Traveling Repairperson, Unrelated Machines, and Other Stories About Average Completion Times
TTraveling Repairperson, Unrelated Machines, andOther Stories About Average Completion Times
Marcin Bienkowski ! Institute of Computer Science, University of Wrocław, Poland
Artur Kraska ! Institute of Computer Science, University of Wrocław, Poland
Hsiang-Hsuan Liu ! Utrecht University, Netherlands
Abstract
We present a unified framework for minimizing average completion time for many seemingly disparate online scheduling problems, such as the traveling repairperson problems (TRP), dial-a-ride problems(DARP), and scheduling on unrelated machines.We construct a simple algorithm that handles all these scheduling problems, by computing andlater executing auxiliary schedules, each optimizing a certain function on already seen prefix ofthe input. The optimized function resembles a prize-collecting variant of the original schedulingproblem. By a careful analysis of the interplay between these auxiliary schedules, and later employingthe resulting inequalities in a factor-revealing linear program, we obtain improved bounds on thecompetitive ratio for all these scheduling problems.In particular, our techniques yield a 4-competitive deterministic algorithm for all previouslystudied variants of online TRP and DARP, and a 3-competitive one for the scheduling on unrelatedmachines (also with precedence constraints). This improves over currently best ratios for theseproblems that are 5 .
14 and 4, respectively. We also show how to use randomization to furtherreduce the competitive ratios to 1 + 2 / ln 3 < .
821 and 1 + 1 / ln 2 < . Theory of computation → Online algorithms; Theory of compu-tation → Scheduling algorithms
Keywords and phrases traveling repairperson problem, dial-a-ride, machine scheduling, unrelatedmachines, minimizing completion time, competitive analysis, factor-revealing LP
Funding
Supported by Polish National Science Centre grant 2016/22/E/ST6/00499.
In the traveling repairperson problem (TRP) [38], requests arrive in time at points of a metricspace and they need to be eventually serviced. In the same metric, there is a mobile server,that can move at a constant speed. The server starts at a distinguished point called theorigin. A request is considered serviced once the server reaches its location; we call suchtime its completion time . The goal is to minimize the sum (or equivalently the average) ofall completion times. We focus on a weighted variant, where all requests have non-negativeweights and the goal is to minimize the weighted sum of completion times.A natural and well-studied extension of the TRP problem is a so-called dial-a-ride problem (DARP) [20], where each request has a source and a destination and the goal is to transportan object between these two points. There, the server may have a fixed capacity limitingthe number of objects it may carry simultaneously; this capacity may be also infinite. Forthe finite-capacity case, one can also distinguish between preemptive variant, where objects a r X i v : . [ c s . D S ] F e b TRP, Unrelated Machines, and Other Stories About Average Completion Times can be unloaded at some points of the metric space (different than their destination) andnon-preemptive variant, where such unloading is not allowed.A seemingly disparate problem is scheduling on m unrelated machines [24]. There,weighted jobs arrive in time, each with a vector of size m describing execution times of thejob when assigned to a given machine. A single machine can execute at most one job ata time. The goal is to assign each job (at or after its arrival) to one of the machines tominimize the weighted sum of completion times. This problem comes in two flavors: inthe preemptive one, job execution may be interrupted and picked up later, while in thenon-preemptive one, such interruption is not possible. As an extension, each job may haveprecedence constraints, i.e., can be executed only once some other jobs are completed. Online Algorithms.
Our focus is on natural online scenarios of TRP, DARP [22], andmachine scheduling [25]. There, an online algorithm
Alg , at time t , knows only requests/jobsthat arrived before or at time t . The number of requests/jobs is also not known by an algorithma priori. We say that an online algorithm Alg is c -competitive if for any request/job sequence I it holds that cost Alg ( I ) ≤ c · cost Opt ( I ), where Opt is a cost-optimal offline solution for I . For a randomized algorithm Alg , we replace its cost by its expectation. The competitiveratio of
Alg is the infimum over all values c such that Alg is c -competitive [15].In this paper, we present a unified framework for handling such online scheduling problemswhere the cost is the weighted sum of completion times. We present an algorithm Mimic that yields substantially improved competitive ratios for all the problems described above.
The currently best algorithms for the TRP, the DARP, and machine scheduling on unrelatedmachines share a common framework. Namely, each of these algorithms works in phasesof geometrically increasing lengths. In each phase, it computes and executes an auxiliaryschedule for the requests presented so far. (In the case of the TRP and DARP, the serveradditionally returns to the origin afterward.) The auxiliary schedule optimizes a certainfunction, such as maximizing the weight of served requests [8,16,25,29,33,34] or minimizing thesum of completion times with an additional penalty for non-served requests [28]. Moreover,known randomized algorithms are also based on a common idea: they delay the executionof the deterministic algorithm by a random offset [16, 28, 33, 34]. We call these approaches phase based . The currently best results are gathered in Table 1.
Traveling Repairperson and Dial-a-Ride Problems.
The online variant of the TRP hasbeen first investigated by Feuerstein and Stougie [22]. By adapting an algorithm for thecow-path problem problem [7], they gave a 9-competitive solution for line metrics. Theresult has been improved by Krumke et al. [33], who gave a phase-based deterministicalgorithm
Interval attaining competitive ratio of 3 + 2 √ < .
829 for an arbitrary metricspace. A slightly different algorithm with the same competitive ratio was given by Jailletand Wagner [29]. Bienkowski and Liu [8] applied postprocessing to auxiliary schedules,serving heavier requests earlier, and improved the ratio to 5 .
429 on line metrics. Finally,Hwang and Jaillet proposed a phase-based algorithm
Plan-And-Commit [28]. They give Computing such auxiliary schedule usually involves optimally solving an NP-hard task. This is typicalfor the area of online algorithms, where the focus is on information-theoretic aspects and not oncomputational complexity. Algorithms presented in this paper also aim at minimizing the achievablecompetitive ratio rather than minimizing the running time. . Bienkowski, A. Kraska and H.-H. Liu 3 deterministic randomizedlower upper lower upperTRP 2 .
