Mechanism Design for Online Real-Time Scheduling

Transcription

1 Mechanism Design or Online Real-Time Scheduling Ryan Porter Computer Science Department Stanord University Stanord, CA ABSTRACT For the problem o online real-time scheduling o jobs on a single processor, previous work presents matching upper and lower bounds on the competitive ratio that can be achieved by a deterministic algorithm. However, these results only apply to the non-strategic setting in which the jobs are released directly to the algorithm. Motivated by emerging areas such as grid computing, we instead consider this problem in an economic setting, in which each job is released to a separate, sel-interested agent. The agent can then delay releasing the job to the algorithm, inlate its length, and declare an arbitrary value and deadline or the job, while the center determines not only the schedule, but the payment o each agent. For the resulting mechanism design problem in which we also slightly strengthen an assumption rom the non-strategic setting, we present a mechanism that addresses each incentive issue, while only increasing the competitive ratio by one. We then show a matching lower bound or deterministic mechanisms that never pay the agents. Categories and Subject Descriptors I.2.11 Artiicial Intelligence: Distributed Artiicial Intelligence Multiagent systems; J.4 Social and Behavioral Sciences: Economics; F.1.2 Computation by Abstract Devices: Modes o Computation Online computation General Terms Algorithms, Economics, Design, Theory Keywords Mechanism Design, Game Theory, Online Algorithms, Scheduling The author would like to acknowledge Yoav Shoham or his inluential comments on drats o the paper, and Yossi Azar or a useul discussion. This work was supported in part by DARPA grant F Permission to make digital or hard copies o all or part o this work or personal or classroom use is granted without ee provided that copies are not made or distributed or proit or commercial advantage and that copies bear this notice and the ull citation on the irst page. To copy otherwise, to republish, to post on servers or to redistribute to lists, requires prior speciic permission and/or a ee. EC 04, May 17 20, 2004, New York, New York, USA. Copyright 2004 ACM /04/ $ INTRODUCTION We consider the problem o online scheduling o jobs on a single processor. Each job is characterized by a release time, a deadline, a processing time, and a value or successul completion by its deadline. The objective is to maximize the sum o the values o the jobs completed by thr respective deadlines. The key challenge in this online setting is that the schedule must be constructed in real-time, even though nothing is known about a job until its release time. Competitive analysis 6, 10, with its roots in 12, is a well-studied approach or analyzing online algorithms by comparing them against the optimal oline algorithm, which has ull knowledge o the input at the beginning o its execution. One interpretation o this approach is as a game between the designer o the online algorithm and an adversary. First, the designer selects the online algorithm. Then, the adversary observes the algorithm and selects the sequence o jobs that maximizes the competitive ratio: the ratio o the value o the jobs completed by an optimal oline algorithm to the value o those completed by the online algorithm. Two papers paint a complete picture in terms o competitive analysis or this setting, in which the algorithm is assumed to know k, the maximum ratio between the value densities value divided by processing time o any two jobs. For k = 1, 4 presents a 4-competitive algorithm, and proves that this is a lower bound on the competitive ratio or deterministic algorithms. The same paper also generalizes the lower bound to 1 + k 2 or any k 1, and 15 then presents a matching 1 + k 2 -competitive algorithm. The setting addressed by these papers is completely nonstrategic, and the algorithm is assumed to always know the true characteristics o each job upon its release. However, in domains such as grid computing see, or example, 7, 8 this assumption is invalid, because buyers o processor time choose when and how to submit thr jobs. Furthermore, sellers not only schedule jobs but also determine the amount that they charge buyers, an issue not addressed in the non-strategic setting. Thus, we consider an extension o the setting in which each job is owned by a separate, sel-interested agent. Instead o bng released to the algorithm, each job is now released only to its owning agent. Each agent now has our dierent ways in which it can manipulate the algorithm: it decides when to submit the job to the algorithm ater the true release time, it can artiicially inlate the length o the job, and it can declare an arbitrary value and deadline or the job. Because the agents are sel-interested, they will choose to manipulate the algorithm i doing so will cause 61

2 thr job to be completed; and, indeed, one can ind examples in which agents have incentive to manipulate the algorithms presented in 4 and 15. The addition o sel-interested agents moves the problem rom the area o algorithm design to that o mechanism design 17, the science o crating protocols or sel-interested agents. Recent years have seen much activity at the interace o computer science and mechanism design see, e.g., 9, 18, 19. In general, a mechanism dnes a protocol or interaction between the agents and the center that culminates with the selection o an outcome. In our setting, a mechanism will take as input a job rom each agent, and return a schedule or the jobs, and a payment to be made by each agent to the center. A basic solution concept o mechanism design is incentive compatibility, which, in our setting, requires that it is always in each agent s best interests to immediately submit its job upon release, and to truthully declare its value, length, and deadline. In order to evaluate a mechanism using competitive analysis, the adversary model must be updated. In the new model, the adversary still determines the sequence o jobs, but it is the sel-interested agents who determine the observed input o the mechanism. Thus, in order to achieve a competitive ratio o c, an online mechanism must both be incentive compatible, and always achieve at least 1 o the c value that the optimal oline mechanism achieves on the same sequence o jobs. The rest o the paper is structured as ollows. In Section 2, we ormally dne and review results rom the original, non-strategic setting. Ater introducing the incentive issues through an example, we ormalize the mechanism design setting in Section 3. In Section 4 we present our irst main result, a 1 + k competitive mechanism, and ormally prove incentive compatibility and the competitive ratio. We also show how we can simpliy this mechanism or the special case in which k = 1 and each agent cannot alter the length o its job. Returning the general setting, we show in Section 5 that this competitive ratio is a lower bound or deterministic mechanisms that do not pay agents. Finally, in Section 6, we discuss related work other than the directly relevant 4 and 15, beore concluding with Section NON-STRATEGIC SETTING In this section, we ormally dne the original, non-strategic setting, and recap previous results. 