, he greedy algorihm for he L ln m norm can be hough of a uing ubexonenial weighing. hird, i doe no ue eimae of he oimal makean or doubling. 4 Randomized online algorihm In hi ecion we reen he fir randomized online algorihm for load balancing on unrelaed machine ha bea he lower bound on he comeiivene of deerminiic one under L norm. Our algorihm i called Balance, ue a vecor α of oiive value, and work a follow. When a new ob i arrive, Balance comue a robabiliy diribuion on he machine, i.e., robabiliie x i ha ob i i aigned o machine. hen, i imly ca a die ha ha one face for each machine wih x i > 0 wih x i being he robabiliy ha he face correonding o machine i he oucome of he die caing and aign ob i o he machine correonding o he oucome of die caing. Moivaed by he greedy algorihm, a naive way o comue he robabiliie in each e could be o [ ] minimize he increae in IE Λ n IE[Λ n ] due o he deciion a he e. Unforunaely, i i no hard o lighly modify he roof of heorem 3.2 and conruc inance where uch an algorihm mimic he greedy algorihm and canno achieve a

6 beer comeiive raio. Inead, algorihm Balance i waching all he inegral momen of he machine load and, a each e, i make i deciion rying o balance he increae in each of hem. In order o do o, i define aroriae game a each e, comue equilibria for hem, and make i deciion according o hee equilibria. A each e i, he algorihm conider a aricular non-aomic congeion game wih a uni amoun of flow ha ha o be roued from a ource node o a deinaion node which are conneced wih m arallel link. Each link correond o machine of he load balancing inance and ha an aroriaely defined non-decreaing laency funcion f i. Algorihm Balance comue an ɛ-aroximae Wardro equilibrium wih ɛ α 2 m, i.e., a flow vecor x i uch ha for any wo link, [m] wih x i > 0, i hold ha f i x i f i x i ɛ. hi alo yield 4.5 x i f i x i ɛ f i x i for any link [m]. In order o define he laency funcion on he link, he algorihm kee rack of auxiliary funcion g i for i [n], [m], and 0,,..., defined a follow. A he fir e, he algorihm e g z zw i if > 0, and g 0 z for any [m] and z 0. For i >, during e i, he algorihm define he auxiliary funcion g i for [m] and 0,..., uing he auxiliary funcion defined in he reviou e, he vecor of robabiliie x i comued a he reviou e, and he load of ob i on he machine, a follow: g i z z 0 +g i, x i, g i, x i, w i Alhough comlicaed a fir glance, he auxiliary funcion have been defined in uch a way ha g i x i IE[Λ i ]. hi i aed in he following lemma ogeher wih oher roerie of he auxiliary funcion g i ha will be ueful in our analyi. Lemma 4.. For any i > 0, [m], and 0,...,, he following roerie hold:. g i z zg i g i 0+g i 0, for any z g i 0 g i 0w i. 3. g i x i IE[Λ i ] IE[Λ n ]. 