414 [22] 5 .
14 [28] 2 .
333 [33] 3 .
641 [28]DARP 3 [22] 5 . ∗ [28] 2 .
410 [33] 3 . ∗ [28] k -TRP 2 [14] 5 . ∗ [28] 2 [14] 3 . ∗ [28] k -DARP 2 [14] 5 . ∗ [28] 2 [14] 3 . ∗ [28] k -TRP, k -DARP (all variants) . scheduling on unrelated machines 1.309 [46] 4 [25] 1 .
157 [40] 2 .
886 [16] . Table 1
Previous and current bounds on the competitive ratios for the TRP and the DARPproblems. Asterisked results were not given in the referenced papers, but they are immediateconsequences of the arguments therein. All bounds for k -TRP and k -DARP hold for an arbitrarynumber of servers k . Bounds proven in the current paper are given in boldface. a computer-based upper bound of 5 .
14 for the competitive ratio and an analytical upperbound of 5 . Interval and
Plan-And-Commit achieve ratios of 3 .
874 [33,34] and 3 .
641 [28], respectively. Interestingly, the latter bound is not a direct randomizationof the deterministic algorithm, but uses a different parameterization, putting more emphasison penalizing requests not served by auxiliary schedules.The phase-based algorithm
Interval extends in a straightforward fashion to the DARPproblem with an arbitrary assumption on the server capacity, both for the preemptiveand non-preemptive variants: all the details of the solved problem are encapsulated in thecomputations of auxiliary schedules [33]. In the same manner,
Interval can be enhanced tohandle multiple servers [14]. Although this was not explicitly stated in [28], the algorithm
Plan-And-Commit can be extended in the same way.From the impossibility side, Feuerstein and Stougie [22] gave a lower bound for the TRP(that also holds already for a line) of 1 + √ > .
414 and the bound of 7 / .
410 [33].The authors of [23] claimed a lower bound of 3 for randomized k -DARP (for any k ). Thiscontradicts the upper bound we present in this paper. In Appendix C, we pinpoint a flaw intheir argument. TRP and DARP: Related Results.
Both online TRP and DARP problems were consideredunder different objectives, such as minimizing the total makespan (when the TRP becomesonline TSP) [3–6, 9–11, 13, 18, 30, 31, 36] or maximum flow time [26, 32, 35].The offline variants of TRP and DARP have been extensively studied both from thecomputational hardness (see, e.g., [21, 38]) and approximation algorithms perspectives. Inparticular, the TRP, also known as the minimum latency problem problem, is NP-hard alreadyon weighted trees [41] (where the closely related traveling salesperson problem [12] becomestrivial) and the best known approximation factor in general graphs is 3.59 [17]. For somemetrics (Euclidean plane, planar graphs or weighted trees) the TRP admits a PTAS [2, 43].
TRP, Unrelated Machines, and Other Stories About Average Completion Times
Machine Scheduling on Unrelated Machines.
The first online algorithm for the schedulingon unrelated machines ( R | r j | P w j C j in the Graham et al. notation [24]) was given byHall et al. [25]. They gave 8-competitive polynomial-time algorithm, which would be 4-competitive if the polynomial-time requirement was lifted. Chakrabarti et al. showed how torandomize this algorithm, achieving the ratio of 2 / ln(2) < .
886 [16]. They also observe thatboth algorithms can handle precedence constraints. The currently best deterministic lowerof 1.309 is due to Vestjens [46], and the best randomized one of 1.157 is due to Seiden [40].
Machine Scheduling: Related Results.
While for unrelated machines, the results have notbeen beaten for the last 25 years, the competitive ratios for simpler models were improvedsubstantially. For example, for parallel identical machines, a sequence of papers lowered theratio to 1 .
791 [19, 37, 39, 42].The problem has also been studied intensively in the offline regime. Both weightedpreemptive and non-preemptive variants were shown to be APX-hard [27, 44]. On thepositive side, 1 . . The phase-based algorithms for DARP variants and machine scheduling on unrelated machinesboth execute auxiliary schedules, but the ones for the DARP variants need to bring theserver back to the origin between schedules. We call the latter action resetting . To providea single algorithm for all these scheduling variants, we define a class of resettable scheduling problems.We assume that jobs are handled by executor , which has a set of possible states. And attime 0, it is in a distinguished initial state . An input to the problem consists of a sequenceof jobs I released over time. Each job r is characterized with its arrival time a ( r ), itsweight w ( r ), and possibly other parameters that determine its execution time. The executorcannot start executing job r before its arrival time a ( r ). We will slightly abuse the notationand use I to also denote the set of all jobs from the input sequence. There is a problem-specific way of executing jobs and we use s Alg ( r ) to denote the completion time of a job byan algorithm Alg . The cost of an algorithm is defined as the weighted sum of job completiontimes, cost
Alg ( I ) = P r ∈I w ( r ) · s Alg ( r ).For any time τ , let I τ be the set of jobs that appear till τ . An auxiliary τ -schedule isa problem-specific way of feasibly executing a subset of jobs from I τ . Such schedule startsat time 0, terminates at time τ , and leaves no job partially executed. We require that thefollowing properties hold for any resettable scheduling problem. Delayed execution.
At any time t , if the executor is the initial state, it can execute an ar-bitrary auxiliary τ -schedule (for τ ≤ t ). Such action takes place in time interval [ t, t + τ ].Any job r that would be completed at time z ∈ [0 , τ ] by the τ -schedule started at time 0is now completed exactly at time t + z (unless it has been already executed before). Resetting executor.
Assume that at time t , the executor was in the initial state, and thenexecuted a τ -schedule, ending at time t + τ . Then, it is possible to reset the executorusing extra γ · τ time, where γ is a parameter characteristic to a problem. That is, attime t + (1 + γ ) · τ , the executor is again in its initial state. Learning minimum.
We define min( I ) to be the earliest time at which Opt may completesome job. We require that min( I ) become known to an online algorithm at or beforetime min( I ) and that min( I ) > . Bienkowski, A. Kraska and H.-H. Liu 5 We call scheduling problems that obey these restrictions γ -resettable. Example 1: Machine Scheduling is 0-Resettable.