2.1 Formulation There exists a single processor on which jobs can execute, and N jobs, although this number is not known beorehand. Each job i is characterized by a tuple θ i = r i,d i,l i,v i, which denotes the release time, deadline, length o processing time required, and value, respectively. The space Θ i o possible tuples is the same or each job and consists o all θ i such that r i,d i,l i,v i R + thus, the model o time is continuous. Each job is released at time r i,atwhichpoint its three other characteristics are known. Nothing is known about the job beore its arrival. Each deadline is irm or, hard, which means that no value is obtained or a job that is completed ater its deadline. Preemption o jobs is allowed, and it takes no time to switch between jobs. Thus, job i is completed i and only i the total time it executes on the processor beore d i is at least l i. Let θ =θ 1,...,θ N denote the vector o tuples or all jobs, and let θ i =θ 1,...,θ i 1,θ i+1,...,θ N denotethe same vector without the tuple or job i. Thus, θ i,θ i denotes a complete vector o tuples. Dne the value density ρ i = v i l i o job i to be the ratio o its value to its length. For an input θ, denote the maximum and minimum value densities as ρ min =min i ρ i and ρ max = max i ρ i. The importance ratio is then dned to be ρmax ρ min, the maximal ratio o value densities between two jobs. The algorithm is assumed to always know an upper bound k on the importance ratio. For simplicity, we normalize the range o possible value densities so that ρ min =1. An online algorithm is a unction :Θ 1... Θ N O that maps the vector o tuples or any number N to an outcome o. An outcome o O is simply a schedule o jobs on the processor, recorded by the unction S : R + {0, 1,...,N}, which maps each point in time to the active job, or to 0 i the processor is idle. To denote the total elapsed time that a job has spent on the processor at time t, we will use the unction e it = t µsx =idx, whereµ is an indicator unction that 0 returns 1 i the argument is true, and zero otherwise. A job s laxity at time t is dned to be d i t l i + e it, the amount o time that it can remain inactive and still be completed by its deadline. A job is abandoned i it cannot be completed by its deadline ormally, i d i t + e it <l i. Also, overload S ande i so that they can also take a vector θ as an argument. For example, Sθ, t is shorthand or the St o the outcome θ, and it denotes the active job at time t when the input is θ. Since a job cannot be executed beore its release time, the space o possible outcomes is restricted in that Sθ, t = i implies r i t. Also, because the online algorithm must produce the schedule over time, without knowledge o uture inputs, it must make the same decision at time t or inputs that are indistinguishable at this time. Formally, let θt denote the subset o the tuples in θ that satisy r i t. The constraint is then that θt =θ t implies Sθ, t =Sθ,t. The objective unction is the sum o the values o the jobs that are completed by thr respective deadlines: W o, θ = i vi µe iθ, d i l i. Let W θ = max o O W o, θ denote the maximum possible total value or the proile θ. In competitive analysis, an online algorithm is evaluated by comparing it against an optimal oline algorithm. Because the oline algorithm knows the entire input θ at time 0 but still cannot start each job i until time r i, it always achieves W θ. An online algorithm is strictly c-competitive i there does not exist an input θ such that c W θ,θ <W θ. An algorithm that is c-competitive is also said to achieve a competitive ratio o c. We assume that there does not exist an overload period o ininite duration. A period o time t s,t is overloaded i the sum o the lengths o the jobs whose release time and deadline both all within the time period exceeds the duration o the interval ormally, i t t s i t s r i,d i t li. Without such an assumption, it is not possible to achieve a inite competitive ratio Previous Results In the non-strategic setting, 4 presents a 4-competitive algorithm called TD 1 version 2 or the case o k = 1, while 15 presents a 1+ k 2 -competitive algorithm called D over or the general case o k 1. Matching lower bounds or deterministic algorithms or both o these cases were shown 62

3 in 4. In this section we provide a high-level description o TD 1 version 2 using an example. TD 1 version 2 divides the schedule into intervals, each o which begins when the processor transitions rom idle to busy call this time t b, and ends with the completion o a job. The irst active job o an interval may have laxity; however, or the remainder o the interval, preemption o the active job is only considered when some other job has zero laxity. For example, when the input is the set o jobs listed in Table 1, the irst interval is the complete execution o job 1 over the range 0.0, 0.9. No preemption is considered during this interval, because job 2 has laxity until time 1.5. Then, a new interval starts at t b =0.9 when job 2 becomes active. Beore job 2 can inish, preemption is considered at time 4.8, when job 3 is released with zero laxity. In order to decide whether to preempt the active job, TD 1 version 2 uses two more variables: t e and p loss. The ormer records the latest deadline o a job that would be abandoned i the active job executes to completion or, i no such job exists, the time that the active job will inish i it is not preempted. In this case, t e =17.0. The value t e t b represents the an upper bound on the amount o possible execution time lost to the optimal oline algorithm due to the completion o the active job. The other variable, p loss, is equal to the length o the irst active job o the current interval. Because in general this job could have laxity, the oline algorithm may be able to complete it outside o the range t b,t e. 1 I the algorithm completes the active job and this job s length is at least te t b +p loss, then the 4 algorithm is guaranteed to be 4-competitive or this interval note that k = 1 implies that all jobs have the same value density and thus that lengths can used to compute the competitive ratio. Because this is not case at time 4.8 since te t b +p loss = > 4.0 =l 4 4 2, the algorithm preempts job 2 or job 3, which then executes to completion. Job r i d i l i v i Table 1: Input used to recap TD 1 version 2 4. The up and down arrows represent r i and d i, respectively, while the length o the box equals l i. 3. MECHANISM DESIGN SETTING However, alse inormation about job 2 would cause TD 1 version 2 to complete this job. For example, i job 2 s deadline were declared as ˆd 2 =4.7, then it would have zero laxity at time 0.7. At this time, the algorithm would preempt job 1 or job 2, because te t b +p loss = > 0.9 =l Job 2 would then complete beore the arrival o job While it would be easy to alter the algorithm to recognize that this is not possible or the jobs in Table 1, our example does not depend on the use o p loss. 2 While we will not describe the signiicantly more complex In order to address incentive issues such as this one, we need to ormalize the setting as a mechanism design problem. In this section we irst present the mechanism design ormulation, and then dne our goals or the mechanism. 