4. g i z u g i 0 u uz g i z u g i g i 0, for any u and z 0. Proof. Proerie and 2 follow rivially by he definiion of g i. he inequaliy in Proery 3 i obviou. In order o rove he equaliy in Proery 3, we will ue inducion on i. For i, if 0, we have IE[Λ 0 ] g 0x by definiion. Alo, if > 0, hen Λ i w wih robabiliy x and 0 wih robabiliy x i, i.e., IE[Λ ] x w g x by definiion. Now aume ha IE[Λ i ] g i x i for any i < i and 0,...,. hen, he random variable Λ i equal Λ i, + w i wih robabiliy x i and Λ i, wih robabiliy x i. Hence, uing lineariy of execaion, he inducive hyohei, and he definiion of auxiliary funcion g i, we have [ ] IE[Λ i] IE x i Λ i, + w i + x i Λ i, [ ] IE x i Λ i,w i + Λ i, 0 x i 0 x i 0 +g i, x i, g i x i. IE[Λ i,]w i + IE[Λ i,] g i, x i, w i Proery 4 clear hold if z 0 or if g i i a conan funcion. In order o rove i for z > 0 and when i i no conan, conider he funcion hz z + c u for ome non-negaive conan c. Since h i convex in [0, + for u, i derivaive a oin z 2 > 0 i no maller han he loe of he line croing oin 0, h0 and z 2, hz 2, i.e., hz 2 h0 z 2 uz 2 + c u which yield hz 2 h0 uz 2 + c u z 2. Proery 4 hen follow by eing z 2 zg i g i 0 and c g i 0 ince by Proery hz 2 zg i g i 0 + g i 0 g i z. he laency funcion aociaed wih link i defined a f i z α g i z / 0 g i zw i. hi comlee he decriion of he algorihm. So far, i hould have become clear ha he algorihm doe no ue he oucome of reviou random choice in order o make deciion a any e. Inead, i ue he momen of he random variable denoing he load on each machine which in urn deend only on he robabiliy diribuion of reviou random choice. hi nice roery make he analyi of he algorihm

7 racable. We are now ready o rove he following aemen which characerize he comeiivene of algorihm Balance in erm of and he arameer vecor α. Lemma 4.2. Le 2 be an ineger. If here exi oiive number ξ > 0 for 0,..., uch ha he value of vecor α aify α ξ 4.6 ξ 0 κ + α κ ξ κ κ ξ α κ+ for,...,, hen algorihm α, -Balance i β/α / -comeiive again obliviou adverarie for load balancing on unrelaed machine under he L norm, where β α ξ 0 ξ. Proof. We wih o how ha he execaion of / he random variable X Λ n i a mo β /. α l By Hölder inequaliy on execaion of non-negaive random variable, we have ha IE[X] IE[X ] / and, hence, i uffice o how ha [ ] / IE Λ n β /. α l By lineariy of execaion, we have α IE Λ n α IE [ Λ ] 4.7 n α IE [ Λ ] / n α IE [ Λ /. n] We will work on he fir um a he righ-hand ide of 4.7. Uing he roerie of funcion g i, we obain 4.8 n n α IE [ Λ ] / n α IE [ Λ ] / [ ] i IE Λ / i, α g i x i / g i 0 /. Nex we ue a echnical lemma which follow by he definiion of funcion f i and he roerie of funcion g i. Lemma 4.3. A each e i [n] and for each machine [m], i hold ha α g i x i / g i 0 / x i f i x i x 2 iα 2. Proof. We ue he roerie, 2, and 4 of funcion g i from Lemma 4. o obain x i α g i x i / g i 0 / x i x i x i 0 α g ix i / g i g i 0 α g ix i / α g ix i / 0 0 α g ix i / x i f i x i x 2 i 0 x i f i x i x 2 i 0 g i x i g i 0 w i α g ix i / w i g i g i 0 u0 α g ix i / u x i f i x i x 2 iα 0 x i f i x i x 2 iα 2. g iu 0w u i g i0 0w i g i 0w i g i x i w i he la inequaliy follow ince g i0 0 and w i. Now, working wih 4.8 and uing Lemma 4.3 and he inequaliy x2 i /m ince x i, we obain 4.9 α IE [ Λ ] / n

