Competitive Algorithms for an Online Rent or Buy Problem with Variable Demand

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1 Competitive Algorithms for Olie Ret or Buy Prolem with Vrile Demd Roh Kodilm High Techology High School, Licroft, NJ Astrct We cosider geerliztio of the clssicl Ski Retl Prolem motivted y pplictios i cloud computig We develop determiistic d proilistic olie lgorithms for ret/uy decisio prolems with time-vryig demd We show tht these lgorithms hve competitive rtios of d 8 respectively We lso further estlish the optimlity of these lgorithms Itroductio The Ski Retl prolem is the coicl exmple of clss of olie ret or uy prolems [9] I the trditiol ski retl prolem, m visits ski resort, ot kowig how my dys he will e le to ski The m c either ret skis t cost of r dollrs per dy, or uy the skis permetly for dollrs The dilemm rises from the fct tht the umer of dys ville for skiig is ot kow i dvce; if there re my dys of skiig it is etter to uy the skis to void pyig the retl fee every dy, ut if there re oly few dys ville for skiig it my e more ecoomicl to ret the skis isted Applictios of the ski-retl prolem iclude schedulig tsks o computer (the system must decide if it should do tsk immeditely y pyig high cost i processig time, or if it should wit d py retl cost for keepig the tsk i witig)[7] d cchig dt (the system decides if it should red lock of dt durig every pss t cost of us cycle, or if it should pss over the dt lock d use severl us cycles to ccess the dt if it is eeded lter) [] Severl extesios of the Ski Retl prolem hve lredy ee studied, icludig vrits where the plyer c switch ewtee two retig optios t cost [], d more complex scerios where there re more th two optios to choose etwee [6] I these cses, e-competitive lgorithm c e foud A turl extesio of these situtios - where there re multiple retig d multiple uyig optios - hs lso ee studied [] Algorithms hve lso ee developed to provide strtegy whe the retl cost r is free to vry with time [] Aother more complex extesio studied is how to llocte cpcity ville through retig or uyig to grph s edges to llow for sufficiet flow etwee sources d siks [] These vrits dd complexity to the ski retl prolem, d s result the competitive rtio vries s fuctio of the ew prmeters itroduced y ech vritio We cosider geerliztio of the ski-retl prolem where demd vries cross time The motivtio for solvig this prolem is the ide of cloud urstig tht is used to ddress the computig Copyright SIAM Uuthorized reproductio of this rticle is prohiited

2 eeds for eterprise [] The mout of computig tht the eterprise requires (defied i terms of umer of computer servers) vries cross time The eterprise c meet this requiremet y uyig these servers d cretig privte etwork or c ret servers temporrily from cloud service provider (pulic cloud) like Amzo This ide of meetig computig requiremet usig comitio of iterl d exterl cloud resources is clled cloud urstig The prolem tht the eterprise hs to solve is to decide whe d how my servers to uy d how my to ret from the cloud Typiclly the computig resource requiremet c vry sigifictly over time It would e prohiitively expesive to hdle the pek lods y uyig servers Similrly, it my ot e ecoomicl to ret cpcity from the pulic cloud to hdle the etire computig requiremet Sice the computig requiremet will, i geerl, ot e kow hed of time, the uy or ret decisio hs to e mde i olie mer There re severl prcticl cosidertios tht go ito solvig the prolem i the rel world ut the model tht we cosider i this pper strcts importt spect of the uy/ret decisio Moreover, this strctio models other olie ret or uy decisio prolems with vrile demd Note tht if the demd is oe uit for set of cosecutive itervls d the ecomes zero the it models the stdrd ski-retl prolem Prolem Defiitio We ow formlly defie the prolem tht we study i this pper sider demds for some good tht rrives to system We ssume tht demds rrive oce per time period For simplicity, we cll ech time itervl dy The demd for dy i will e represeted y di, with di The demd c e stisfied i two wys The user c ret item t cost of r per dy, d this item will stisfy oe uit of demd for oe dy The user c lso uy item t cost of r, i which cse the item will stisfy oe uit of demd for dy i d ll susequet dys The system ojective is to determie the uy d ret optios to stisfy the demd t miiml cost There re two versios of the prolem tht c e cosidered Offlie Prolem: I the offlie versio of the prolem, ll of the demds re kow to the user hed of time, d the cost optimiztio is therefore performed with complete kowledge of ll the demds t the egiig Olie Prolem: I the olie versio of the prolem, oly the vlues of d r re kow iitilly, ut ech demd di is give to the user o dy i The olie user lso does ot kow the umer of time periods durig which demds will rrive The user must perform the cost optimiztio without y kowledge of future demds The optiml offlie cost provides lower oud o the cost chieved y y olie lgorithm sice the offlie lgorithm hs more kowledge th the olie lgorithm I this pper, we re iterested i derivig lgorithms to solve the olie prolem d mesurig their performce Mesurig the Performce of Olie Algorithms We mesure the performce of olie lgorithm y comprig the cost icurred y tht lgorithm to tht of the cost icurred y the optimum offlie lgorithm The optiml cost will e fuctio of the iput vector of demds The demd for dys c e represeted s dimesiol vector D (d, d,, d ) Let (D) represet the cost of olie lgorithm d