3.1 Formulation There exists a center, who controls the processor, and N agents, where the value o N is unknown by the center beorehand. Each job i is owned by a separate agent i. The characteristics o the job dne the agent s type θ i Θ i. At time r i,agenti privately observes its type θ i, and has no inormation about job i beore r i. Thus, jobs are still released over time, but now each job is revealed only to the owning agent. Agents interact with the center through a direct mechanism Γ = Θ 1,...,Θ N,g, in which each agent declares a job, denoted by ˆθ i =ˆr i, ˆd i, ˆl i, ˆv i, and g :Θ 1... Θ N O maps the declared types to an outcome o O. Anoutcome o =S,p 1,...,p N consists o a schedule and a payment rom each agent to the mechanism. In a standard mechanism design setting, the outcome is enorced at the end o the mechanism. However, since the end is not well-dned in this online setting, we choose to model returning the job i it is completed and collecting a payment rom each agent i as occurring at ˆd i, which, according to the agent s declaration, is the latest relevant point o time or that agent. That is, even i job i is completed beore ˆd i, the center does not return the job to agent i until that time. This modelling decision could instead be viewed as a decision by the mechanism designer rom a larger space o possible mechanisms. Indeed, as we will discuss later, this decision o when to return a completed job is crucial to our mechanism. Each agent s utility, u igˆθ,θ i=v i µe iˆθ, d i l i µ ˆd i d i p iˆθ, is a quasi-linear unction o its value or its job i completed and returned by its true deadline and the payment it makes to the center. We assume that each agent is a rational, expected utility maximizer. Agent declarations are restricted in that an agent cannot declare a length shorter than the true length, since the center would be able to detect such a lie i the job were completed. On the other hand, in the general ormulation we will allow agents to declare longer lengths, since in some settings it may be possible add unnecessary work to a job. However, we will also consider a restricted ormulation in which this type o lie is not possible. The declared release time ˆr i is the time that the agent chooses to submit job i to the center, and it cannot precede the time r i at which the job is revealed to the agent. The agent can declare an arbitrary deadline or value. To summarize, agent i can declare any type ˆθ i =ˆr i, ˆd i, ˆl i, ˆv i such that ˆl i l i and ˆr i r i. While in the non-strategic setting it was suicient or the algorithm to know the upper bound k on the ratio ρmax ρ min, in the mechanism design setting we will strengthen this assumption so that the mechanism also knows ρ min or, equivalently, the range ρ min,ρ max o possible value densities. 3 D over, we note that it is similar in its use o intervals and its preerence or the active job. Also, we note that the lower bound we will show in Section 5 implies that alse inormation can also bent a job in D over. 3 Note that we could then orce agent declarations to satisy ρ min ˆv i ρ ˆli max. However, this restriction would not 63

4 While we eel that it is unlikely that a center would know k without knowing this range, we later present a mechanism that does not depend on this extra knowledge in a restricted setting. The restriction on the schedule is now that Sˆθ, t =i implies ˆr i t, to capture the act that a job cannot be scheduled on the processor beore it is declared to the mechanism. As beore, preemption o jobs is allowed, and job switching takes no time. The constraints due to the online mechanism s lack o knowledge o the uture are that ˆθt =ˆθ t implies Sˆθ, t = Sˆθ,t, and ˆθ ˆd i=ˆθ ˆd i implies p iˆθ =p iˆθ or each agent i. The setting can then be summarized as ollows. Overview 1 o the Setting: or all t do The center instantiates Sˆθ, t i, or some i s.t. ˆr i t i i, r i = t then θ i is revealed to agent i i i, t r i and agent i has not declared a job then Agent i can declare any job ˆθ i,s.t. ˆr i = t and ˆl i l i i i, ˆd i = t e iˆθ, t l i then Completed job i is returned to agent i i i, ˆd i = t then Center sets and collects payment p iˆθ rom agent i 3.2 Mechanism Goals Our aim as mechanism designer is to maximize the value o completed jobs, subject to the constraints o incentive compatibility and individual rationality. The condition or dominant strategy incentive compatibility is that or each agent i, regardless o its true type and o the declared types o all other agents, agent i cannot increase its utility by unilaterally changing its declaration. Dnition 1. A direct mechanism Γ satisies incentive compatibility IC i i, θ i,θ i, ˆθ i : u igθ i, ˆθ i,θ i u igθ i, ˆθ i,θ i From an agent perspective, dominant strategies are desirable because the agent does not have to reason about ther the strategies o the other agents or the distribution rom the which other agent s types are drawn. From a mechanism designer perspective, dominant strategies are important because we can reasonably assume that an agent who has a dominant strategy will play according to it. For these reasons, in this paper we require dominant strategies, as opposed to a weaker equilibrium concept such as Bayes-Nash, under which we could improve upon our positive results. 4 decrease the lower bound on the competitive ratio. 