An Improved Randomized Truthful Mechanism for Scheduling Unrelated Machines

Symposium on Theoretical Aspects of Computer Science 2008 (Bordeaux), pp. 527-538 www .stacs-conf .org AN IMPR O VED RANDOMIZED TR UTHFUL MECHANISM F OR SCHEDULING UNRELA TED MACHINES PINY AN LU 1 AND CHAN GYUAN Y U 1 1 Institute for Theoretical Computer S cience, Tsinghua Universit y , Beijing, 100084, P .R. China E-mail addr ess : {lpy, yucy05}@mail s.tsinghua.edu.cn URL : http://www.itcs.ts inghua.edu.cn/ { LuPY,YuCY } Abstra ct. W e study the sc heduling problem on unrelated mac hines in the mechanism design setting. This problem w as p rop osed and studied in the seminal pap er of Nisan and Ronen [NR99], where they ga ve a 1 . 75-approximati on rand omized truthful mechanism for the case of tw o mac hines. W e improve this result b y a 1 . 6 737-approximation rand omized truthful mec h an ism. W e also generali ze our result to a 0 . 8 368 m -approximation mechanism for task scheduling with m machines, which improve the previous b est u p p er b ound of 0 . 875 m [MS07]. 1. In tro duction Mec hanism design has b ecome an activ e area of r esearc h b oth in C omputer Science and Game T heory . In the m ec hanism design setting, pla yers are selfish and wish to maximize their o wn utilities. T o deal with the selfishn ess of the pla yers, a mec han ism should b oth satisfy some game-theoretic al requiremen ts such as truthfulness and some compu tational prop erties suc h as go o d appr oximation r atio . The study of their alg orithm ic asp ect wa s ini- tiated b y Nisan and Ronen in their seminal pap er “Algorithmic Mec hanism Design” [NR99]. The fo cus of that pap er was on the sc h eduling problem on u nrelated mac hines, f or whic h the standard mec hanism design to ols ( VCG mec hanisms [Clarke7 1 , Gro ve s1973 , Vic krey61 ])do not su ffice. T h ey prov ed that no deterministic mec hanism can h av e an app ro ximation r atio b etter than 2 f or this pr oblem. This b ound is tigh t for the case of tw o m achines. Ho w- ev er if w e allo w randomized mechanisms, this b ound can b e b eaten. In particular they ga v e a 1 . 75-a p pro ximation rand omized tr uthful mec hanism for the case of t wo m ac h ines. Since then, many researc h er s ha ve studied the scheduling problem on un related mac hines in mec hanism design setting [JP99, Sourd01, S S02, S X02, GMW07, C KV07, CKK 07, MS07]. Ho w eve r their mec h anism remains the b est to the b est of our kno wledge. In a recen t pap er [MS07], Mu’alem and Sc hapir a prov ed a lo wer b ound of 1 . 5 for this setting. So to explore the exact b ound b et ween 1 . 5 and 1 . 75 is an in teresting op en problem in this area. In this pap er, 1998 ACM Subje ct Classific ation: algorithmic mechanism design. Key wor ds and phr ases: t ru thful mechanism, scheduling. Supp orted b y the National Natural Science F oundation of China Gran t 605530 01 an d the N ational Basic Researc h Program of China Grant 2007CB80 7900, 2007CB8079 01. c  P . Lu and C . Y u CC  Creative Comm ons Attr ibution-NoDer ivs License 528 P . LU A ND C. YU w e imp ro ve the upp er b ound from 1 . 75 to 1 . 6737. F orm ally w e give a 1 . 6737-appro ximation randomized tru thful mec hanism for task sc h eduling with t wo mac hines. Using similar tec h- niques of [MS07], w e also generalize our result to a 0 . 8368 m -appro ximation mechanism for task sc hedu ling with m m ac hines. Let us describ e th e problem more carefully . Th ere are m mac hin es and n tasks, and eac h mac hin e is con trolled by an agen t. W e u s e t i j to denote the r u nning time of task j on mac hine i , which is also called th e t yp e v alue of the agen t(mac hin e) i on task j . The ob jectiv e is to minimize the co mp letion time of the last assignmen t (the makesp an ). Unlik e in the classical optimization p r oblem, the sc h eduling designer do es not kno w t i j . Eac h selfish agen t i holds his/her o wn t yp e v alues (the t i j s). In order to motiv ate the agen ts to r ep ort their tru e v alue t i j s, the m echanism needs to pa y the agen ts. S o a mechanism consists of an allo cation algorithm and a paymen t algorithm. A mechanism is called truthful when telling one’s true v alue is among the optimal str ategies for eac h agen t, no m atter how other agen ts b eha ve. Here the utilit y of eac h agen t is th e p a yment he/she gets min u s the load of tasks allo cated to h is/her mac hine. When rand omness is inv olved, there are t wo ve r sions of truthfuln ess: in the stronger v ersion, i.e. universal ly truthfulness , th e mec hanism remains truthful eve n if the agen ts kno w the rand om b its; in the weak er ve rs ion, i.e. truthfulness in exp e ctation , an agen t maximizes his/her exp ecte d utilit