3 Cof f (D) represet the cost of the optimum offlie lgorithm opertig o iput D The competitive rtio ρ of the olie lgorithm is the defied s ρ mx D (D) Cof f (D) I other words, the competitive rtio is the worst cse rtio of the costs of the olie lgorithm to the optimum offlie lgorithm over ll demd vectors over y umer of dys Note tht the costs icurred y the lgorithms d hece the competitive rtio will e fuctio of the per-dy retl cost r, the cost of uyig d the demd set D Note tht without loss of geerlity we c ssume tht the per-dy retl cost r This c e doe y resclig, tht is, settig the vlue of to /r For the ese of presettio, we lso ssume tht is itegrl This is purely doe for coveiece d ll the results i the pper c pplied to the cse where the rescled vlue of is ot itegrl y settig to de We use the followig ottio i the lysis of the lgorithms: sider the demds for the first k dys (d,d,, dk ) Let π e permuttio of the set {,,, k} such tht dπ() dπ() dπ(k) Assume tht ties re roke ritrrily We use hk to deote the th highest of the first k demds If > k, the hk Therefore hk dπ() We defie Sk {π(), π(), π()} to deote the idices of the highest vlues out of the k rrivls We ow stte simple result tht is used i Sectio Lemm Let Sk deote the idices of the highest demds out of the first k rrivls d hk deote the th highest vlue out of the first k rrivls If dk > hk the i Sk di di + dk hk i Sk Proof Sice dk > hk, idex k Sk d idex with vlue hk is ot i the set Sk We ow study the offlie optimiztio prolem d derive the optiml offlie cost Offlie Prolem I the offlie prolem, ll of the demds re kow hed of time Sice the vlue of does ot vry over time d ll demds re kow iitilly, we c ssume tht the optimum offlie solutio uys ll the eeded goods iitilly I other words, if optimum offlie solutio uys xt uits o some dy t, the the solutio vlue does ot icrese if xt is ought iitilly The cost of uyig still remis xt for these xt items d there is opportuity to use these items t some dy efore dy t d perhps reduce the retl costs Therefore, the offlie lgorithm cosists of purchsig x items t the egiig d retig ll other items s eeded As efore, let the demds form the vector D The optimum offlie lgorithm is clled Aof f (D) d the totl cost it icurs is defied s Cof f (D) Let x deote the umer of items ought iitilly d yi, the umer reted o dy