4 A possible argument against the need or incentive compatibility is that an agent s lie may actually improve the schedule. In act, this was the case in the example we showed or the alse declaration ˆd 2 =4.7. However, i an agent lies due to incorrect belies over the uture input, then the lie could instead make the schedule the worse or example, i job 3 were never released, then job 1 would have been unnecessarily abandoned. Furthermore, i we do not know the belies o the agents, and thus cannot predict how they will lie, then we can no longer provide a competitive guarantee or our mechanism. While restricting ourselves to incentive compatible direct mechanisms may seem limiting at irst, the Revelation Principle or Dominant Strategies see, e.g., 17 tells us that i our goal is dominant strategy implementation, then we can make this restriction without loss o generality. The second goal or our mechanism, individual rationality, requires that agents who truthully reveal thr type never have negative utility. The rationale behind this goal is that participation in the mechanism is assumed to be voluntary. Dnition 2. A direct mechanism Γ satisies individual rationality IR i i, θ i, ˆθ i, u igθ i, ˆθ i,θ i 0. Finally, the social welare unction that we aim to maximize is the same as the objective unction o the non-strategic setting: W o, θ = i vi µe iθ, d i l i. As in the nonstrategic setting, we will evaluate an online mechanism using competitive analysis to compare it against an optimal oline mechanism which we will denote by Γ oline. An oline mechanism knows all o the types at time 0, and thus can always achieve W θ. 5 Dnition 3. An online mechanism Γ is strictly c- competitive i it satisies IC and IR, and i there does not existaproileoagenttypesθ such that c W gθ,θ <W θ. 4. RESULTS In this section, we irst present our main positive result: a 1+ k competitive mechanism Γ 1. Ater providing some intuition as to why Γ 1 satisies individual rationality and incentive compatibility, we ormally prove irst these two properties and then the competitive ratio. We then consider a special case in which k = 1 and agents cannot lie about the length o thr job, which allows us to alter this mechanism so that it no longer requires ther knowledge o ρ min or the collection o payments rom agents. Unlike TD 1 version 2 and D over, Γ 1 gives no preerence to the active job. Instead, it always executes the available job with the highest priority: ˆv i + k e iˆθ, t ρ min. Each agent whose job is completed is then charged the lowest value that it could have declared such that its job still would have been completed, holding constant the rest o its declaration. By the use o a payment rule similar to that o a secondprice auction, Γ 1 satisies both IC with respect to values and IR. We now argue why it satisies IC with respect to the other three characteristics. Declaring an improved job i.e., declaring an earlier release time, a shorter length, or a later deadline could possibly decrease the payment o an agent. However, the irst two lies are not possible in our setting, while the third would cause the job, i it is completed, to be returned to the agent ater the true deadline. This is the reason why it is important to always return a completed job at its declared deadline, instead o at the point at which it is completed. 5 Another possibility is to allow only the agents to know thr types at time 0, and to orce Γ oline to be incentive compatible so that agents will truthully declare thr types at time 0. However, this would not aect our results, since executing a VCG mechanism see, e.g., 17 at time 0 both satisies incentive compatibility and always maximizes social welare. 64

5 Mechanism 1 Γ 1 Execute Sˆθ, according to Algorithm 1 or all i do i e iˆθ, ˆd i ˆl i {Agent i s job is completed} then p iˆθ arg min v i 0e iˆr i, ˆd i, ˆl i,v i, ˆθ i, ˆd i ˆl i else p iˆθ 0 Algorithm 1 or all t do Avail {i t ˆr i e iˆθ, t < ˆl i e iˆθ, t+ ˆd i t ˆl i} {Set o all released, non-completed, non-abandoned jobs} i Avail then Sˆθ, t arg max i Avail ˆv i + k e iˆθ, t ρ min {Break ties in avor o lower ˆr i } else Sˆθ, t 0 It remains to argue why an agent does not have incentive to worsen its job. The only possible eects o an inlated length are delaying the completion o the job and causing it to be abandoned, and the only possible eects o an earlier declared deadline are causing to be abandoned and causing it to be returned earlier which has no eect on the agent s utility in our setting. On the other hand, it is less obvious why agents do not have incentive to declare a later release time. Consider a mechanism Γ 1 that diers rom Γ 1 in that it does not preempt the active job i unless there exists another job j such that ˆv i + k l iˆθ, t ρ min < ˆv j.notethat as an active job approaches completion in Γ 1, its condition or preemption approaches that o Γ 1. However, the types in Table 2 or the case o k =1show why an agent may have incentive to delay the arrival o its job under Γ 1. Job 1 becomes active at time 0, and job 2 is abandoned upon its release at time 6, because = v 1+l 1 >v 2 = 13. Then, at time 8, job 1 is preempted by job 3, because = v 1 + l 1 <v 3 = 22. Job 3 then executes to completion, orcing job 1 to be abandoned. However, job 2 had more wght than job 1, and would have prevented job 3 rom bng executed i it had been the active job at time 8, since = v 2 + l 2 >v 3 = 22. Thus, i agent 1 had alsely declared ˆr 1 = 20, then job 3 would have been abandoned at time 8, and job 1 would have completed over the range 20, 30. Job r i d i l i v i Table 2: Jobs used to show why a slightly altered version o Γ 1 would not be incentive compatible with respect to release times. Intuitively, Γ 1 avoids this problem because o two properties. First, when a job becomes active, it must have a greater priority than all other available jobs. Second, because a job s priority can only increase through the increase o its elapsed time, e iˆθ, t, the rate o increase o a job s priority is independent o its characteristics. These two properties together imply that, while a job is active, there cannot exist a time at which its priority is less than the priority that one o these other jobs would have achieved by executing on the processor instead. 4.1 Proo o Individual Rationality and Incentive Compatibility Ater presenting the trivial proo o IR, we break the proooicintolemmas. Theorem 1. Mechanism Γ 1 satisies individual rationality. Proo. For arbitrary i, θ i, ˆθ i, ijobi is not completed, then agent i pays nothing and thus has a utility o zero; that is, p iθ i, ˆθ i =0andu igθ i, ˆθ i,θ i = 0. On the other hand, i job i is completed, then its value must exceed agent i s payment. Formally, u igθ i, ˆθ i,θ i=v i arg min v i 0e ir i,d i,l i,v i, ˆθ i,d i l i 0 must hold, since v i = v i satisies the condition. To prove IC, we need to show that or an arbitrary agent i, and an arbitrary proile ˆθ i o declarations o the other agents, agent i can never gain by making a alse declaration ˆθ i θ i, subject to the constraints that ˆr i r i and ˆl i l i. We start by showing that, regardless o ˆv i, i truthul declarations o r i, d i,andl i do not cause job i to be completed, then worse declarations o these variables that is, declarations that satisy ˆr i r i, ˆl i l i and ˆd i d icannever cause the job to be completed. We break this part o the proo into two lemmas, irst showing that it holds or the release time, regardless o the declarations o the other variables, and then or length and deadline. Lemma 2. In mechanism Γ 1, the ollowing condition holds or all i, θ i, ˆθ i: ˆv i, ˆl i l i, ˆdi d i, ˆr i r i, ˆri, ˆd i, ˆl i, ˆv i, ˆθ i, ˆd i ˆli = ri, ˆd i, ˆl i, ˆv i, ˆθ i, ˆd i ˆli Proo. Assume by contradiction that this condition does not hold that is, job i is not completed when r i is truthully declared, but is completed or some alse declaration ˆr i r i. We irst analyze the case in which the release time is truthully declared, and then we show that job i cannot be completed when agent i delays submitting it to the center. Case I: Agent i declares ˆθ i =r i, ˆd i, ˆl i, ˆv i. First, dne the ollowing three points in the execution o job i. Let t s =argmin t Sˆθ i, ˆθ i,t=i bethetimethat job i irst starts execution. Let t p =argmin t>t s Sˆθ i, ˆθ i,t i be the time that job i is irst preempted. Let t a =argmin t ˆθ i, ˆθ i,t+ ˆd i t<ˆl i be the time that job i is abandoned. 65