For the machine scheduling problem,the executor is always in the initial state, and no resetting is necessary. As we may assumethat processing of any job takes positive time, min( I ) > I . Example 2 : DARP Problems are 1-Resettable.
For the DARP variants, the executorstate is the position of the algorithm server, with the origin used as the initial state. Jobsare requests for transporting objects and an auxiliary τ -schedule is a fixed path of length τ starting at the origin, augmented with actions of picking and dropping particular objects. Itis feasible to execute a τ -schedule starting at any time t when the server is at the origin. Insuch case, jobs are completed with an extra delay of t . Furthermore, right after serving the τ -schedule, the distance between the server and the origin is at most τ . Thus, it is possibleto reset the executor to the initial state within extra time 1 · τ .Finally, as we may assume that there are no requests that arrive at time 0 with bothstart and destination at the origin, min( I ) > I . In this paper, we provide a deterministic routine
Mimic and its randomized version thatsolves any γ -resettable scheduling problem. It achieves a deterministic ratio of 3 + γ anda randomized one of 1 + (1 + γ ) / ln(2 + γ ).That is, for 1-resettable scheduling problems (the DARP variants with arbitrary servercapacity, an arbitrary number of servers, and both in the preemptive and non-preemptivesetting, or the TRP problem with an arbitrary number of servers), this gives a solutionwhose ratios are at most 4 and 1 + 2 / ln 3 < . / ln 2 < . Challenges and Techniques.
Mimic works in phases of geometrically increasing lengths.At the beginning of each phase, at time τ , it computes an auxiliary τ -schedule that optimizesthe total completion time of jobs seen so far with an additional penalty for non-completedjobs: they are penalized as if they were completed at time τ . Then, within the phase itexecutes this schedule and afterward it resets the executor. We obtain a randomized variantby delaying the start of Mimic by an offset randomly chosen from a continuous distribution.Admittedly, this idea is not new, and in fact, when we apply
Mimic to the TRP problem,it becomes a slightly modified variant of
Plan-And-Commit [28]. Hence, the main technicalcontribution of our paper is a careful and exact analysis of such an approach. The crux hereis to observe several structural properties and relations among schedules produced by
Mimic in consecutive phases, carefully following the overlaps of the job sets completed by them. Onthis basis, and for a fixed number Q of phases, we construct a maximization linear program In the variants with k servers, the executor state is a k -tuple describing positions of all servers. In the preemptive variants, preemption is allowed inside an auxiliary schedule, provided that aftera τ -schedule terminates, each job is either completed or untouched. TRP, Unrelated Machines, and Other Stories About Average Completion Times (LP), whose optimal value upper-bounds the competitive ratio of
Mimic . Roughly speaking,the LP encodes, in a sparse manner, an adversarially created input. To upper bound itsvalue, we explicitly construct the solution to its dual (minimization) program and show thatits value is at most 4 for any number of phases Q .Bounding the competitive ratio for the randomized version of Mimic is substantiallymore complicated as we need to combine the discrete world of an LP with uncountably manyrandom choices of the algorithm. To tackle this issue, we consider an intermediate solution
Disc which approximates the random choice of
Mimic to a given precision, choosing anoffset randomly from a discrete set of M values. This way, we upper-bound the ratio of Mimic by 1 + (1 /M ) · P Mj =1 (2 + γ ) j/M . This bound holds for an arbitrary value of M , andthus by taking the limit, we obtain the desired bound on the competitive ratio. Interestingly,we use the same LP for analyzing both deterministic and randomized solution.Due to space limitations, some proofs have been moved to the appendix. To describe our approach for the γ -resettable scheduling, we start with defining auxiliaryschedules used by our routine Mimic . The parameter γ will be used to define partitioning oftime into phases. Both our deterministic and randomized solutions will run Mimic , however,the randomized one will execute it for a random choice of parameters.
Auxiliary Schedules.
As introduced already in Subsection 1.2, an (auxiliary) τ -schedule A describes a sequence of job executions, has the total duration τ , and may be executedwhenever the executor is in the initial state. For the preemptive variants, we assume thatonce such a schedule terminates, each job is processed either completely or not at all.For a fixed input I , and a τ -schedule A , we use R ( A ) to denote the set of jobs that wouldbe served by A if it was executed from time 0, i.e., in the interval [0 , τ ). For any set of jobs R ⊆ R ( A ), let w ( R ) = P r ∈ R w ( r ) and cost A ( R ) = P r ∈ R w ( r ) · s A ( r ) . (1)Note that if a schedule A serves all jobs from the input ( R ( A ) = I ), then cost A ( R ( A ))coincides with the cost of an algorithm that executes schedule A at time 0.Recall that I τ ⊆ I denote the set of jobs that arrive till time τ . For any τ -schedule A ,we define its value as val τ ( A ) = cost A ( R ( A )) + τ · w ( I τ \ R ( A )) . (2)The value corresponds to the actual cost of completing jobs from I τ by schedule A ininterval [0 , τ ), but we charge A for unprocessed jobs as if they were completed at time τ . ▶ Definition 1.
For any τ ≥ , let S τ be the τ -schedule minimizing function val τ . Ties arebroken arbitrarily, but in a deterministic fashion. Routine MIMIC.
For solving the γ -resettable scheduling problem, we define routine Mimic ( γ, ω ), where ω ∈ [ − ,
0] is an additional parameter that controls the initial delay.Our deterministic algorithm is simply
Mimic ( γ, ω uniformly at random from therange ( − , Mimic ( γ, ω ) and returns its actions. . Bienkowski, A. Kraska and H.-H. Liu 7 S τ (1) S τ (2) S τ (3) Mimic α α α Figure 1
An example execution of algorithm
Mimic (1 ,
0) applied for the TRP problem. Weassume that min( I ) = 1. Within time interval [ τ k = α k , · α k ) of phase k + 1, Mimic executes a τ k -schedule S τ ( k ) that optimizes function val τ ( k ) . Afterwards within time interval [2 · α k , τ k +1 = 3 · α k ) Mimic resets its state to the initial one (the server of TRP returns to the origin).