4 i The miiml cost c e foud y settig up the followig lier progrm: miimize x + yi i suject to x + yi di i [,] x yi i [,] Where [,] represets the itegers etwee d, iclusive of oth edpoits The vriles x d yi must e itegrl, ut sice the lier progrmmig relxtio ove leds to itegrl optiml solutios we solve the lier prolem isted of the iteger progrm The ojective fuctio of this lier progrm represets the totl cost of the lgorithm Cof f (D), which we wt to miimize The first costrit esures tht the demd is met ech dy The other costrits esure tht x d yi re o-egtive To fid the optiml solutio, let us tke the dul of the lier progrm The dul [8] is mximize suject to i di zi zi i zi, i [,] This prolem is solved y ssigig zi for the highest demds Recll tht hk represets thepth highest demd vlue mog the first k rrivls The optiml solutio to the dul prolem is i S di If priml solutio c e costructed with the sme ojective fuctio vlue s the dul, tht priml solutio is optiml y strog dulity To costruct priml solutio with the sme optiml vlue, the vriles x d yi re set so tht x h ( yi di h if i / S if i S This correspods to uyig h items immeditely d retig y more items if di > h The cost Cof f (D) is the just the optiml vlue of the priml ojective fuctio, so Cof f (D) di () i S As this vlue is equl to tht of the dul ojective fuctio, it is optiml y strog dulity Offlie Algorithm Exmple Let us cosider the exmple where, d D is {,,,,8,,6} The lgorithm suggests tht the user should uy h h7 items immeditely I this cse, h7 items should e ought Doig so 6

5 dds cost of to the cost icurred With regrds to the retig costs, it is oly ecessry to ret o dys i whe di >, so o items will e reted for i,,,,6 Three items must e reted o dy i d o dy i 7, so totl of items t totl cost of must e reted The totl cost, which is the sum of uyig d retig costs, is therefore + 9 Determiistic Olie Algorithm We ow cosider the olie optimiztio prolem I the olie optimiztio prolem, the demd is give oe dy t time d the ret/uy decisios hve to e mde o demd rrivl Therefore, o dy k, oly the demds d, d,,dk re kow We ow outlie lgorithm Ao d lyze its competitive rtio I the descriptio of the lgorithm we use x k to deote the umer of items ought o dy k d y k to deote the umer of items reted o dy k Recll tht we use hk to deote the th highest demd mog the first k demds Algorithm Ao for k to do if k< the x k y k dk else Compute hk x k hk hk y k mx, dk hk ed The olie lgorithm Ao works s follows: For the first dys, the olie lgorithm rets ll the demds From ech dy k owrds, the olie lgorithm first computes hk It the uys hk hk items It is esy to see tht the totl umer of items tht hve ee ought up to (d icludig) dy k is hk If dk > hk, it rets dk hk items to meet the demd dk I the ext theorem, we derive closed form expressio for the cost icurred y the olie lgorithm fter processig k demds Theorem For y k, the cost of the olie lgorithm fter k demds hve ee processed, k deoted y, is give y k di + ( )hk () i Sk Proof We prove the theorem vi iductio o k whe k Before the rrivl the olie P, lgorithm oly rets Therefore fter processig rrivl, the totl cost will e i di Whe k, the the olie lgorithm uys h items ech t cost d rets d h icurrig totl cost of h + d h i dy Therefore the totl cost from dy oe util fter rrivl is processed 7