y by telling the tru e type v alue. Our mec hanisms prop osed in this pap er are un iv ersally tru thful. No w w e can talk ab out th e h igh lev el idea of the tec h nical part. Here w e only talk ab out the allo cation algorithms, and the corresp onding p a yment algorithms, which mak e the mec h anism truthful, will b e giv en later. First we describ e Nisan and Ronen’s mec hanism [NR99]. In their mec hanism, eac h task is allocated indep end en tly . F or a particular task j , if the tw o v alues t 1 j and t 2 j are relativ ely close to eac h other, sa y t 1 j /t 2 j ∈ [3 / 4 , 4 / 3], th en they allo cate task j randomly to mac h ine 1 or 2 with equal probability; if one is m u c h higher then the other, sa y t 1 j /t 2 j > 4 / 3 or t 2 j /t 1 j > 4 / 3, the task j is allocated to the more efficien t mac hine. The main idea of our mec hanism is to partition the tasks into three catego ries rather than t wo. So we n eed t wo threshold v alues, sa y α, β , where 1 < β < α and a biased p robabilit y r , where 1 / 2 < r < 1. If th e t w o v alues are relativ ely close to eac h other, sa y t 1 j /t 2 j ∈ [1 /β , β ], or one is muc h higher then the other, sa y t 1 j /t 2 j > α or t 2 j /t 1 j > α , w e do the same thin gs as Nisan and Ronen’s mec hanism. In the remaining case, one is significan tly larger than th e other, b ut how ever still do es not dominate, sa y t 1 j /t 2 j ∈ [ β , α ] or t 2 j /t 1 j ∈ [ β , α ]. In this case, w e allo cate the task j to the more efficien t one with a higher probabilit y r ( r > 1 / 2) and the less effici ent one with a lo w er probabilit y 1 − r . Th e mechanism is quite simple, so it is v ery computationally efficien t. In tuitiv ely our mec hanism will give b etter approximati on ratios by c h o osing suitable parameters α, β and r . T h is is indeed tru e. W e can p ro ve an improv ed appr o ximation ratio of 1 . 6737 by c ho osing α = 1 . 4844 , β = 1 . 1854 , r = 0 . 7932. Ho w eve r, the pro of is quite inv olv ed. One reason is th at the situation for the n ew case (mid dle case) is m ore complicated than the original t wo. The main reason is that their app roac h b ecomes infeasible in the analysis of our mec hanism. The pr o of in Nisan and Ronen’s p ap er is basically case b y case, bu t unfortun ately the num b er of sub ca ses increases doub le exp onential ly with the n u m b er of task types. So we int r o duce s ome sub stan tial new pr o of tec hniqu es to o vercome this. W e also think this tec hn iques may further impro ve the upp er b ou n d. TRUTHFUL MECHANISM FOR SCHEDULING UNRELA TED MA CHINES 529 1.1. Related W ork Sc hedu ling on unr elated mac hines is one of the most fun damen tal sc heduling Prob- lems. F or this NP-h ard optimization pr oblem, there is a p olynomial time algorithm with appro ximation ratio of 2[LST87]. Esp ecially if the n umb er of machines is b ounded b y some constan t, Angel, Bampis and Konono v ga ve an FPT AS[ABK01]. Ho we ver there is no corresp onding p aymen t strategy to mak e either of the ab o ve allocation algorithms tr uthful. The stud y of this problem in the mec han ism d esign setting is initiated by Nisan and Ronen. In their pap er [NR99], they gav e a 1 . 75-appro ximation randomized truthful mec ha- nism for t wo mac h ines. Th is result was generali zed by Mu’alem and Schapira to a 0 . 875 m - appro ximation randomized mechanism for m mac h ines[MS07]. W e impro ved the tw o upp er b ound s to 1 . 6 737 and 0 . 8368 m resp ectiv ely . F or the lo wer b oun d side, Nisan and Ronen ga v e a lo wer b oun d of 2 for deterministic v ersion. Th is b ound was impro ved by Ch risto doulou, K outsoupias and Vidali to 1 + √ 2 for 3 or m ore mac hines [CK V07]. F or the randomized version, Mu’alem and Sc h apira ga v e a lo w er b ound of 2 − 1 /m [MS07]. This also h olds for the weak er notion of truthfu ln ess, i.e., truthfuln ess in exp ectation. La vi and Swarm y considered a restricted v arian t, where eac h task j on ly h as t wo v alues of run ning time , and gav e a 3-appro ximation rand omized truthful mechanism [LS07]. Th ey first use the cycle monotonicit y in designing mechanisms. In [CKK07], Christo d oulou, Koutsoupias and Ko v´ acs considered the fractional v ersion of this problem, in whic h eac h task can b e sp lit among the m achines. F or this v ersion, they ga v e a lo we r b ound of 2 − 1 /m and an up p er b ound of ( m + 1) / 2. W e remark that these tw o b ound s are closed for the case of tw o mac hin es as in the in tegral deterministic v ersion. So to explore the exact b oun d for the randomized ve r sion seems v ery in teresting and desirable. W e b eliev e that our wo rk in this p ap er is an imp ortan t step to w ard this ob jectiv e. 2. Problem and Definitions