8 n n x i f i x i α 2 x i f i x i α 2. m Since he flow vecor x i i an α 2 m -aroximae Wardro equilibrium for he congeion game conidered a e i, we ue 4.5 and 4.9 o obain a relaion of he lef-hand ide of 4.9 o he deciion in he oimal aignmen, i.e., α IE [ Λ / n] x 2 i n y i f i x i. By ubiuing f i, uing he roery g i x i IE[Λ n ] for any i [n], [m], and 0,...,, and ince n y iwi l for any [m] and [], we obain a more clear relaion o he oimal aignmen: 4.0 n 0 α IE [ Λ ] / n α g i x i y i w i n α 0 y i w i n 0 y i w i 0 α α g i x i / IE [ Λ ] / [ ] n IE Λ n IE [ Λ ] / [ ] n IE Λ n IE [ Λ ] / [ ] n IE Λ n l. In order o uer-bound he righ-hand ide of 4.0, we will ue he following echnical lemma. Lemma 4.4. Le γ, δ be non-negaive ineger uch ha γ + δ and ζ, ζ 2 > 0. hen, 4. z γ zδ 2z γ δ 3 γ ζγ ζ δ 2z + δ ζγ ζδ 2 z 2 for any z, z 2, z γ + δ ζ γ ζδ 2z 3 Given ζ and ζ 2, wha Lemma 4.4 eenially doe i o bound he lef-hand ide of 4. wih a olynomial of he form c z + c 2z 2 + c 3z 3 o ha he bound i igh when z ζ and z 2 ζ 2. We aly Lemma 4.4 o each erm a he righ-hand ide of 4.0. In aricular, alying Lemma 4.4 wih z IE [ /, Λn] z 2 IE [ Λ n] /, z3 l, ζ ξ, ζ 2 ξ, γ, and δ, we obain IE [ Λ ] / [ ] n IE Λ n l and, hence, 4.0 yield α IE [ Λ ] / n 0 +ξ + ξ κ+ + α IE [ Λ n α ξ α κ κ α ξ 0 + ξ + ξ ξie [ Λ ] / n ξ ξ ξl IE [ Λ ] / n ξ ξie [ Λ ] / n ] / + ξ ξ 0 ξκ κ ξ α IE [ Λ ] / n + β ξ l l ξ l IE [ Λ ] / n where he la inequaliy follow by 4.6[ and by he ] definiion of β. Hence, 4.7 yield IE Λ n β α l and Lemma 4.2 follow. By alying Lemma 4.2, we can eaily obain he following reul for he L 2 norm. Corollary 4.. Algorihm α, 2-Balance wih α α 2 i 5-comeiive again obliviou adverarie for load balancing on unrelaed machine under he L 2 norm. Proof. Alying Lemma 4.2 wih ξ 0 ξ ξ 2 2, we have ha inequaliy 4.6 i aified and β 5. For larger norm, comuing he be oible value for vecor α o ha here exi oiive ξ ha

9 aify he condiion 4.6 of Lemma 4.2 and imulaneouly minimize β i no eay ince hee condiion are quie comlicaed. So, in order o comue good value for he arameer of algorihm Balance, we reaon a follow. We will denoe by ˆα he correonding arameer vecor. Fir, we normalize vecor ˆα by eing ˆα. We gue aroriae value ˆξ for ξ and require ha he condiion 4.6 of Lemma 4.2 are aified wih equaliy. hi yield a yem of linear equaion in which he ˆα for,..., are he unknown and which can be rereened a Aˆα b where A i an uer riangular marix wih A, ˆξ ˆξ 0 for,...,, A,κ 0 for,..., and κ,...,, and κ A,κ ˆξ κ κ ˆξ for,..., and κ +,...,, and b ˆξ for,...,. Of coure, we mu gue he ˆξ o ha hi yem of linear equaion yield oiive value for he arameer ˆα. he following lemma i roof i omied rovide a ufficien condiion for hi. Lemma 4.5. If ˆξ i increaing in, and ˆξ, hen he yem of linear equaion Aˆα b ha a unique oiive oluion. We have ued ˆξ and have olved he correonding yem of linear equaion for inegral value of u o 37 by imlemening a imle back ubiuion rouine in C. he limiaion on come from he range of number he double daa ye of C can rereen. Our reul ugge ha, for he aricular elecion of he arameer vecor ˆα, he comeiive raio of algorihm ˆα, -Balance i alway beer han he lower bound for deerminiic online algorihm and a mo.222. A a corollary, uing Lemma 2., we obain imroved comeiivene for online makean minimizaion a well when he number of machine i u o an aronomically large conan. he nex aemen ummarize he dicuion of hi ecion. heorem 4.. Algorihm ˆα, -Balance ha comeiive raio a mo.222 again obliviou adverarie for load balancing under he L norm for inegral 37. Algorihm ˆα, ln m -Balance ha comeiive raio a mo.222e ln m log m again obliviou adverarie for makean minimizaion on m e 37 unrelaed machine. 