Internally,
Mimic ( γ, ω ) uses a parameter α = 2 + γ . It splits time into phases in thefollowing way. For any k , let τ k = τ ( k ) = min( I ) · α k + ω . The k -th phase (for k ≥
1) startsat time τ k − = min( I ) · α k − ω and ends at time τ k = min( I ) · α k + ω . The time interval[0 , τ ) = [0 , α ω · min( I )) does not belong to any phase. As α ω · min( I ) ≤ min( I ), no jobscan be completed within this interval, by the definition of min( I ) (see Subsection 1.2). Mimic does nothing till the end of phase 1 (till time τ = α ω · min( I )). Since ω ≥ − τ ≥ min( I ). As Mimic learns the value of min( I ) latest at time min( I ), it canthus correctly identify the value of τ before or at time τ .For a phase k + 1, where k ≥ Mimic behaves in the following way. We ensure that attime τ k , at the beginning of phase k + 1, Mimic is in its initial state. At this time,
Mimic computes the τ k -schedule S τ ( k ) (see Definition 1), executes it within time interval [ τ k , · τ k ]and afterwards, it resets its state to the initial one. The execution of S τ ( k ) will not beinterrupted or modified when new jobs arrive within phase k + 1. The resetting part takestime γ · τ k , and is thus finished at time (2 + γ ) · τ k = α · τ k = τ k +1 when the next phasestarts. An illustration is given in Figure 1. As mentioned in the introduction, we introduce an additional intermediate algorithm
Disc ,whose analysis will allow us to bound the competitive ratios of both our deterministic andrandomized solution. For an integer ℓ , we use [ ℓ ] to denote the set { , . . . , ℓ − } . Disc ( γ, M, β ) solves the γ -resettable scheduling problem, and is additionally parameter-ized with a positive integer M , and a real number β ∈ (0 , /M ]. Disc ( γ, M, β ) first choosesa random integer m ∈ [ M ]. Then it executes Mimic ( γ, ω = − m/M + β ) and returns itsactions. The main result of this paper is the following bound. ▶ Theorem 2.
For any γ , any positive integer M , and any β ∈ (0 , /M ] , the competitiveratio of Disc ( γ, M, β ) for the γ -resettable scheduling is at most /M ) · P Mj =1 (2 + γ ) j/M . ▶ Corollary 3.
For any γ , the competitive ratio of our deterministic solution is γ andthe ratio of randomized one at most γ ) / ln(2 + γ ) . Proof.
Let R M = 1 + (1 /M ) · P Mj =1 α j/M . First, we note that Disc ( γ, M = 1 , β = 1) choosesdeterministically m = 0 and executes Mimic ( γ, ω = − TRP, Unrelated Machines, and Other Stories About Average Completion Times our deterministic algorithm. Hence, by Theorem 2, the corresponding competitive ratio is atmost R = 3 + γ .For analyzing our randomized algorithm, we observe that instead of choosing a random ω ∈ ( − , m ∈ [ M ] and a random real β ∈ (0 , /M ] and set ω = − m/M + β . Thus, for any fixed integer M , our randomized algorithm is equivalentto choosing random β ∈ (0 , /M ] and running Disc ( γ, M, β ).Fix any input I . By Theorem 2, E m [ cost Disc ( γ,M,β ) ( I )] ≤ R M · cost Opt ( I ) holds for any β ∈ (0 , /M ], where the expected value is taken over random choice of m . Clearly, this relationholds also when β is chosen randomly, i.e., E ω [ cost Mimic ( γ,ω ) ] = E γ E m [ cost Disc ( γ,M,β ) ( I )] ≤ R M · cost Opt ( I ). As the bound is valid for any M , and the competitive ratio of ourrandomized algorithm is at most inf M ∈ N { R M } = lim M →∞ R M = 1 + (1 + γ ) / ln(2 + γ ). ◀ In this section, we build relations useful for analyzing the performance of
Disc ( γ, M, β ) onany instance I of the γ -resettable scheduling problem.We start by presenting structural properties of schedules S τ . We note that even if thereexists a τ -schedule A that completes all jobs from I , S τ may leave some jobs untouched.However, a sufficiently long schedule S τ completes all jobs. ▶ Lemma 4.
Fix any input I . There exists value T I , such that for any τ ≥ T I , S τ completesall jobs of I and is an optimal (cost-minimal) solution for I . Sub-phases.
Recall that the algorithm
Disc ( γ, M, β ) chooses a random integer m ∈ [ M ],and executes Mimic ( γ, ω = − m/M + β ). To compare Disc executions for differentrandom choices, we introduce sub-phases. Recall that α = 2 + γ ; let δ = α /M .Recall that the k -th phase of Mimic starts at time τ k − and ends at time τ k , where τ k = min( I ) · α k − m/M + β = min( I ) · α β − · δ m + k · M . For any q , we define η q = η ( q ) = min( I ) · α β − · δ q . (3)In these terms, τ k = η m + k · M . We define the q -th sub-phase (for q ≥
0) as the time intervalstarting at time η q − and ending at time η q . Then, phase k of Disc ( γ, M, β ) consists ofexactly M sub-phases, numbered from ( k − · M + m + 1 to k · M + m . An example ofphases and sub-phases is given in Figure 2. We emphasize that the start and the end ofa sub-phase is a deterministic function of the parameters of Disc , while the start and end ofa phase depend additionally on the value m ∈ [ M ] that Disc chooses randomly.Recall that our deterministic algorithm is equivalent to
Mimic ( γ, ≡ Disc ( γ, , m = 0, and thus η q = τ q for any q , i.e., each phase consists of one sub-phase, andtheir indexes coincide. Sub-phases vs Auxiliary Schedules.