In this section w e review some defi n itions and results on mec hanism design and sched- uling problem. More details can b e found in[NR99]. In a mec han ism d esign p roblem, there are us ually some resources to distribute among n agent s. Ev ery agen t i has a t yp e v alue t i , which denotes his/her preference on the resources. L et t = ( t i ) i ∈ [ m ] denote the ve ctor of all agen ts’ type v alues and t − i denote the v ector of all agen ts’ t yp e v ectors exce pt agen t i ’s. Receiving all the t yp e v alues t from agen ts, the m ec hanism will pro d u ce an output o ( t ) = ( x ( t ) , p ( t )). Here x ( t ) sp ecifies the allocation of the r esources and is p r o duced by an allo cation algorithm. p ( t ) sp ecifies the pa yment to agent s and is p ro duced by an pa ym ent algorithm. Eve ry a gent i has a v aluation v i ( x, t i ), whic h describ e his/her p reference on the outpu t allo cation. Th e agen t i ’s ob jectiv e is maximizing his/h er utilit y function u i , where u i = v i + p i , and p i is the paymen t obtai n ed from the mec hanism. The mechanism’s ob jectiv e is to maximize an ob jectiv e fun ction g ( o, t ). F ormally , we ha v e the follo wing defin itions. Definition 2.1. A mec hanism is a pair of Algorithms M = ( X , P ). • A l lo c ation Algorithm X : Its input is m agent s’ t yp e vect ors, t 1 , t 2 , · · · , t m , whic h are rep orted b y the agent s. and its output is x = ( x 1 , · · · , x m ), where x i = ( x i j ) j ∈ [ n ] ∈ { 0 , 1 } n are allo cation vecto r of agent i . 530 P . LU A ND C. YU • Payment Algorithm P : It outputs a p a ymen t v ector p = ( p 1 , · · · , p m ), whic h de- p ends b oth on agen ts’ type v ectors an d allo cation vec tors pro du ced by allo cation algorithm. A mec hanism is deterministic if b oth the allo cation algorithm and pa yment algorithm are deterministic. When at least one of them uses random bits, it is called a r andomize d me chanism . In order to in cr ease u tilit y , an agen t may lie w h en rep orting his/her typ e v alues. But for some mec hanisms, no agent can increase his/her u tilit y b y lying. This nice p r op ert y of a mec hanism is called truthfulness . W e give the formal defin itions of truthfulness. Definition 2.2. A deterministic mec h an ism is truthful iff for ev ery a gent, rep orting his/her tru e t yp e v alues is among the b est strategies to maximize his/her utilit y , no matter ho w other agen ts acts. A randomized mec hanism is truthful in exp ectation iff no agen t can increase his/her exp ected utilit y b y lying. A randomized mec h anism is univ ersally truthful iff it remains truthfu l ev en if the agents kno w th e r andom bits. F rom no w on, we will only fo cus on tru thful mec h anisms. The most imp ortan t p os- itiv e result in mec h anism design is generaliz ed Vickrey-Cla rke-Gro ves(V CG) mec hanism [Vic krey61, Grov es1973 , Clark e71]. Man y kno wn tru th ful mec hanisms are all in V CG fam- ily . The mec hanism s of V CG family usually apply to mec hanism design problem in which the ob jectiv e fun ction is the (wei ghted) su m of all agen ts’ v aluations. T o b e formal, w e hav e Definition 2.3. [NR99] A mec hanism M = ( X , P ) b elongs to weig hted VCG family if there are real num b ers (weigh ts) β 1 , · · · , β n > 0, s uc h that: (1) the problem’s ob jectiv e f unction satisfies g ( o, t ) = P i β i v i ( t i , o ) . (2) o ( t ) ∈ ar gmax o ( g ( o, t ). (3) p i ( t ) = 1 β i P i ′ 6 = i β i ′ v i ′ ( t i ′ , o ( t )) + h i ( t − i ), where h i () is an arb itrary function of t − i . Theorem 2.4. ( [Rob erts79] ) A weighte d VCG me chanism is truthful. No w w e sp ecify these mechanism n otions in the pr oblem of sc hed u ling unrelated ma- c hines. Assu me there are n tasks to b e allocated to m machines, eac h of w h ic h is controlle d b y an agent . Eac h agen t i ’s t yp e v alue is t i = ( t i j ) j ∈ [ n ] , w here t i j denotes th e time to p erform task j on mac hine i . W e use a binary arr a y x i = ( x i j ) j ∈ [ n ] to sp ecify the allo cation of tasks to mac hine i . x i j is 1 if task j is allo cated to mac hine i and otherw ise 0. Let x = ( x i ) i ∈ [ m ] denote the allocation of all the tasks. F or an allo cation x , agen t i ’s v aluation is v i = − x i · t i , wh ere x i · t i = P n j =1 x i j t i j . Definition 2.5. Giv en any allo cation x of the tasks, the longest r unning time of the ma- c hines is called the make sp an of the allocation. F ormally , mak espan( x ) = max i ∈ [ m ] x i · t i . The ob jectiv e of the mechanism is to minimize the (exp ected) makespan of the alloca- tion. This is n ot the (w eigh ted) su m of all agen ts’ v aluations. So we can not apply V CG mec hanism h ere. Ho wev er, w e r emark that if ther e is only one task, the m akespan can b e view ed as the sum of all agents’ v aluations. W e w ill use this observ ation in our analysis. F rom [NR99] and [MS07], we kno w that there is no optimal tru th ful mechanism for this problem, eve n if w e allo w sup er-p olynomial r unning time and r andomness. So we will try to find a tru thful mec hanism with go o d appr o ximation ratio. TRUTHFUL MECHANISM FOR SCHEDULING UNRELA TED MA CHINES 531 Definition 2.6. L et t M ( t ) b e the (exp ected) mak esp an of the mechanism M on instances t and t opt ( t ) b e the optimal mak espan of instance t . W e sa y mec hanism M has approximati on ratio c iff f or an y instance t , t M ( t ) /t opt ( t ) ≤ c . 