6 I t s and t p are undned because job i never becomes active, then let t s = t p = t a. Also, partition the jobs declared by other agents beore t a into the ollowing three sets. X = {j ˆr j <t p j i} consists o the jobs other than i that arrive beore job i is irst preempted. Y = {j t p ˆr j t a ˆv j > ˆv i + k e iˆθ i, ˆθ i, ˆr j} consists o the jobs that arrive in the range t p,t a and that when they arrive have higher priority than job i note that we are make use o the normalization. Z = {j t p ˆr j t a ˆv j ˆv i + k e iˆθ i, ˆθ i, ˆr j} consists o the jobs that arrive in the range t p,t a and that when they arrive have lower priority than job i. We now show that all active jobs during the range t p,t a must be ther i or in the set Y. Unless t p = t a in which case this property trivially holds, it must be the case that job i has a higher priority than an arbitrary job x X at time t p, since at the time just preceding t p job x was available and job i was active. Formally, ˆv x + k e xˆθ i, ˆθ i,t p < ˆv i + k e iˆθ i, ˆθ i,t p musthold. 6 We can then show that, over the range t p,t a, no job x X runs on the processor. Assume by contradiction that this is not true. Let t t p,t a be the earliest time in this range that some job x X is active, which implies that e xˆθ i, ˆθ i,t =e xˆθ i, ˆθ i,t p. We can then show that job i has a higher priority at time t as ollows: ˆv x+ k e xˆθ i, ˆθ i,t =ˆv x+ k e xˆθ i, ˆθ i,t p < ˆv i + k e iˆθ i, ˆθ i,t p ˆv i + k e iˆθ i, ˆθ i,t, contradicting the act that job x is active at time t. A similar argument applies to an arbitrary job z Z, starting at it release time ˆr z >t p, since by dnition job i has a higher priority at that time. The only remaining jobs that can be active over the range t p,t a ar and those in the set Y. Case II: Agent i declares ˆθ i =ˆr i, ˆd i, ˆl i, ˆv i, where ˆr i >r i. We now show that job i cannot be completed in this case, given that it was not completed in case I. First, we can restrict the range o ˆr i that we need to consider as ollows. Declaring ˆr i r i,t s would not aect the schedule, since t s would still be the irst time that job i executes. Also, declaring ˆr i >t a could not cause the job to be completed, since d i t a < ˆl i holds, which implies that job i would be abandoned at its release. Thus, we can restrict consideration to ˆr i t s,t a. In order or declaring ˆθ i to cause job i to be completed, a necessary condition is that the execution o some job y c Y must change during the range t p,t a, since the only jobs other than i that are active during that range are in Y. Let t c = argmin t t p,t a y c Y,Sˆθ i, ˆθ i,t = y c Sˆθ i, ˆθ i,t y c be the irst time that such a change occurs. We will now show that or any ˆr i t s,t a, there cannot exist a job with higher priority than y c at time t c, contradicting Sˆθ i, ˆθ i,t y c. First note that job i cannot have a higher priority, since there would have to exist a t t p,t c such that y 6 For simplicity, when we give the ormal condition or a job x to have a higher priority than another job y, we will assume that job x s priority is strictly greater than job y s, because, in the case o a tie that avors x, uture ties would also be broken in avor o job x. Y,Sˆθ i, ˆθ i,t=y Sˆθ i, ˆθ i,t=i, contradicting the dnition o t c. Now consider an arbitrary y Y such that y y c. In case I, we know that job y has lower priority than y c at time t c ; that is, ˆv y + k e yˆθ i, ˆθ i,t c < ˆv y c + k e y cˆθ i, ˆθ i,t c. Thus, moving to case II, job y must replace some other job beore t c.sinceˆr y t p, the condition is that there must exist some t t p,t c such that w Y {i}, Sˆθ i, ˆθ i,t= w Sˆθ i, ˆθ i,t=y. Since w Y would contradict the dnition o t c,weknowthatw = i. That is, the job that y replaces must be i. By dnition o the set Y, we know that ˆv y > ˆv i + k e iˆθ i, ˆθ i, ˆr y. Thus, i ˆr y t, thenjobi could not have executed instead o y in case I. On the other hand, i ˆr y >t,thenjoby obviously could not execute at time t, contradicting the existence o such a time t. Now consider an arbitrary job x X. We know that in case I job i has a higher priority than job x at time t s, or, ormally, that ˆv x + k e xˆθ i, ˆθ i,t s < ˆv i + k e iˆθ i, ˆθ i,t s. We also know that ˆv i+ k e iˆθ i, ˆθ i,t c < ˆv y c + k e y cˆθ i, ˆθ i,t c. Since delaying i s arrival will not aect the execution up to time t s, and since job x cannot execute instead o a job y Y at any time t t p,t c by dnition o t c, the only way or job x s priority to increase beore t c as we move rom case I to II is to replace job i over the range t s,t c. Thus, an upper bound on job x s priority when agent i declares ˆθ i is: ˆv x+ k e xˆθ i, ˆθ i,t s +e iˆθ i, ˆθ i,t c e iˆθ i, ˆθ i,t s < ˆv i+ k e iˆθ i, ˆθ i,t s +e iˆθ i, ˆθ i,t c e iˆθ i, ˆθ i,t s = ˆv i + k e iˆθ i, ˆθ i,t c < ˆv y c + k e y cˆθ i, ˆθ i,t c. Thus, even at this upper bound, job y c would execute instead o job x at time t c. A similar argument applies to an arbitrary job z Z, starting at it release time ˆr z.since the sets {i},x,y,z partition the set o jobs released beore t a, we have shown that no job could execute instead o job y c, contradicting the existence o t c, and completing the proo. Lemma 3. In mechanism Γ 1, the ollowing condition holds or all i, θ i, ˆθ i: ˆv i, ˆl i l i, ˆdi d i, ri, ˆd i, ˆl i, ˆv i, ˆθ i, ˆd i ˆli = ri,d i,l i, ˆv i, ˆθ i, ˆd i li Proo. Assume by contradiction there exists some instantiation o the above variables such that job i is not completed when l i and d i are truthully declared, but is completed or some pair o alse declarations ˆl i l i and ˆd i d i. Note that the only eect that ˆd i and ˆl i have on the execution o the algorithm is on whether or not i Avail. Speciically, they aect the two conditions: e iˆθ, t < ˆl i and e iˆθ, t+ ˆd i t ˆl i. Because job i is completed when ˆli and ˆd i are declared, the ormer condition or completion must become alse beore the latter. Since truthully declaring l i ˆl i and d i ˆd i will only make the ormer condition become alse earlier and the latter condition become alse later, the execution o the algorithm will not be aected when moving to truthul declarations, and job i will be completed, a contradiction. We now use these two lemmas to show that the payment or a completed job can only increase by alsely declaring worse ˆl i, ˆd i,andˆr i. 66