5 Exenion and oen roblem Our load balancing algorihm for L norm can be adaed o work wih imilar comeiivene by making he analyi lighly more comlicaed in nework rouing when he obecive i o minimize he oal laency or he maximum laency er link and he delay funcion on he nework link are olynomial; he adaaion include aroximaing he Wardro equilibrium of more general non-aomic congeion game in nework. Concerning oen roblem, here are ill many. Fir, i would be inereing o heoreically rove uer bound on he comeiivene of algorihm Balance for any value of. We conecure ha hee bound will be only marginally larger han hoe comued in Secion 4. Furhermore, he queion abou he limi of randomizaion for online load balancing under he L norm i very inereing even for he L 2 norm. Finally, alhough we have made ome rogre on online makean minimizaion on unrelaed machine, he ga beween our uer bound and he lower bound of [9] remain. For hi roblem, i i even widely oen wheher randomizaion i really neceary in order o obain he be oible comeiivene. Our reul in Secion 4 ugge ha game equilibria can be ueful in online combinaorial oimizaion. We lan o inveigae hi relaion more yemaically in he fuure by conidering differen online roblem. Reference [] S. Alber. On randomized online cheduling. In Proceeding of he 34h Annual ACM Symoium on heory of Comuing SOC 02, , [2] N. Alon, Y. Azar, G. J. Woeginger and. Yadid. Aroximaion cheme for cheduling. In Proceeding of he 8h Annual ACM-SIAM Symoium on Dicree Algorihm SODA 97, , 997. [3] J. Ane, Y. Azar, A. Fia, S. Plokin, and O. Waar. On-line rouing of virual circui wih alicaion o load balancing and machine cheduling. Journal of he ACM, 443, , 997. [4] A. Avidor, Y. Azar, and J. Sgall. Ancien and new algorihm for load balancing in he L norm. Algorihmica, 29, , 200. [5] B. Awerbuch, Y. Azar, and A. Eein. he rice of rouing unliable flow. In Proceeding of he 37h Annual ACM Symoium on heory of Comuing SOC 05, , [6] B. Awerbuch, Y. Azar, E. F. Grove, M.-Y. Kao, P. Krihnan, and J. S. Vier. Load balancing in he L norm. In Proceeding of he 36h Annual IEEE Symoium on Foundaion of Comuer Science FOCS 95, , 995. [7] Y. Azar and A. Eein. Convex rogramming for cheduling unrelaed arallel machine. In Proceeding

10 of he 37h Annual ACM Symoium on heory of Comuing SOC 05, , [8] Y. Azar, L. Eein, Y. Richer, and G. Woeginger. Allnorm aroximaion algorihm. Journal of Algorihm, 522, , [9] Y. Azar, J. Naor, and R. Rom. he comeiivene of on-line aignmen. Journal of Algorihm, 82, , 995. [0] Y. Baral, A. Fia, H. J. Karloff and R. Vohra. New algorihm for an ancien cheduling roblem. Journal of Comuer and Syem Science, 53, , 995. [] M. Beckmann, C. B. McGuire, and C. B. Winen. Sudie in he Economic of ranoraion, Yale Univeriy Pre, 956. [2] P. Berman, M. Charikar, and M. Karinki. On-line load balancing for relaed machine. Journal of Algorihm, 35,. 