We now identify the times when auxiliary schedulesare computed by
Disc ( γ, M, β ). Recall that at the beginning of any phase k + 1 (where k ≥ τ k = η m + k · M , Disc computes and executes schedule S η ( m + k · M ) . Let T I be the threshold guaranteed by Lemma 4 and we define K I as the smallest integer satisfying η ( K I · M ) ≥ T I . Note that K I is a deterministic function of input I .For any choice of m ∈ [ M ], the schedule S η ( m + K I · M ) completes all jobs. This scheduleis executed by Disc in phase K I + 1, and thus Disc terminates latest in phase K I + 1.Summing up, Disc ( γ, M, β ) executes schedules S η ( m + M ) , S η ( m +2 M ) , . . . , S η ( m + K I · M ) . At . Bienkowski, A. Kraska and H.-H. Liu 9 η η η η η η η η η η η η η η η m = 0 m = 1 m = 2 min( I ) τ τ τ τ τ τ τ τ τ τ τ τ τ τ τ η − Figure 2
Example of phases (green) and sub-phases (black) of algorithm
Disc ( γ, M = 3 , β ) forall possible choices of m The time interval lengths are in logarithmic scale. The starts and ends ofsub-phases are deterministic functions of γ , M , and β , but the start of a phase depends additionallyon the integer m ∈ [ M ] chosen randomly by Disc . Sub-phase 0 is not contained in any phase, butwill be used in our analysis. the beginning of the first phase,
Disc does nothing, but for notational ease, we assumethat in the first phase, it also computes and executes a dummy schedule S η ( m ) , which doesnot complete any job. For succinctness, we use A q = S η ( q ) . In these terms, Disc ( γ, M, β )executes schedules A m + k · M for k ∈ [ K I + 1].Let Q = K I · M + ( M − Disc range from 0to Q . For any schedule A q , we define the set of indexes of preceding schedules P ( q ) = { q ′ , q ′ + M, . . . , q − M } , where q ′ = q mod M . Fresh and Stale Requests.
We assume that no jobs are completed by an online algorithmwhile it is resetting the executor, and we assume that the execution of schedule A q maycomplete only jobs from set R ( A q ). It is however important to note that R ( A q ) and R ( A q − M )may overlap significantly, in which case the execution of schedule A q serves only these jobsfrom R ( A q ) that have not been served already. To further quantify this effect, for q ∈ [ Q + 1],we define the set of fresh jobs of schedule A q as R F ( A q ) = R ( A q ) \ S ℓ ∈ P ( q ) R ( A ℓ ) . (4)The remaining jobs from R ( A q ) are called stale and are denoted R S ( A q ) = R ( A q ) \ R F ( A q ).For succinctness, we define the following shorthand notations for their weights: w F q = w ( R F ( A q )) , w S q = w ( R S ( A q )) , w q = w ( R ( A q )) = w F q + w S q . (5) ▶ Lemma 5.
For any q ∈ [ Q + 1] , it holds that w S q ≤ P ℓ ∈ P ( q ) w F ℓ . This relation becomesequality for q ≥ K I · M . Proof.
By a simple induction, it can be shown that U ℓ ∈ P ( q ) R F ( A ℓ ) = S ℓ ∈ P ( q ) R ( A ℓ ) forany q ∈ [ Q + 1]. Then, using the definition of stale jobs, R S ( A q ) ⊆ S ℓ ∈ P ( q ) R ( A ℓ ) = U ℓ ∈ P ( q ) R F ( A ℓ ). Applying weight to both sides yields w S q ≤ P ℓ ∈ P ( q ) w F ℓ .Next, we show that this relation can be reversed for q ≥ K I · M (i.e., for the sched-ule executed in the last phase of Disc ). For such q , A q completes all jobs, and thus S ℓ ∈ P ( q ) R ( A ℓ ) ⊆ R ( A q ) = R F ( A q ) ⊎ R S ( A q ). By the definition of fresh jobs, R F ( A q ) doesnot contain any job from S ℓ ∈ P ( q ) R ( A ℓ ), and thus S ℓ ∈ P ( q ) R ( A ℓ ) ⊆ R S ( A q ). This impliesthat U ℓ ∈ P ( q ) R F ( A ℓ ) = S ℓ ∈ P ( q ) R ( A ℓ ) ⊆ R S ( A q ). After applying weights to both sides, weobtain w S q ≥ P ℓ ∈ P ( q ) w F ℓ as desired. ◀ Jobs Completed in Sub-phases.
For further analysis, we refine our notions when a job iscompleted. For a η q -schedule A q , let R j ( A q ) be the set of jobs completed in sub-phase j ≤ q ,i.e., within interval [ η j − , η j ). As η − ≤ η m − ≤ min( I ) (cf. (3)), no job can be completedwithin the interval [0 , η − ) (before sub-phase 0). Hence, R ( A q ) = U qj =0 R j ( A q ).We partition sets R F ( A q ) and R S ( A q ) analogously, defining sets R F j ( A q ) and R S j ( A q ) (for0 ≤ j ≤ q ), such that R F ( A q ) = U qj =0 R F j ( A q ) and R S ( A q ) = U qj =0 R S j ( A q ). For succinctness,for 0 ≤ j ≤ q , we introduce the following shorthand notations: w F qj = w ( R F j ( A q )), w S qj = w ( R S j ( A q )), and w qj = w ( R j ( A q )) = w F qj + w S qj ; g F qj = cost A q ( R F j ( A q )), g S qj = cost A q ( R S j ( A q )), and g qj = cost A q ( R j ( A q )) = g F qj + g S qj . ▶ Lemma 6.
For any ≤ q < ℓ ≤ Q , it holds that P qj =0 ( g qj − g ℓj )+ P qj =0 η q · ( w ℓj − w qj ) ≤ . Proof.
For any η q -schedule B , it holds that val η ( q ) ( B ) = cost B ( R ( B ))+ η q · w (cid:0) I η ( q ) \ R ( B ) (cid:1) = P qj =0 cost B ( R j ( B )) + η q · w ( I η ( q ) ) − η q · P qj =0 w ( R j ( B )).Fix any ℓ ≤ Q and let A qℓ be the η q -schedule consisting of the first q sub-phases of η ℓ -schedule A ℓ . Since A q is a minimizer of val η ( q ) , it holds that val η ( q ) ( A q ) ≤ val η ( q ) ( A qℓ ).Thus, P qj =0 g qj − η q · P qj =0 w qj ≤ P qj =0 g ℓj − η q · P qj =0 w ℓj . ◀ Costs of DISC and OPT.