7 Lemma 4. In mechanism Γ 1, the ollowing condition holds or all i, θ i, ˆθ i: ˆl i l i, ˆdi d i, ˆr i r i, arg min ˆri, ˆd i, ˆl i,v i, ˆθ i, ˆd i ˆli v i 0 arg min ri,d i,l i,v i, ˆθ i,d i li v i 0 Proo. Assume by contradiction that this condition does not hold. This implies that there exists some value v i such that the condition e iˆr i, ˆd i, ˆl i,v i, ˆθ i, ˆd i ˆl i holds, but e ir i,d i,l i,v i, ˆθ i,d i l i does not. Applying Lemmas 2 and 3: e iˆr i, ˆd i, ˆl i,v i, ˆθ i, ˆd i ˆl i = e ir i, ˆd i, ˆl i,v i, ˆθ i, ˆd i ˆl i = e ir i,d i,l i,v i, ˆθ i,d i l i, a contradiction. Finally, the ollowing lemma tells us that the completion o a job is monotonic in its declared value. Lemma 5. In mechanism Γ 1, the ollowing condition holds or all i, ˆθ i, ˆθ i: ˆv i ˆv i, ˆri, ˆd i, ˆl i, ˆv i, ˆθ i, ˆd i ˆli = ˆri, ˆd i, ˆl i, ˆv i, ˆθ i, ˆd i ˆli The proo, by contradiction, o this lemma is omitted because it is essentially identical to that o Lemma 2 or ˆr i.in case I, agent i declares ˆr i, ˆd i, ˆl i, ˆv i and the job is not completed, while in case II he declares ˆr i, ˆd i, ˆl i, ˆv i and the job is completed. The analysis o the two cases then proceeds as beore the execution will not change up to time t s because the initial priority o job i decreases as we move rom case I to II; and, as a result, there cannot be a change in the execution o a job other than i over the range t p,t a. We can now combine the lemmas to show that no proitable deviation is possible. Theorem 6. Mechanism Γ 1 satisies incentive compatibility. Proo. For an arbitrary agent i, we know that ˆr i r i and ˆl i l i hold by assumption. We also know that agent i has no incentive to declare ˆd i >d i, because job i would never be returned beore its true deadline. Then, because the payment unction is non-negative, agent i s utility could not exceed zero. By IR, this is the minimum utility it would achieve i it truthully declared θ i. Thus, we can restrict consideration to ˆθ i that satisy ˆr i r i, ˆl i l i,and ˆd i d i. Again using IR, we can urther restrict consideration to ˆθ i that cause job i to be completed, since any other ˆθ i yields a utility o zero. I truthul declaration o θ i causes job i to be completed, then by Lemma 4 any such alse declaration ˆθ i could not decrease the payment o agent i. On the other hand, i truthul declaration does not cause job i to be completed, then declaring such a ˆθ i will cause agent i to have negative utility, since v i < arg min v i 0 r i,d i,l i,v i, ˆθ i, ˆd i l i arg minv i 0 ˆr i, ˆd i, ˆl i,v i, ˆθ i, ˆd i ˆl i holds by Lemmas 5 and 4, respectively. 4.2 Proo o Competitive Ratio The proo o the competitive ratio, which makes use o techniques adapted rom those used in 15, is also broken into lemmas. Having shown IC, we can assume truthul declaration ˆθ = θ. Since we have also shown IR, in order to prove the competitive ratio it remains to bound the loss o social welare against Γ oline. Denote by 1, 2,...,F the sequence o jobs completed by Γ 1. Divide time into intervals I =t open,t close, one or each job in this sequence. Set t close to be the time at which job is completed, and set t open = t close 1 or 2, and t open 1 = 0 or = 1. Also, let t begin be the irst time that the processor is not idle in interval I. Lemma 7. For any interval I, the ollowing inequality holds: t close t begin k v Proo. Interval I begins with a possibly zero length period o time in which the processor is idle because there is no available job. Then, it continuously executes a sequence o jobs 1, 2,...,c, where each job i in this sequence is preempted by job i + 1, except or job c, which is completed thus, job c in this sequence is the same as job is the global sequence o completed jobs. Let t s i be the time that job i begins execution. Note that t s 1 = t begin. Over the range t begin,t close, the priority v i+ k e iθ, t o the active job is monotonically increasing with time, because this unction linearly increases while a job is active, and can only increase at a point in time when preemption occurs. Thus, each job i>1 in this sequence begins execution at its release time that is, t s i = r i, because its priority does not increase while it is not active. We now show that the value o the completed job c exceeds the product o k and the time spent in the interval on jobs 1 through c 1, or, more ormally, that the ollowing condition holds: v c k c 1 h=1 e hθ, t s h+1 e h θ, t s h. To show this, we will prove by induction that the stronger condition v i k i 1 h=1 e hθ, t s h+1 holds or all jobs i in the sequence. Base Case: For i =1,v 1 k 0 h=1 e hθ, t s h+1 =0, since the sum is over zero elements. Inductive Step: For an arbitrary 1 i<c, we assume that v i k i 1 h=1 e hθ, t s h+1 holds. At time t s i+1, we know that v i+1 v i + k e iθ, t s i+1 holds, because t s i+1 = r i+1. These two inequalities together imply that v i+1 k i h=1 e hθ, t s h+1, completing the inductive step. We also know that t close t s c l c v c must hold, by the simpliying normalization o ρ min = 1 and the act that job c s execution time cannot exceed its length. We can thus by: t close bound the total execution time o I t close t s c+ c 1 h=1 e hθ, t s h+1 e h θ, t s h 1+ 1 t begin = k v.. Because the processor is idle ater t close F, any such job i would become active at some time t t close F, which would lead to the completion o some job, creating a new interval and contradicting the act that I F is the last one. We now consider the possible execution o uncompleted jobs by Γ oline. Associate each job i that is not completed by Γ 1 with the interval during which it was abandoned. All jobs are now associated with an interval, since there are no gaps between the intervals, and since no job i can be abandoned ater the close o the last interval at t close F 67