6 is give y di + h + d h i di + ( )h i S This estlishes the se step of the iductio Let us ow ssume tht the cost formul holds util rrivl k is processed, ie, k di + ( )hk i Sk Whe rrivl k is processed, the lgorithm uys hk hk items t cost of hk hk It the rets (if ecessry) dk hk items We cosider two seprte cses: If dk hk, the the demd dk is less th the totl umer of items lredy purchsed d Sk Sk d d therefore o items will I this cse hk hk P e ought or reted k k k therefore i S k di + ( )h If dk > hk, the hk > hk Moreover dk hk items will e d hk hk items will e ought t cost of hk hk reted Therefore the totl cost icurred y the olie lgorithm fter demd k is processed is k k + hk hk + dk hk di + ( )hk + hk hk + dk hk i Sk di + ( )hk + ( )hk hk + dk i Sk di + ( )hk + dk hk i Sk di + ( )hk i Sk where the lst equlity follows from Lemm Note tht the totl cost icurred y the olie lgorithm is idepedet of the order i which the demds rrive ito the system It is just fuctio of the ordered set of demds We ow c give the competitive rtio of the olie lgorithm Theorem Give y o-egtive -vector D of demds, let Cof f (D) e the cost icurred y the offlie lgorithm d (D) e the cost icurred y the olie lgorithm The ρ (D) Cof f (D) 8

7 Proof If the umer of demds <, the the optiml offlie solutio is to ret ll the demds d the olie lgorithm lso does the sme The competitive rtio ρ if < If, the from Equtios () d (), we c write P i S di + ( )h P Cof i S di f ( )h + P i S di ( )h + h ( ) + The iequlity follows from the fct tht di h for ll i S Algorithm Ao c lso e show to e the est possile determiistic olie lgorithm to solve this prolem, or, equivletly, tht o other determiistic olie lgorithm c hve competitive rtio lower th This follows directly from the proof of the optimlity of the stdrd item retl prolem [9] d we give the proof for completeess Theorem No determiistic olie lgorithm opertig with the sme demds s Ao to crete ret/uy schedule to solve the ski-retl prolem with demds c hve competitive rtio less th Proof Let us costruct simple exmple for which o determiistic lgorithm c hve competitive rtio lower th We cosider specil cse of our prolem where the demds re oe up to some dy fter which the demd ecomes zero sider olie lgorithm for this prolem Assume tht the olie lgorithm uys oe uit of the item o dy m I this cse the cost of the olie lgorithm is m + where m is the cost of retig for the first m dys d is the cost of uyig o dy m Assume tht the demd geertor sets the demds to zero from dy m + The optiml offlie cost to this prolem is mi{m, } Therefore the competitive rtio is m + mx{m, } + mi{m, } mi{m, } mi{m, } The iequlity follows from the fct tht the rtio is miimized whe m Olie Determiistic Algorithm Exmple Let us gi cosider the exmple where, d D is {,,,,8,,6} Let t represet the curret dy For the first two dys, while t <, we will oly ret to stisfy the demds At ll further dys t, we will uy x ht ht items d ret y ht dt items, s i the tle elow: 9

8 t 6 7 x t y t t Thus the totl cost icurred y the olie lgorithm is 9, which is withi fctor of of the optiml offlie cost of 9 The rtio of the olie d offlie costs i this exmple is Proilistic Olie Algorithm It hs ee show tht the est olie determiistic lgorithm will icur o more th out twice the cost of offlie lgorithm tht kows ll the demds i dvce A proilistic olie lgorithm rdomizes over multiple determiistic strtegies c potetilly chieve etter expected performce We first defie how the performce of proilistic olie lgorithms is mesured Assume tht the lgorithm hs set A of determiistic strtegies tht it rdomizes over Assume tht it uses strtegy A with proility p Sice the proilistic lgorithm rdomizes over multiple determiistic strtegies, we hve to weight the competitive rtio chieved with strtegy with the proility tht strtegy is used i order to get the expected competitive rtio ρ The expected competitive rtio is give y C (D) o ρ mx p D Cof f (D) A Note tht the performce of the proilistic olie lgorithm is mesured over the worst cse demd iput From the worst-cse lysis of the determiistic lgorithm i the previous sectio, ovious wekess of the determiistic lgorithm is tht it ofte uys too little I geerl, the proilistic lgorithm we re out to descrie will ttempt to correct this y uyig more items th the determiistic lgorithm I the determiistic lgorithm, the user will try to uy items so tht the totl mout ought is equl to the th highest demd see so fr t ech time itervl The proilistic lgorithm will, isted of uyig oly up to the th highest, will uy up to the th highest demd where the vlue of is chose rdomly etwee d By choosig the distriutio of crefully, the proilistic lgorithm ttempts the miimize the expected (D) represet the cost of the olie lgorithm which rets for the first competitive rtio Let demds d the chooses the th highest demd ech dy O dy k, the lgorithm uys hk hk items d rets mx{, dk hk } items Lemm 6 Let (D) represet the cost of the olie lgorithm tht uys to the th highest demd ech dy d rets if ecessry whe the demd vector is D The (D) di + ( )h () i S Proof The proof follows exctly the sme rgumet s i the lst sectio for the determiistic olie lgorithm We replce with keepig i mid tht the cost of uyig the items is still