Finally, we can express costs of
Disc and
Opt using the newlyintroduced notions. ▶ Lemma 7.
For any input I , parameters M and β ∈ (0 , /M ] , it holds that E [ cost Disc ( I )] =(1 /M ) · P Qq =0 P qj =0 η q · w F qj + g F qj . Proof.
Recall that
Disc chooses random m ∈ [ M ] and then at time η q it executes schedule A q ,for all q ∈ { m, m + M, . . . , m + K I · M } . When Disc executes A q , it completes jobs from R F ( A q ). By the delayed execution property of the resettable scheduling (cf. Subsection 1.2),each job r ∈ R F ( A q ) is completed at time η q + s A q ( r ). Thus, the cost of executing A q by Disc is equal to P r ∈ R F ( A q ) ( η q · w ( r ) + s A q ( r )) = η q · w ( R F ( A q )) + cost A q ( R F ( A q )) = η q · w F q + P qj =0 g F qj = P qj =0 (cid:0) η q · w F qj + g F qj (cid:1) . For any q ∈ [ Q + 1], the probability that Disc executes A q is equal to 1 /M , and thus the lemma follows. ◀▶ Lemma 8.
For any input I and any q ∈ { Q − M + 1 , Q − M + 2 , . . . , Q } , it holds that cost Opt ( I ) = P qj =0 g qj . Proof.
Recall that for such choice of q , schedules A q serve all job of I achieving optimalcost. Therefore, cost Opt ( I ) = cost A q ( R ( A q )) = P qj =0 cost A q ( R j ( A q )) = P qj =0 g qj . ◀ Now we show that the
Disc -to-
Opt cost ratio on an arbitrary input I can be upper-boundedby a value of a linear (maximization) program.Assume we fixed γ and any input I to the γ -resettable scheduling problem. We alsofix parameters of Disc : an integer M and β ∈ (0 , /M ]. These choices imply the valuesof Q and η q for any q . This allows us to define the linear program P γ, I ,M,β whose goal is tomaximize P Qq =0 P qj =0 η q · w F qj + g F qj (6) . Bienkowski, A. Kraska and H.-H. Liu 11 subject to the following constraints: P qj =0 g qj ≤ Q − M + 1 ≤ q ≤ Q (7) P qj =0 w S qj − P ℓ ∈ P ( q ) P ℓj =0 w F ℓj ≤ ≤ q ≤ Q − M (8) P ℓ ∈ P ( q ) P ℓj =0 w F ℓj − P qj =0 w S qj ≤ Q − M + 1 ≤ q ≤ Q (9) P qj =0 ( g qj − g ℓj ) + P qj =0 η q · ( w ℓj − w qj ) ≤ ≤ q < ℓ ≤ Q (10) η j − · w S qj − g S qj ≤ ≤ j ≤ q ≤ Q (11) g F qj − η j · w F qj ≤ ≤ j ≤ q ≤ Q (12) η j − · w F qj − g F qj ≤ ≤ j ≤ q ≤ Q (13)and non-negativity of all variables. In (10), we treat w qj and g qj not as variables, but asshorthand notations for w F qj + w S qj and g F qj + g S qj , respectively.The intuition behind this LP formulation is that instead of creating the whole input I ,the adversary only chooses the values of variables w F qj , w S qj , g F qj and g S qj that satisfy somesubset of inequalities (inequalities that have to be satisfied if these variables were created onthe basis of actual input I ). This intuition is formalized below. ▶ Lemma 9.
Fix any γ , any input I for γ -resettable scheduling, and parameters of Disc :integer M and β ∈ (0 , /M ] . Then, E [ cost Disc ( I )] / cost Opt ( I ) ≤ P ∗ γ, I ,M,β /M , where P ∗ γ, I ,M,β is the value of the optimal solution to P γ, I ,M,β . Dual Program and Competitive Ratio.
By Lemma 9, the optimal value of P γ, I ,M,β isan upper bound on the competitive ratio of Disc . By the weak duality, such upper-bound isgiven by any feasible solution to the dual program D γ, I ,M,β that we present below. D γ, I ,M,β uses variables ξ q , B q , C q , D ℓq , F qj , G qj , and H qj , corresponding to inequalities(7)–(11) from P γ, I ,M,β , respectively. In the formulas below, we use L q = M · K + ( q mod M )and S ( q ) = { q + M, q + 2 · M, . . . , L q − M } . For succinctness of the description, we introducetwo auxiliary variables for any 0 ≤ j ≤ q ≤ Q : U qj = P Qℓ = q +1 D ℓq − P q − ℓ = j D qℓ and V qj = P q − ℓ = j η ℓ · D qℓ − P Qℓ = q +1 η q · D ℓq . (14)The goal of D γ, I ,M,β is to minimize P Qq = Q − M +1 ξ q (15)subject to the following constraints (in all of them, we omitted the statement that they holdfor all j ∈ { , . . . , q } ): U qj + G qj − H qj ≥ ≤ q ≤ Q − M (16) U qj − F qj ≥ ≤ q ≤ Q − M (17) U qj + G qj − H qj + ξ q ≥ Q − M + 1 ≤ q ≤ Q (18) U qj − F qj + ξ q ≥ Q − M + 1 ≤ q ≤ Q (19) V qj + η j − · H qj − η j · G qj + C L q − P ℓ ∈ S ( q ) B ℓ ≥ η q for all 0 ≤ q ≤ Q − M (20) V qj + η j − · F qj + B q ≥ ≤ q ≤ Q − M (21) V qj − η j · G qj + η j − · H qj ≥ η q for all Q − M + 1 ≤ q ≤ Q (22) V qj + η j − · F qj − C q ≥ Q − M + 1 ≤ q ≤ Q (23) ▶ Lemma 10.