8 The ollowing lemma is equivalent to Lemma 5.6 o 15, but the proo is dierent or our mechanism. Lemma 8. For any interval I and any job i abandoned in I, the ollowing inequality holds: v i 1 + kv. Proo. Assume by contradiction that there exists a job i abandoned in I such that v i > 1 + kv. At t close, the priority o job is v + k l < 1 + kv. Because the priority o the active job monotonically increases over the range t begin,t close, job i would have a higher priority than the active job and thus begin execution at some time t t begin,t close. Again applying monotonicity, this would imply that the priority o the active job at t close exceeds 1 + kv, contradicting the act that it is 1 + kv. As in 15, or each interval I,wegiveΓ oline the ollowing git : k times the amount o time in the range t begin,t close that it does not schedule a job. Additionally, we give the adversary v, since the adversary may be able to complete this job at some uture time, due to the act that Γ 1 ignores deadlines. The ollowing lemma is Lemma 5.10 in 15, and its proo now applies directly. Lemma With the above gits the total net gain obtained by the clairvoyant algorithm rom scheduling the jobs abandoned during I is not greater than 1 + k v. The intuition behind this lemma is that the best that the adversary can do is to take almost all o the git o k t close t begin intuitively, this is equivalent to executing jobs with the maximum possible value density over the time that Γ 1 is active, and then begin execution o a job abandoned by Γ 1 right beore t close. By Lemma 8, the value o this job is bounded by 1 + k v. We can now combine the results o these lemmas to prove the competitive ratio. Theorem 10. Mechanism Γ 1 is 1+ k competitive. Proo. Using the act that the way in which jobs are associated with the intervals partitions the entire set o jobs, we can show the competitive ratio by showing that Γ 1 is 1+ k competitive or each interval in the sequence 1,...,F. Over an arbitrary interval I, the oline algorithm can achieve at most t close t begin k+v +1+ kv, rom the two gits and the net gain bounded by Lemma 9. Applying Lemma 7, this quantity is then bounded rom above by 1+ 1 k v k+v +1+ kv = 1+ k 2 +1 v. Since Γ 1 achieves v, the competitive ratio holds. 4.3 Special Case: Unalterable length and k=1 While so ar we have allowed each agent to lie about all our characteristics o its job, lying about the length o the job is not possible in some settings. For example, a user may not know how to alter a computational problem in a way that both lengthens the job and allows the solution o the original problem to be extracted rom the solution to the altered problem. Another restriction that is natural in some settings is uniorm value densities k = 1,whichwas the case considered by 4. I the setting satisies these two conditions, then, by using Mechanism Γ 2, we can achieve a competitive ratio o 5 which is the same competitive ratio as Γ 1 or the case o k = 1 without knowledge o ρ min and without the use o payments. The latter property may be necessary in settings that are more local than grid computing e.g., within a department but in which the users are still sel-interested. 7 Mechanism 2 Γ 2 Execute Sˆθ, according to Algorithm 2 or all i do p iˆθ 0 Algorithm 2 or all t do Avail {i t ˆr i e iˆθ, t <l i e iˆθ, t+ ˆd i t l i} i Avail then Sˆθ, t arg max i Avail l i + e iˆθ, t {Break ties in avor o lower ˆr i } else Sˆθ, t 0 Theorem 11. When k = 1, and each agent i cannot alsely declare l i, Mechanism Γ 2 satisies individual rationality and incentive compatibility. Theorem 12. When k = 1, and each agent i cannot alsely declare l i, Mechanism Γ 2 is 5-competitive. Since this mechanism is essentially a simpliication o Γ 1, we omit proos o these theorems. Basically, the act that k = 1 and ˆl i = l i both hold allows Γ 2 to substitute the priority l i+e iˆθ, t or the priority used in Γ 1; and, since ˆv i is ignored, payments are no longer needed to ensure incentive compatibility. 5. COMPETITIVE LOWER BOUND We now show that the competitive ratio o 1 + k achieved by Γ 1 is a lower bound or deterministic online mechanisms. To do so, we will appeal to third requirement on a mechanism, non-negative payments NNP, which requires that the center never pays an agent ormally, i, ˆθ, p iˆθ i 0. Unlike IC and IR, this requirement is not standard in mechanism design. We note, however, that both Γ 1 and Γ 2 satisy it trivially, and that, in the ollowing proo, zero only serves as a baseline utility or an agent, and could be replaced by any non-positive unction o ˆθ i. The proo o the lower bound uses an adversary argument similar to that used in 4 to show a lower bound o 1 + k 2 in the non-strategic setting, with the main novelty lying in the perturbation o the job sequence and the related incentive compatibility arguments. We irst present a lemma relating to the recurrence used or this argument, with the proo omitted due to space constraints. Lemma 13. For any k 1, or the recurrence dned by l i+1 = λ l i k i h=1 l h and l 1 =1,where1 + k 2 1 < λ < 1 + k 2, there exists an integer m 1 such that m 1 l m+k h=1 l h l m >λ. 7 While payments are not required in this setting, Γ 2 can be changed to collect a payments without aecting incentive compatibility by charging some ixed raction o l i or each job i that is completed. 68

9 Theorem 14. There does not exist a deterministic online mechanism that satisies NNP and that achieves a competitive ratio less than 1 + k Proo. Assume by contradiction that there exists a deterministic online mechanism Γ that satisies NNP and that achieves a competitive ratio o c =1+ k 2 +1 ɛ or some ɛ > 0 and, by implication, satisies IC and IR as well. Since a competitive ratio o c implies a competitive ratio o c + x, or any x>0, we assume without loss o generality that ɛ<1. First, we will construct a proile o agent types θ using an adversary argument. Ater possibly slightly perturbing θ to assure that a strictness property is satisied, we will then use a more signiicant perturbation o θ to reach a contradiction. We now construct the original proile θ. Pick an α such α ck+3k that 0 <α<ɛ, and dne δ =. The adversary uses two sequences o jobs: minor and major. Minor jobs i are characterized by l i = δ, v i = k δ, and zero laxity. The irst minor job is released at time 0, and r i = d i 1 or all i>1. The sequence stops whenever Γ completes any job. Major jobs also have zero laxity, but they have the smallest possible value ratio that is, v i = l i. The lengths o the major jobs that may be released, starting with i =1,are determined by the ollowing recurrence relation. l i+1 = c 1+α l i k i h=1 l h l 1 = 1 The bounds on α imply that 1 + k 2 1 <c 1+α< 1+ k 2, which allows us to apply Lemma 13. Let m be the lm+k m 1 h=1 smallest positive number such that l h l m >c 1+α. The irst major job has a release time o 0, and each major job i>1 has a release time o r i = d i 1 δ, just beore the deadline o the previous job. The adversary releases major job i m i and only i each major job j<iwas executed continuously over the range r i,r i+1. No major job is released ater job m. In order to achieve the desired competitive ratio, Γ must complete some major job, because Γ oline can always at least complete major job 1 or a