3. Our Mech anism and the A nalysis In this section, w e giv e a truthful sc h eduling mec hanism for 2 mac hin es case , and sho w that its appro ximation ratio is 1 . 6737. Th en we generalize our result to the m mac hines case as in [MS07] and obtain a 0 . 8368 m -appro ximation rand omized truthful mec hanism. 3.1. Generalized Randomly Biased Mec hanism P arameters: Real n umb ers α > β ≥ 1 > r ≥ 1 2 . (Here we c ho ose α = 1 . 4844 , β = 1 . 1854 , r = 0 . 7932.) Input: Th e rep orted t yp e v ectors t = ( t 1 , t 2 ). Output: A r andomized allocation x = ( x 1 , x 2 ), and a paymen t p = ( p 1 , p 2 ). Allo cation and P a yment algorithm: x 1 j ← 0 , x 2 j ← 0 , j = 1 , 2 · · · , n ; p 1 ← 0; p 2 ← 0. F or eac h task j = 1 , 2 · · · , n d o s j ←          α, with probability 1 − r , β , with probability r − 1 / 2 , 1 /β , with probability r − 1 / 2 , 1 /α, with probability 1 − r . if t 1 j < s j t 2 j , x 1 j = 1 , p 1 ← p 1 + s j t 2 j ; else x 2 j = 1 , p 2 ← p 2 + s − 1 j t 1 j . Theorem 3.1. The Gener alize d R andomly Biase d M e chanism (GBM for short) is univer- sal ly truthful and c an achieve a 1 . 6737 -app r oximation solution for task sche duling with two machines. W e will pro ve this theorem in the follo wing t w o sub sections. In 3.2, w e will pro ve that our mec hanism is universally truthful. Then we analyze its approximat ion ratio in 3.3 3.2. T ruthfulness Lemma 3.2. The Gener alize d R andomly Biase d Me c hanism is universal ly truthful. Pr o of. T o pro v e that the GBM is univ ersally truthful, w e only need to pro ve th at it is truthful when the random sequence s j is fi xed. Since the utilit y of an agen t equals the sum of the utilities obtained f rom eac h task and our mec hanism is task-indep enden t, we only need co ns ider the ca se of one task. I n this case, sa y s j is fixed and there is only one task j , the m ec hanism is exactly the V CG mechanism with w eight (1 , s j ). Since a weigh ted V CG is truthful, the GBM mec hanism is universally truthful. 532 P . LU A ND C. YU 3.3. Estimation of the Appro ximation Ratio If th is su bsection, we will estimate the app ro ximation ratio of our GBM mechanism. Since we already pr o v ed that GBM is unive r s ally truthfu l in 3.2, w e only need to fo cu s on the allo cation algorithms of GBM. So w e can restate the allo cation algorithms for GBM in an equiv alen t bu t more unders tandable wa y . In tuitive ly we should assign one task with larger p robabilit y to the mac h ine wh ic h has smaller t yp e v alue(run ning time) on it. The idea of our mec hanism is to partition all the tasks in to several t yp es acco r ding to the r atio of tw o agen ts’ t yp e v alues. F or differen t t yp es of ta sks , we use differen t biased probabilities to allo cate them. T o b e formal, w e h a v e the follo wing definition. Definition 3.3. F or a ta sk j , w e ca ll it an h -task iff t i j t 3 − i j > α for some i ∈ { 1 , 2 } ; we called it an m -task iff β < t i j t 3 − i j ≤ α for some i ∈ { 1 , 2 } ; we call it an l -task if t i j t 3 − i j ≤ β for an y i ∈ { 1 , 2 } . Then, we ha v e th e follo wing claim. Claim 3.4. The GBM me chanism al lo c ates the tasks in the same way as the fol lowing al lo c ating algorithm do es. • F or h -task, we al lo c ate it to the machine with lower typ e value. • F or m -task, we al lo c ate it to the mor e efficient machine with pr ob ability r and to the less efficient machine with pr ob ability 1 − r . • F or l -task, we al lo c ate it to two machines with e qual pr ob abilities. Pr o of. F or eac h task j , we consider the probabilit y th at it is allo cated to mac hin e 1 in GBM. According to the ratio of t 1 j t 2 j , w e h a v e the follo wing 5 cases: • Case 1: t 1 j ≥ αt 2 j , then P r ( x 1 j = 1) = 0 • Case 2: β ≤ t 1 j < αt 2 j , then P r ( x 1 j = 1) = 1 − r • Case 3: β − 1 ≤ t 1 j < β t 2 j , then P r ( x 1 j = 1) = (1 − r ) + ( r − 1 2 ) = 1 2 • Case 4: α − 1 ≤ t 1 j < β − 1 t 2 j , then P r ( x 1 j = 1) = (1 − r ) + ( r − 1 2 ) + ( r − 1 2 ) = r • Case 5: t 2 j < α − 1 t 2 j , then P r ( x 1 j = 1) = (1 − r ) + ( r − 1 2 ) + ( r − 1 2 ) + (1 − r ) = 1 The pr obabilities that task j is assigned to mac hine 1 b y t wo algorithms are alwa ys the same, so the lemma is true. Remark 3.5. This claim only sa ys that the (d istribution of ) allocation pro duced by the t wo metho ds are the same. Ho we ver if we use this allo cation algorithm sta ted in the claim, w e can only mak e th e mec hanism truthfu l in exp ectation. As in [NR99], w e obtain the follo wing crucial claim, whic h can help us cut the n umb er of tasks. The pro of of this claim is similar , and we put it in the Ap p endix. Claim 3.6. T o analyze the p erformanc e of the gener alize d r andomize d b iase d me chanism, we only ne e d c onsider the fol lowing c ases: (1) F or e ach h - task j , the r atio of the two machines’ typ e value is arbitr arily close to α . So we c an assume it e quals α . (2) If O P T al lo c ates an l -task j to machine i , then t 3 − i j /t i j = β . (3) If O P T al lo c ates an m -task j to machine i which has smal ler typ e value, th en t 3 − i j /t i j = α . TRUTHFUL MECHANISM FOR SCHEDULING UNRELA TED MA CHINES 533 (4) If O P T al lo c ates an m -task j to machine i which has bigger typ e value, then t 3 − i j /t i j = β − 1 . (5) One of the machines is mor e efficient than the other on al l h -tasks. We ass ume it’s machine 1 . (6) Ther e ar e at most 8 tasks A, B , C , D , E , F , G, H . In O P T , tasks A, C , E , G ar e al lo c ate d to machine 1 , and the others to machine 2 . T asks A, B ar e h -tasks. T asks C, D , E , F ar e m -tasks and tasks G, H ar e l -tasks. F rom the ab o ve analysis, we know that we only n eed to consider the reduced case as describ ed in Figure 1. t yp e task t 1 j t 2 j opt-alloc g b m-alloc(prob ab ility) h 1 A a αa 1 1 : 0 h 2 B b αb 2 1 : 0 m 1 1 C c αc 1 r : (1 − r ) m 1 2 D d β d 2 r : (1 − r ) m 2 1 E β e e 1 (1 − r ) : r m 2 2 F αf f 2 (1 − r ) : r l 1 G g β g 1 1 2 : 1 2 l 2 H β h h 2 1 2 : 1 2 Figure 1: T he Reduced Case. No w we can estimate the appr o ximation r atio based on this r educed case. Lemma 3.7. The al lo c ation pr o duc e d by GBM is a 1 . 6737 -ap pr oximation solution for the task sche duling pr oblem with two machines. Pr o of. L et t opt b e the mak e-span of an optimal solution and let t g bm b e the exp ected mak espan of allocations pr o duced b y GBM. W e wan t to sho w that t g bm ≤ 1 . 6737 t opt . F rom the allocation of th e optimal solution, we ha v e that t opt = max { a + c + β e + g , αb + β d + f + h } . No w we will estimate the exp ected mak espan of our m ec hanism t g bm . First we in tro d uce some n otation whic h will b e u sed in th e follo wing analysis. W e will treat the same name X ( X = A, B , · · · , H ) as a random v ariable, whic h denotes the assignmen t of the task X . F or example, C = 2 means that our mechanism assigns the task C to the second mac h ine. Then the last column in Figure 1 can also b e viewed as the d istr ibution of the rand om v ariable X ( X = A, B , · · · , H ). F or example P r ( C = 1) = r and P r ( C = 2) = 1 − r . Sin ce our mec hanism assigns eac h task indep enden tly , the rand om v ariables are also indep enden t of eac h other. More pr ecisely , for any X , Y ∈ { A, B , · · · , H } , i, j ∈ { 1 , 2 } and X 6 = Y , we hav e P r ( X = i, Y = j ) = P r ( X = i ) P r ( Y = j ) . W e use a random v ariable M to denote the mac hin e finishing last. More p recisely , M = 1 means the completion time of the fi r st mac hine is not earlier than th e second mac hine, otherwise w e ha ve M = 2. No w w e compute the con tribution of eac h task to t g bm . Let the j -th task b e X . Th en its con trib u tion to t g bm con tains t wo parts. First part is fr om t 1 j . t 1 j con tributes to t g bm 534 P . LU A ND C. YU iff our mec hanism assigns task X to 1 (e.t. X = 1) and the m ac h ine 1 fin ish es later (e.t. M = 1). The situation for t 2 j is similar. T o sum up, the contribution of the j -th task X to t g bm is P r ( M = 1 , X = 1) t 1 j + P r ( M = 2 , X = 2) t 2 j . F or example, the con trib u tion of task C to t g bm is P r ( M = 1 , C = 1) c + αP r ( M = 2 , C = 2) c = ( P r ( M = 1 , C = 1) + αP r ( M = 2 , C = 2)) c. Similarly , w e can compute the contribution of eac h task to t g bm easily . T o simplify the notation, we use C x ( x = a, b, · · · , h ) to denote the co efficien t of x in t g bm . So we ha v e t g bm = C a a + C b b + C c c + C d d + C e e + C f f + C g g + C h h. where C a = P r ( M = 1) , C b = P r ( M = 1) , C c = P r ( M = 1 , C = 1) + αP r ( M = 2 , C = 2) , C d = P r ( M = 1 , D = 1) + β P r ( M = 2 , D = 2) , C e = β P r ( M = 1 , E = 1) + P r ( M = 2 , E = 2) , C f = αP r ( M = 1 , F = 1) + P r ( M = 2 , F = 2) , C g = P r ( M = 1 , G = 1) + β P r ( M = 2 , G = 2) , C h = β P r ( M = 1 , H = 1) + P r ( M = 2 , H = 2) . Since t g bm = C a a + C b b + C c c + C d d + C e e + C f f + C g g + C h h = ( C a a + C c c + C e β β e + C g g ) + ( C b α αb + C d β β d + C f f + C h h ) ≤ max ( C a , C c , C e β , C g )( a + c + β e + g ) + max ( C b α , C d β , C f , C h )( αb + β d + f + h ) ≤ max ( C a , C c , C e β , C g ) t opt + max ( C b α , C d β , C f , C h ) t opt So the p erformance of our mec hanism is b ound ed b y max ( C a , C c , C e β , C g ) + max ( C b α , C d β , C f , C h ) . W e will giv e b ound f or ev ery p ossible sum betw een { C a , C c , C e β , C g } and { C b α , C d β , C f , C h } . First C a + C b α = P r ( M = 1) + P r ( M = 1) α ≤ 1 + 1 α . So C a + C b α is b ounded b y 1 + 1 α . Later w e will c ho ose suitable parameter α so that th is v alue is n ot to o big. No w we analyze a more complicated case, sa y C c + C f . Subs tituting C c and C f , we ha ve C c + C f = P r ( M = 1 , C = 1)+ αP r ( M = 2 , C = 2)+ αP r ( M = 1 , F = 1)+ P r ( M = 2 , F = 2) . TRUTHFUL MECHANISM FOR SCHEDULING UNRELA TED MA CHINES 535 Here w e use P ij k to d enote the j oin t d istribution of three rand om v ariables M , C, F (e.t. P ij k = P r ( M = i, C = j, F = k ) ). Then we can rewrite the form u la as follo wing P 111 + P 112 + α ( P 221 + P 222 ) + α ( P 111 + P 121 ) + ( P 212 + P 222 ) . Then w e recom bine th e terms as follo win g ( P 111 + P 112 + P 212 ) + α ( P 111 + P 121 + P 221 ) + (1 + α ) P 222 . The first term is b ounded by P r ( C = 1) (sin ce P r ( C = 1) = P 111 + P 112 + P 212 + P 211 ); similarly the s econd term is b ounded by αP r ( F = 1); the third term is b oun ded b y (1 + α ) P r ( C = 2 , F = 2). So we can b oun d C c + C f b y P r ( C = 1) + αP r ( F = 1) + (1 + α ) P r ( C = 2 , F = 2) = r + α (1 − r ) + (1 + α ) r (1 − r ) = 2 r + α − r 2 − αr 2 . Similarly we can b oun d the remaining 14 sums as follo ws. Some of pro ofs are slightly more complicated but all of them are along similar lines. W e only list the b ounds here, and the details are omitted here due to th e space limitation. (1) C a + C d β ≤ 1 + r β . (2) C a + C f ≤ 1 + (1 − r ) α. (3) C a + C h ≤ 1 + β 2 . (4) C c + C b α ≤ 1 + 1 α . (Here we use the assump tion that α ≤ 1 + 1 α .) (5) C c + C d β ≤ r 2 β + 1 + r 2 + α − r + αr. (6) C c + C h ≤ 1 2 + 1 2 r + α − αr + 1 2 β r . (7) C e β + C b α ≤ 1 + 1 α . (8) C e β + C d β ≤ (1 − r ) + 1 β r + (1 + 1 β ) r (1 − r ) ≤ C c + C f . (9) C e β + C f ≤ r 2 β + 1 + r 2 + α − r + αr. (10) C e β + C h ≤ 1 + r 2 β + 1 2 β − 1 2 r . (11) C g + C b α ≤ 1 + 1 α . (Here w e use the assum p tion that α ≤ 1 + 1 α .) (12) C g + C d β ≤ 1 + r 2 β + 1 2 β − 1 2 r . (13) C g + C f ≤ 1 2 + 1 2 r + α − αr + 1 2 β r . (14) C g + C h ≤ 3 4 + 3 4 β . T o sum up , we hav e 9 differen t b ounds: 1 + 1 α , 2 r + α − r 2 − αr 2 , 1 + r β , 1 + (1 − r ) α , 1 + β 2 , r 2 β + 1 + r 2 + α − r + αr , 1 2 + 1 2 r + α − αr + 1 2 β r , 1 + r 2 β + 1 2 β − 1 2 r , 3 4 + 3 4 β , and one assumption that α ≤ 1 + 1 α . W e w ant to c ho ose suitable parameter α, β , r such that the assum ption is satisfied and the maximal b ound is as small as p ossible. T his can b e easily done numerically b y a mathematical to ol suc h as Matlab. W e can choose α = 1 . 4844 , β = 1 . 1854 , r = 0 . 7932. Substituting these v alues, w e can ve rif y that all the b oun ds are less than 1 . 6737. S o w e pro ved that our mec h an ism has an app ro ximate ratio of 1 . 673 7. 3.4. An Improv ed Mechanism for m Mac hines As an application of our main result, we turn to the case of m mac hines. In [NR99], Nisan and Ronen ga v e a truth ful deterministic mec hanism that ac hieves an m -approximat ion. Recen tly , Mu ’alem and Sc hapira [MS07] generalized Nisan and Ronen’s truth ful r an d omized mec hanism for 2 machines to the case of m mac hines. They partitioned the m machines into 536 P . LU A ND C. YU t wo sets of mac hines with equal size, S 1 and S 2 . Then they construct a new in stance with only t wo machines, w ith t yp e v alues t i j = min a ∈ S i t a j , i = 1 , 2. Applying the mec hanism f or 2 mac hines case, They sho wed a unive rsally tru thful randomized mec hanism that obtains an appro ximation of 0 . 875 m . Using this idea and our imp r o v ed result for tw o mac hines case, w e can imp ro ve th e ratio from 0 . 875 m to 0 . 