08-2, [3] A. Borodin and R. El-Yaniv. Online comuaion and comeiive analyi. Cambridge Univeriy Pre, 998. [4] I. Caragianni, M. Flammini, C. Kaklamani, P. Kanellooulo, and L. Mocardelli. igh bound for elfih and greedy load balancing. In Proceeding of he 33rd Inernaional Colloquium on Auomaa, Language, and Programming ICALP 06, LNCS 405, Sringer, Par I, , [5] A. K. Chandra and C. K. Wong. Wor-cae analyi of a lacemen algorihm relaed o orage allocaion. SIAM Journal on Comuing, 43, , 975. [6] G. Chriodoulou and E. Kououia. he rice of anarchy of finie congeion game. In Proceeding of he 37h Annual ACM Symoium on heory of Comuing SOC 05, , [7] G. Chriodoulou, V. S. Mirrokni, and A. Sidirooulo. Convergence and aroximaion in oenial game. In Proceeding of he 23rd Annual Symoium on heoreical Aec of Comuer Science SACS 06, LNCS 3884, Sringer, , [8] R. A. Cody and E. G. Coffman, Jr. Record allocaion for minimizing execed rerieval co on crumlike orage device. Journal of he ACM, 23,. 03-5, 976. [9] L. Eein and J. Sgall. Aroximaion cheme for cheduling on uniformly relaed and idenical arallel machine. Algorihmica, 39, , [20] S. Ficher, H. Räcke, and B. Vöcking. Fa convergence o Wardro equilibria by adaive amling mehod. In Proceeding of he 38h Annual ACM Symoium on heory of Comuing SOC 06, , [2] G. Galambo and G. Woeginger. An online cheduling heuriic wih beer wor cae raio han graham li cheduling. SIAM Journal on Comuing, 222, , 993. [22] M. R. Garey and D. S. Johnon. Comuer and Inracabiliy, W. H. Freeman and Comany, 979. [23] R. L. Graham. Bound for cerain muliroceor anomalie. Bell Syem echnical Journal, 45, , 966. [24] D. S. Hochbaum and D. B. Shmoy. Uing dual aroximaion algorihm for cheduling roblem heoreical and racical reul. Journal of he ACM, 34, , 987. [25] D. S. Hochbaum and D. B. Shmoy. A olynomial aroximaion cheme for cheduling on uniform roceor: uing he dual aroximaion aroach. SIAM Journal on Comuing, 73, , 988. [26] E. Horowiz and S. Sahni. Exac and aroximae algorihm for cheduling non-idenical roceor. Journal of he ACM, 23, , 976. [27] E. Kououia and C. Paadimiriou. Wor-cae equilibria. In Proceeding of he 6h Inernaional Symoium on heoreical Aec of Comuer Science SACS 99, LNCS 563, Sringer, , 999. [28] V. S. Anil Kumar, M. V. Marahe, S. Parhaarahy, and A. Srinivaan. Aroximaion algorihm for cheduling on mulile machine. In Proceeding of he 46h Annual IEEE Symoium on Foundaion of Comuer Science FOCS 05, , [29] J. K. Lenra, D. B. Shmoy, and E. ardo. Aroximaion algorihm for cheduling unrelaed arallel machine. Mahemaical Programming, 46, , 990. [30] Y. Neerov and A. Nemirokii. Inerior-oin olynomial algorihm in convex rogramming. SIAM Sudie in Alied Mahemaic, SIAM, 994. [3] N. Nian,. Roughgarden, E. ardo, and V. V. Vazirani. Algorihmic game heory, Cambridge Univeriy Pre, 2007, o aear. [32]. Roughgarden and E. ardo. How bad i elfih rouing? Journal of he ACM, 492: , [33] S. S. Seiden. Online randomized muliroceor cheduling. Algorihmica, 282, 73-26, [34] D. B. Shmoy and E. ardo. An aroximaion algorihm for he generalized aignmen roblem. Mahemaical Programming, 62, , 993. [35] S. Suri, C. óh and Y. Zhou. Selfih load balancing and aomic congeion game. Algorihmica, 47, , [36] J. G. Wardro. Some heoreical aec of road raffic reearch. In Proceeding of he Iniue of Civil Engineer, P. II, , 952.

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