value o 1, and Γ can α complete at most one minor job or a value o < 1. c+3 c Also, in order or this job to be released, the processor time preceding r can only be spent executing major jobs that are later abandoned. I <m, then major job +1 will be released and it will be the inal major job. Γ cannot complete job + 1, because r + l = d >r +1. Thereore, θ consists o major jobs 1 through +1 or,, i = m, plus minor jobs rom time 0 through time d. We now possibly perturb θ slightly. By IR, we know that v p θ. Since we will later need this inequality to be strict, i v = p θ, then change θ to θ, where r = r, but v, l,andd are all incremented by δ over thr respective values in θ. By IC, job must still be completed by Γ or the proile θ,θ. I not, then by IR and NNP we know that p θ,θ = 0, and thus that u gθ,θ,θ = 0. However, agent could then increase its utility by alsely declaring the original type o θ, recving a utility o: u gθ,θ,θ =v p θ =δ>0, violating IC. Furthermore, agent must be charged the same amount that is, p θ,θ =p θ, due to a similar incentive compatibility argument. Thus, or the remainder o the proo, assume that v >p θ. We now use a more substantial perturbation o θ to com- to be identical to θ, except that d = d +1 + l, allowing job to be completely executed ater job + 1 is completed. I = m, then instead set d = d +l. IC requires that or the proile θ,θ, Γ still executes job continuously over the range r,r +l, thus preventing job +1 rom bng completed. Assume by contradiction that this were not true. Then, at theoriginaldeadlineod, job is not completed. Consider plete the proo. I < m, then dne θ the possible proile θ,θ,θ x, which diers rom the new proile only in the addition o a job x which has zero laxity, r x = d,andv x = l x = maxd d, c +1 l + l +1. Because this new proile is indistinguishable rom θ,θ to Γ beore time d, it must schedule jobs in the same way until d. Then, in order to achieve the desired competitive ratio, it must execute job x continuously until its deadline, which is by construction at least as late as the new deadline d o job. Thus, job will not be completed, and, by IR and NNP, it must be the case that p θ,θ,θ x = 0andu gθ,θ,θ x,θ = 0. Using the act that θ is indistinguishable rom θ,θ,θ xuptotimed,iagent alsely declared his type to be the original θ, then its job would be completed by d and it would be charged p θ. Its utility would then increase to u gθ,θ,θ x,θ = v p θ > 0, contradicting IC. While Γ s execution must be identical or both θ,θ and θ,θ, Γ oline can take advantage o the change. I <m, then Γ achieves a value o at most l +δ the value o job i it were perturbed, while Γ oline achieves a value o at least k h=1 l h 2δ+l +1 +l by executing minor jobs until r +1, ollowed by job +1andthenjob we subtract two δ s instead o one because the last minor job beore r +1 may have to be abandoned. Substituting in or l +1,the competitive ratio is then at least: k h=1 l h 2k δ+c 1+α l k h=1 l h+l l +δ c l +ck+3kδ 2k δ l >c. +δ k h=1 l h 2δ+l +1 +l l +δ = = c l +α l 2k δ l +δ I instead = m, then Γ achieves a value o at most l m+δ, while Γ oline achieves a value o at least k m h=1 l h 2δ +l m by completing minor jobs until d m = r m + l m, and then completing job m. The competitive ratio is then k m at least: h=1 l h 2δ+l m = k m 1 h=1 l h 2k δ+kl m+l m l m+δ c 1+α l m 2k δ+kl m l m+δ = c+k 1 lm+αlm 2k δ l m+δ l m+δ > >c. 6. RELATED WORK In this section we describe related work other than the two papers 4 and 15 on which this paper is based. Recent work related to this scheduling domain has ocused on competitive analysis in which the online algorithm uses a aster processor than the oline algorithm see, e.g., 13, 14. Mechanism design was also applied to a scheduling problem in 18. In thr model, the center owns the jobs in an oline setting, and it is the agents who can execute them. The private inormation o an agent is the time it will require to execute each job. Several incentive compatible mechanisms are presented that are based on approximation algorithms or the computationally ineasible optimization problem. This paper also launched the area o algorithmic mechanism design, in which the mechanism must sat- 69

10 isy computational requirements in addition to the standard incentive requirements. A growing sub-ield in this area is multicast cost-sharing mechanism design see, e.g., 1, in which the mechanism must ciently determine, or each agent in a multicast tree, whether the agent recves the transmission and the price it must pay. For a survey o this and other topics in distributed algorithmic mechanism design, see 9. Online execution presents a dierent type o algorithmic challenge, and several other papers study online algorithms or mechanisms in economic settings. For example, 5 considers an online market clearing setting, in which the auctioneer matches buy and sells bids which are assumed to be exogenous that arrive and expire over time. In 2, a general method is presented or converting an online algorithm into an online mechanism that is incentive compatible with respect to values. Truthul declaration o values is also considered in 3 and 16, which both consider multi-unit online auctions. The main dierence between the two is that the ormer considers the case o a digital good, which thus has unlimited supply. It is pointed out in 16 that thr results continue to hold when the setting is extended so that bidders can delay thr arrival. The only other paper we are aware o that addresses the issue o incentive compatibility in a real-time system is 11, which considers several variants o a model in which the center allocates bandwidth to agents who declare both thr value and thr arrival time. A dominant strategy IC mechanism is presented or the variant in which every point in time is essentially independent, while a Bayes-Nash IC mechanism is presented or the variant in which the center s current decision aects the cost o uture actions. 7. CONCLUSION In this paper, we considered an online scheduling domain or which algorithms with the best possible competitive ratio had been ound, but or which new solutions were required when the setting is extended to include sel-interested agents. We presented a mechanism that is incentive compatible with respect to release time, deadline, length and value, and that only increases the competitive ratio by one. We also showed how this mechanism could be simpliied when k = 1 and each agent cannot lie about the length o its job. We then showed a matching lower bound on the competitive ratio that can be achieved by a deterministic mechanism that never pays the agents. Severalopenproblemsremaininthissetting. Onsto determine whether the lower bound can be strengthened by removing the restriction o non-negative payments. 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