8368 m . T o b e self co ntained, we giv e the formal description of the mec hanism h er e. The pro of is similar with [MS07] and omitted here. ✬ ✫ ✩ ✪ P arameters: real n umb ers α > β ≥ 1 > r ≥ 1 2 . Input: the rep orted t yp e v alue vec tors t = ( t 1 , t 2 , · · · , t m ). Output: an randomized allo cation x = ( x 1 , x 2 , · · · , x m ) and a pa yment p = ( p 1 , p 2 , · · · , p m ). Mec hanism: (1) F or eac h mac hin e i , let x i ← ∅ ; p i ← 0. (2) P artition the set of mac hines into t wo s ets S 1 , S 2 with equal size. If m is not ev en, we can add an extra mac hine w ith infinite t yp e v alues on ev ery task. (3) F or eac h task j , Let t a = min i ∈ S 1 t i j , a = ar g min i ∈ S 1 t i j , t a ′ = min i ∈ S 1 −{ a } t i j . Let t b = min i ∈ S 2 t i j , b = ar g min i ∈ S 1 t i j , t b ′ = min i ∈ S 1 −{ a } t i j . (4) Apply our m ec hanism GBM for t wo mac hines case to mac hine a and b on task j . Also the p a ymen t strategy need a little c hange. If a gets th e task, and it will gain a pa yment p a j in GBM, then we pa y it min { p a j , t a ′ j } . If b gets the task, and it will gain a pa yment p b j in GB M, then w e p a y it min { p b j , t b ′ j } . This change is in order to ke ep the mechanism truthful. Theorem 3.8. m-GBM is an universal ly truthful r andomize d me chanism for the sc he duling pr oblem that obtains an app r oximation r atio of 0 . 8369 m when cho osing α = 1 . 484 4 , β = 1 . 1854 , r = 0 . 7932 . 4. Conclusions and Op en Pro blems This is the first impro vemen t since Nisan and Ronen pr op osed the problem and th e 1.75-mec hanism. W e b eliev e it is p ossible to further impro ve the upp er b oun d u sing our tec h nics. A direct op en problem is to close the gap b et w een the lo wer b ound of 1 . 5 and our new upp er b ound of 1 . 67 37. Another more imp ortant direction is to generalize the mec han ism s for 2 mac hine to mec hanisms for m mac hin es in a more cleve r w a y . In the general case, the gap b etw een the b est lo we r b ounds (constants) and th e b est upp er b ounds (Θ( m )) is huge b ot h in d etermin - istic and rand omized versions. Any impro v ement in either direction is highly desirable. References [ABK01] Angel, E., Bampis, E. and Kononov, A. A FPT AS for Approximating the Un related Parallel Mac hines Scheduling Problem with Costs. In Pr o c e e dings of the 9th Annual Eur op e an Symp osium on Algor i thms (A ugust 28 − 31, 2001). F. M. Heide, Ed. L e ctur e Notes In Computer Scienc e , vol. 2161. Springer-V erlag, London, 194 − 205. [CKK07] Christodoulou, G., Koutsoupias, E. and Kov´ acs, A . Mechanism Design for F ractional Scheduling on Unrelated Machines In Pr o c e e dings of ICALP 2007, 40 − 52. TRUTHFUL MECHANISM FOR SCHEDULING UNRELA TED MA CHINES 537 [CKV07] Christodoulou, G., Koutsoupias, E. and Vidali, A. A lo w er b ou n d for sc heduling mechanisms. In Pr o c e e dings of the eighte enth annual ACM-SIAM symp osium on Discr ete al gorithms (SODA ’07), 2007, 1163 − 1170. [Clark e71] Clarke, E. H. Multipart Pricing of Pub lic Go od s. Public Choic e , 11 : 17 − 33 , 1971. [GMW07] Gairing, M., Monien, B., and W o cla w, A. A faster combinatorial ap p ro ximation algorithm for sc hed uling u nrelated parallel machines. The or. Comput. Sci. 380, 1 − 2 (Jun. 2007), 87 − 99. [Gro ves1973 ] Gro ves, T. Incentives in T eams. Ec onometric a , 41 : 617 − 631 , 1973. [JP99] Jansen, K. and Pork olab, L. Imp ro ved approximation schemes for scheduling unrelated parallel ma- chines. I n Pr o c e e di ngs of the Thirty-First Ann ual ACM Symp osi um on the ory of Computing (Atlan ta, Georgia, United States, May 01 − 04, 1999). STOC ’99. ACM Pr ess , New Y ork, NY, 408 − 417. [LS07] La vi, R. and Swa my , C. T ruthful mechanism design for m ulti- d imensional scheduling via cy cle- monotonicit y . In Pr o c e e di ngs 8th A CM Confer enc e on Ele ctr onic Commer c e (EC) , 2007. 252-261 . [LST87] Len stra, J. K., Shmoys, D. B., and T ardos, ´ E. A p proximati on algorithms fo r sc heduling unrelated parallel machines. Math. Pr o gr am. 46 , 3 (Apr. 1990), 259 − 271. [MS07] Mu’alem, A . and S c hap ira, M. Setting low er b ounds on truthfulness. In Pr o c e e dings of the Eighte enth Ann ual ACM-SIAM Symp osium on Discr ete Algorithms (New Orleans, Louisiana, January 07 − 09, 2007). Symp osium on Di scr ete Algorithms. Society for Industrial and Applied Mathematics, Philadel- phia, P A , 1143 − 1152. [NR99] N isan, N. and Ronen, A. Algorithmic mechanism design (extend ed abstract). I n Pr o c e e dings of the Thirty-First A nnual A CM Symp osi um on the ory of Computing (Atlan ta, Georgia, United States, Ma y 01 − 04, 1999). STOC ’99. ACM Pr ess , New Y ork, NY, 129 − 140. [Rob erts79] R ob erts, K. The characteriza tion of implementable choise rules. I n Je an-Jac ques L affont, e ditor, A ggr e gation and R evelation of Pr efer enc es , pages 321 C 349. North-Holland, 1979. P ap ers presented at the first Europ ean Su mmer W orkshop of t h e Econometric So ciety . [Sourd01] S ourd, F. Scheduling T asks on U nrelated Machines: Large Neighborho od Improv ement Pro ce- dures. Journal of Heuristics 7, 6 (Nov. 2001), 519 − 531. [SS02] S c hulz, A . S. and Sk utella, M. Sc h ed uling Unrelated Mac hines b y Randomized Roun ding. SIAM J. Discr et. Math. 15, 4 (A pr. 2002), 450 − 469. [ST93] S hmoys, D. B. and T ardos, ´ E. Scheduling unrelated machines with costs. In Pr o c. F ourth Annual ACM-SIAM Symp osium on Di scr ete Algorithms (Austin, T exas, United States, 1993). Symp osium on Discr ete Algor i thms. So ci ety for Industrial and Applie d M athematics , Philadelphia, P A, 448 − 454. [SX02] S erna, M. and Xh afa, F. A pproximating Scheduling Unrelated Parallel Machines in Par allel. Comput. Optim. Appl . 