For any γ , any input I for γ -resettable scheduling, any positive integer M ,and any β ∈ (0 , /M ] , there exists a feasible solution to D γ, I ,M,β of value at most M + P Mj =1 (2 + γ ) j/M . Proof.
Let ∆ k = P ki =0 δ i = (cid:0) δ k +1 − (cid:1) / ( δ − ξ q = 1 + δ q − Q + M for Q − M + 1 ≤ q ≤ Q,F qj = ξ q for Q − M + 1 ≤ j ≤ q ≤ Q,δ · ∆ M − for 0 ≤ j ≤ Q − M and q = j, ≤ j ≤ Q − M and q ∈ { j + 1 , . . . , j + M } , ,G qj = ∆ q − Q + M − − ∆ q − j for Q − M + 1 ≤ j ≤ q ≤ Q, ∆ q − j − M − for j ≤ q − M , and also B q = η q − M − · (cid:0) δ M +1 + 1 (cid:1) · (cid:0) δ M − (cid:1) for 0 ≤ q ≤ Q − M,C q = η q − M − · (cid:0) δ M +1 + 1 (cid:1) for Q − M + 1 ≤ q ≤ Q,D qj = F q,j +1 − F qj for 0 ≤ j < q ≤ Q,H qj = F qj + G qj − ≤ j ≤ q ≤ Q. In Appendix A, we show that the given values of dual variables are all non-negative, satisfyall constraints (16)–(23) of the dual program D γ, I ,M,β , and yield appropriate objective valueof M + P Mj =1 (2 + γ ) j/M . ◀ Given the lemma above, we may bound the competitive ratio of our algorithm.
Proof of Theorem 2.
Fix any γ , and consider algorithm Disc ( γ, M, β ) for any positiveinteger M , and any β ∈ (0 , /M ]. Fix any input I to γ -resettable scheduling problem. Let P ∗ γ, I ,M,β be the value of an optimal solution to P γ, I ,M,β . By weak duality and Lemma 10, P γ, I ,M,β ≤ M + P Mj =1 (2 + γ ) j/M . Hence, by Lemma 9, E [ cost Disc ( I )] / cost Opt ( I ) ≤ P ∗ γ, I ,M,β /M ≤ /M ) · P Mj =1 (2 + γ ) j/M , as desired. ◀ The analysis of our algorithms is tight as proven below. For the deterministic one, weadditionally show that choosing ω different from 0 does not help. ▶ Theorem 11.
For any γ , there are γ -resettable scheduling problems, such that for any ω ∈ [ − , , the competitive ratio of Mimic ( γ, ω ) is at least γ . ▶ Theorem 12.
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A Verifying Solution to the Dual Program
In this section, we give a complete proof of Lemma 10. First, we restate the chosen valuesof the dual variables, show that they are non-negative, satisfy all inequalities (16)–(23) ofthe dual program D γ, I ,M,β , and that they yield the appropriate objective value. We use thefollowing shorthand notion∆ k = k X i =0 δ i = (cid:0) δ k +1 − (cid:1) / ( δ − . Q − MQ − M + 1 Q Q − M Q F qj qj M ξ q δ · ∆ M − Q − MQ − M + 1 Q Q − M Q G qj qj M q − j − M − ∆ q − Q + M − − ∆ q − j Figure 3
Visual presentation of values assigned to dual variables F qj (left) and G qj (right) for M = 3 and Q = 8. In particular ∆ − = 0. We restate the values of dual variables: ξ q = 1 + δ q − Q + M for Q − M + 1 ≤ q ≤ Q,F qj = ξ q for Q − M + 1 ≤ j ≤ q ≤ Q,δ · ∆ M − for 0 ≤ j ≤ Q − M and q = j, ≤ j ≤ Q − M and q ∈ { j + 1 , . . . , j + M } , ,G qj = ∆ q − Q + M − − ∆ q − j for Q − M + 1 ≤ j ≤ q ≤ Q, ∆ q − j − M − for j ≤ q − M , and also B q = η q − M − · (cid:0) δ M +1 + 1 (cid:1) · (cid:0) δ M − (cid:1) for 0 ≤ q ≤ Q − M,C q = η q − M − · (cid:0) δ M +1 + 1 (cid:1) for Q − M + 1 ≤ q ≤ Q,D qj = F q,j +1 − F qj for 0 ≤ j < q ≤ Q,H qj = F qj + G qj − ≤ j ≤ q ≤ Q. The values of F qj and G qj (for 0 ≤ j ≤ q ≤ Q ) are depicted in Figure 3 for an easierreference. We will extensively use the property that η i · δ j = η i + j for any i and j . Objective Value.
With the above assignment of dual variables the objective value of D γ, I ,M,β is equal to P Qq = Q − M +1 ξ q = M + P Mj =1 δ j = M + P Mj =1 (2 + γ ) j/M as desired. Non-negativity of Variables
Variables ξ q , C q , B q , F qj and G qj are trivially non-negative(for those q and j for which they are defined). The non-negativity of D qj = F q,j +1 − F qj follows as F qj is a non-decreasing function of its second argument (cf. Figure 3).Finally, for showing non-negativity of variable H qj , we consider two cases. If j ≥ q − M ,then F qj ≥
1. Otherwise, j ≤ q − M −
1, and then G qj = ∆ q − j − M − ≥
1. Thus, in eithercase H qj = F qj + G qj − ≥ . Bienkowski, A. Kraska and H.-H. Liu 17 Variables R q and Helper Bounds. We define a few helper notions and identities that areused throughout the proof of dual feasibility. For any q ∈ [ Q + 1], let R q = Q X ℓ = q +1 D ℓq = Q X ℓ = q +1 ( F ℓ,q +1 − F ℓq ) . ▶ Lemma 13. R q = δ · ∆ M − for q ≤ Q − M and R q = 0 otherwise. Proof.