9 6 Pickig the Optiml Distriutio We ow derive the optiml distriutio of i order to miimize the competitive gurtee Towrds this ed, we first mke ottiol simplifictio i order to keep the lysis cle Sice oth the offlie costs s well s the olie cost for y strtegy does ot deped o the order of demds, without loss of geerlity, we ssume tht d d d With the demds ordered i this fshio, we c ow write (D) di + ( )d () di () i Cof f (D) i This due to the fct tht the set S {,,, } The ojective of the lgorithm desiger is to P pick the proilities p where p such ρ is miimized Assume tht the lgorithm desiger hs fixed the proility vector p The dversry for the lgorithm desiger is the demd geertor For fixed proility vector, the demd geertor solves the followig optimiztio prolem: (D) p mx D Cof f (D) I other words, the demd geertor ttempts to fid the demd iput tht mximizes the competitve rtio From Equtios () d (), ote tht the competitive rtio is uchged if ll the demds re scled Therefore, without loss of geerlity, we c ssume tht the demd geertor ormlizes the demds such tht Cof f (D) d from Equtio (), Usig the vlue of p (D) " p # di + ( )d (6) i " di pi + i # p (7) i+ Equtio (7) is just re-rrgemet of the terms i Equtio (6) i order to collect the terms correspodig to di Therefore the optimiztio prolem solved y the demd geertor is mx d p + pj st j+ d d

10 P Let c p + j+ pj e the coefficiet of d i the ojective fuctio The optiml solutio to this prolem is for the demd geertor to set d correspodig to the mximum c Therefore, it is i the iterest of the lgorithm desiger to mke ll the coefficiets c equl I this cse ρ will equl this (commo) vlue of c The lgorithm desiger, picks p such tht c p + pj ρ (8) j+ Equtig the coefficiets c d c+, we get p + pi p+ + i+ pi i+ The equtio c e simplified y ccellig the commo terms o oth sides, givig p + p+ p+, which further simplifies to p p+ This shows tht the proilities p re i geometric progressio with commo rtio r Therefore, p p r,,, P Sice p, we solve for p to get p r r d r r,,,, (9) r With the distriutio ow kow, we c fully outlie the proilistic olie lgorithm Apro (D), usig the sme ottio s used i the descriptio of lgorithm Ao i sectio p Algorithm Apro pick rdomly usig distriutio (9) for k to do if k < the x k y k dk else Compute hk x k hk h k y k mx, dk hk ed

11 We ow estlish the proilistic expected competitive rtio To solve for the expected competitive rtio, we use Equtio (8) whe, d we get ρ p r r r r r r e Iequlity () follows from the fct tht + + e e () e Therefore, we c stte the followig theorem out the proilistic olie lgorithm Theorem 6 The proilistic olie lgorithm Apro (D) chieves expected competitive rtio e ρ of e Sice this prolem geerlizes the stdrd ski-retl prolem, we c stte the followig kow result which follows directly from the ski retl prolem [9] Theorem 6 No proilistic olie lgorithm opertig with the sme iputs s Apro (D) to crete ret/uy schedule to solve the ski-retl prolem with demds c hve expected come petitive rtio less th e 6 Proilistic Algorithm Exmple Let us gi cosider the exmple where, d D is {,,,,8,,6} We ow hve three pure strtegies to use: strtegy, strtegy, d strtegy For strtegy, we will items d ret y t ht dt items, s i the ret if t <, d if t we will uy x t ht ht tle elow: t 6 7 x t y t Cpro (t) 9 For strtegy, we will ret if t <, d if t we will uy x t ht ht items d ret y t ht dt items, s i the tle elow:

12 t 6 7 x t y t Cpro (t) 7 Filly, for strtegy, while t <, we will oly ret At times t, we will uy x t ht ht items d ret y t ht dt items, s i the tle elow: t 6 7 x t y t Cpro (t) The proility tht the user chooses strtegy c e computed from Equtio (9) So, p p p ( ) (( ( ) )) ( ) (( ( ) )) 8 ( ) (( ( ) )) 77, which is ()() + + p Cpro + p Cpro Now, we c fid the expected cost s p Cpro (8)() + (77)(9) 668 We c ow see tht the expected cost of the proilistic lgorithm is lower th tht of the determiistic lgorithm, which hs cost of 9 The expected rtio of the proilistic lgorithm s cost to tht of the offlie lgorithm i this exmple is clusio d Further Reserch I this pper, we cosidered olie uy or ret decisio prolem with vrile demds d we developed A determiistic olie lgorithm with competitive rtio of A proilistic olie lgorithm with expected competitive rtio of e e

13 We further estlished the optimlity of these two lgorithms Curretly, we re studyig vrits of this prolem uder more geerl cost models s well s relistic costrits Refereces [] L AI, WU, LINGIAO HUANG, LONGBO HUANG, P TANG, d J LI, The Multi-shop Ski Retl Prolem, i Computig Reserch Repository, [] M BIENKOWSKI, Ski Retl Prolem with Dymic Pricig, Istitute Of Computer Sciece, Uiversity Of Wroclw, Report /8, 8 [] H FUJIWARA, T KITANO, d T FUJITO, O the Best Possile Competitive Rtio for Multislope Ski Retl, i ISAAC Proceedigs of the d itertiol ferece o Algorithms d Computtio,, pp - [] A GUPTA, A KUMAR, M PAL, d T ROUGHGARDEN, Approximtio vi cost shrig: Simpler d etter pproximtio lgorithms for etwork desig, i Jourl of the ACM (JACM), Vol, No, 7, pp - [] A R KARLIN, M S MANASSEE, L A MCGEOCH, d S OWICKI, Competitive rdomized lgorithms for o-uiform prolems, i Proceedigs of the First Aul ACM-SIAM Symposium o Discrete Algorithms (SODA 9) Society for Idustril d Applied Mthemtics, Phildelphi, PA, USA, 99, pp -9 [6] Z LOTKER, B PATT-SHAMIR, d D RAWITZ, Ret, Lese or Buy: Rdomized Algorithms for Multislope Ski Retl, i Symposium o Theoreticl Aspects of Computer Sciece STACS, Bordeux, Frce, 8, pp - [7] S S SEIDEN, A guessig gme d rdomized olie lgorithms, i Proceedigs of the thirtysecod ul ACM symposium o Theory of computig (STOC ) ACM, New York, NY, USA,, pp 9-6 [8] S LAHAIE, How to tke the Dul of Lier Progrm, <wwwcscolumiedu/coms6998/lpprimerpdf>, 8 [9] M QUEYRANNE, A Itroductio to Competitive Alysis for Olie Optimiztio, <wwwimumedu/ mli/olie Brow-Bg Slidespdf>, [] G TIAN, U SHARMA, T WOOD, S SAHU, d P SHENOY, Segull: itelliget cloud urstig for eterprise pplictios, i Proceedigs of the Useix Aul Techicl ferece, <wwwuseixorg/system/files/coferece/tc/tc-fil7pdf>,