21, 3 (Mar. 2002), 325 − 338. [Vic k rey61] Vickrey , W. Counterspecu lation, Auctions, and Comp etitive Sealed T enders. Journal of Financ e , 16 : 8 − 37 , 1961. App endix Pro of of C laim 3.6 (1) F or h -task j , assume t 1 j < t 2 j . W e can d ecrease t 2 j to αt 1 j , then t g bm will n ot c h ange since GBM alwa ys allo cates task j to agent 1. But this may help O P T , so the app ro ximation ratio can only b e w orse. (2) Increasing t 3 − i j to β t i j will not affect O P T but will increase t g bm . T his is b ecause the probabilit y to allo cate j do es n ot c hange as long as it is still an l -task, and on e t yp e v alue is increased. (3) It is s im ilar with the ab o ve. W e can keep increasing t 3 − i j while j is still m -task. Here β ≤ t 3 − i j /t i j ≤ α , so w e can mak e it equal α . (4) Here β − 1 ≥ t 3 − i j /t i j ≥ α − 1 , so we can increase t 3 − i j unt il this ratio equals β − 1 . (5) This is the same as in [NR99]. W e omit the pro of here. (6) Let h a , l a , a ∈ { 1 , 2 } den ote an h -task or l -task resp ectiv ely w h ic h is allo cated to agen t a in O P T . Let m a b , a, b ∈ { 1 , 2 } den ote an m -task allo cated to agen t b in O P T , on wh ic h 538 P . LU A ND C. YU agen t a has smaller t yp e v alue. So there are 8 types of tasks. W e will pr ov e that any tw o task j 1 , j 2 of the s ame type can b e com b ined into a single task j of the same typ e. Firstly , notice that j 1 , j 2 ha ve the same ratio of the t wo agen ts’ t yp e v alues. so task j still has this ratio, hence th e same t yp e. F urther more, they are all allocated by GBM with the same probabilit y distribution. In one d irection, combining will lea ve t opt unc h an ged. Obviously , com bin ing can only increase t opt b ecause an y allo cation obtained for the n ew ins tance can b e get for the old one. Also t opt can b e ac hieved for the new instance since tw o tasks of the same type are allocated to the s ame agen t. In the other d irection, com bining can only increase t g bm . F or the h -t ask case, t g bm is also un changed b eca us e GBM alw ays allo cate the h - tasks to th e more efficien t agen t. F or th e m -task case, assume j 1 , j 2 are b oth m a b , a, b ∈ { 1 , 2 } . Let Y denote an allocation of all the tasks except task j 1 , j 2 . Let t Y , j 1 ,j 2 (resp. t Y , j ) denote the exp ecte d make -sp an when j 1 , j 2 (resp. j ) are (is) allocated by GBM and all other tasks are allocated according to Y . W e ha v e to sho w that t Y , j 1 ,j 2 ≤ t Y , j . Let T 1 , T 2 denote finish ing time of t w o agen ts resp ectiv ely when allo cation is Y . If agen t i finishes last r egardless of ho w j 1 , j 2 are allo cated, then t Y , j 1 ,j 2 = T i + r i ( t i j 1 + t i j 2 ) = t Y , j Here r i denotes the probability that j 1 , j 2 and j are allo cated to agen t i . Otherw ise, if agen t i finish es last iff b oth j 1 , j 2 are allocated to it, then T 3 − i ≤ T i + t i j 1 + t i j 2 t Y , j 1 ,j 2 = r 2 i ( T i + t i j 1 + t i j 2 ) + r i (1 − r i )( T 3 − i + t 3 − i j 1 + T 3 − i + t 3 − i j 2 ) + (1 − r i ) 2 ( T 3 − i + t 3 − i j 1 + t 3 − i j 2 ) ≤ ( r 2 i + r i (1 − r i ))( T i + t i j 1 + t i j 2 ) + ((1 − r i ) 2 + r i (1 − r i ))( T 3 − i + t 3 − i j 1 + t 3 − i j 2 ) = r i ( T i + t i j 1 + t i j 2 ) + (1 − r i )( T 3 − i + t 3 − i j 1 + t 3 − i j 2 ) = t Y , j Finally assume that t i j 1 ≥ t i j 2 , i = 1 , 2 and consider the last case wh ere the agen t to whic h j 1 is allo cated fi nishes last. In this case t Y , j 1 ,j 2 = r 2 i ( T i + t i j 1 + t i j 2 ) + r i (1 − r i )( T i + t i j 1 ) + r i (1 − r i ) T 3 − i + t 3 − i j 1 ) + (1 − r i ) 2 ( T 3 − i + t 3 − i j 1 + t 3 − i j 2 ) ≤ ( r 2 i + r i (1 − r i ))( T i + t i j 1 + t i j 2 ) + ((1 − r i ) 2 + r i (1 − r i ))( T 3 − i + t 3 − i j 1 + t 3 − i j 2 ) = r i ( T i + t i j 1 + t i j 2 ) + (1 − r i )( T 3 − i + t 3 − i j 1 + t 3 − i j 2 ) = t Y , j The l -task case is sim ilar with m -task case, with r i = 1 2 . This wor k is licensed unde r the Creative Commons Attr ibution-NoDer ivs License. T o view a copy of this license, visit http:// creativecommons.org/licenses/by- nd/3.0/ .