We consider three cases. q ∈ { , . . . , Q − M − } . Then, R q = F q +1 ,q +1 + P Qℓ = q +1 ( F ℓ +2 ,ℓ +1 − F ℓ +1 ,ℓ ) − F Qq = δ · ∆ M − + P Qℓ = q +1 − δ · ∆ M − . q = Q − M . Then, R q = P Qℓ = Q − M +1 ( ξ ℓ −
1) = P Mj =1 δ j = δ · ∆ M − . q ∈ { Q − M + 1 , . . . , Q } . Then, R q = P Qℓ = q +1 ( ξ ℓ − ξ ℓ ) = 0. ◀ Next, we investigate the values of V qj for different q and j . Using its definition (cf. (14)), V qj = q − X ℓ = j η ℓ · D qℓ − Q X ℓ = q +1 η q · D ℓq = q − X ℓ = j η ℓ · ( F q,ℓ +1 − F qℓ ) − η q · R q . (24)Additionally, using H qj = F qj + G qj −
1, we obtain η j · G qj − η j − · H qj = ( η j − η j − ) · G qj + η j − − η j − · F qj . (25)Using the chosen values of G qj , we observe that( η j − η j − ) · G qj = η q + j − Q + M − − η q for Q − M + 1 ≤ j ≤ q,η q − M − − η j − for j ≤ q − M − , . (26)Furthermore, in all the cases, it can be verified that( η j − η j − ) · G qj + η j − − η q − M − ≥ . (27) A.1 Showing inequalities (16) – (19) . We prove that inequalities (16)–(19) hold with equality. In fact, it suffices to show relations(17) and (19): inequalities (16) and (18) follow immediately as we chose H qj = F qj + G qj − U qj (cf. (14)), we obtain U qj = Q X ℓ = q +1 D ℓq − q − X ℓ = j D qℓ = R q − q − X ℓ = j ( F q,ℓ +1 − F qℓ ) = R q − F qq + F qj . Now, we observe that for q ≤ Q − M , it holds that R q − F qq = δ · ∆ M − − δ · ∆ M − = 0, andthus U qj − F qj = 0, which implies (17). On the other hand, for q > Q − M , it holds that R q − F qq = 0 − ξ q , and hence U qj − F qj + ξ q = 0, which implies (19). A.2 Showing inequalities (20) – (21) Within this part, we assume q ≤ Q − M . We start with evaluating some terms that arepresent in (20) and (21). First, we observe that B q = η q − M − · (cid:0) δ M +1 + 1 (cid:1) · (cid:0) δ M − (cid:1) = η q + M − η q + η q − − η q − M − . (28) Second, we compute the term C L q − P ℓ ∈ S ( q ) B ℓ . Recall that S ( q ) = { q + M, q +2 · M, . . . , L q − M } . Thus, C L q − X ℓ ∈ S ( q ) B ℓ = (cid:0) δ M +1 + 1 (cid:1) · η ( L q − M − − ( δ M − · η ( − M − · X ℓ ∈ S ( q ) δ ℓ = ( δ M +1 + 1) · (cid:2) η ( L q − M − − η ( − M − · (cid:0) δ L q − δ q + M (cid:1)(cid:3) = ( δ M +1 + 1) · η q − = η q + M + η q − . (29) ▶ Lemma 14.
Fix any ≤ j ≤ q ≤ Q − M . Then, V qj = η q − η q − − η q + M − η j − · F qj + ( η j − η j − ) · G qj + η j − . Proof.
By the definition, ∆ M − = P M − i =0 δ i , and therefore ( η q − − η q ) · δ · ∆ M − = η q − η q + M .Thus, it suffices to show the following relation V qj = η q − · ( δ · ∆ M − − − η q · δ · ∆ M − − η j − · F qj + ( η j − η j − ) · G qj + η j − . To evaluate V qj using (24), it is useful to trace values F qj , F q,j +1 , . . . , F qq (cf. Figure 3),noting that only the increases of these values contribute to V qj . We also note that for q ≤ Q − M , possible increases are from 0 to 1 (between F q,q − M − and F q,q − M ) and from 1to δ · ∆ M − (between F q,q − and F qq ). We consider three cases, using R q = δ · ∆ M − below. j ≤ q − M −
1. Then, F qj = 0 and V qj = η q − · ( F qq − F q,q − ) + η q − M − · ( F q,q − M − F q,q − M − ) − η q · R q = η q − · ( δ · ∆ M − − − η q · δ · ∆ M − + η q − M − − η j − + η j − − η j − · F qj . The lemma follows as ( η j − η j − ) · G qj = η q − M − − η j − (see (26)). j ∈ { q − M, . . . , q − } . Then F qj = 1, and V qj = η q − · ( F qq − F q,q − ) − η q · R q = η q − · ( δ · ∆ M − − − η q · δ · ∆ M − = η q − · ( δ · ∆ M − − − η q · δ · ∆ M − − η j − · F qj + η j − . The lemma follows as ( η j − η j − ) · G qj = 0 (see (26)). j = q . Then F qj = δ · ∆ M − , and thus V qj = − η q · R q = η q − · δ · ∆ M − − η q · δ · ∆ M − − η j − · F qj = η q − · ( δ · ∆ M − − − η q · δ · ∆ M − − η j − · F qj + η j − . As in the previous case, the lemma follows as ( η j − η j − ) · G qj = 0. ◀ Showing Inequality (20) . We show that (20) holds with equality. Using Lemma 14, (29),and (25) yields V qj + η j − · H qj − η j · G qj + C L q − P ℓ ∈ S ( q ) B ℓ = η q − η q − − η q + M − η j − · F qj + ( η j − η j − ) · G qj + η j − − ( η j − η j − ) · G qj − η j − + η j − · F qj + η q + M + η q − = η q . . Bienkowski, A. Kraska and H.-H. Liu 19 Showing Inequality (21) . Using Lemma 14, (29), and (25) yields V qj + η j − · F qj + B q = η q − η q − − η q + M − η j − · F qj + ( η j − η j − ) · G qj + η j − + η j − · F qj + η q + M − η q + η q − − η q − M − = ( η j − η j − ) · G qj + η j − − η q − M − ≥ . where the last inequality follows by (27). A.3 Showing inequalities (22) – (23) . Within this part, we assume q ≥ Q − M + 1. ▶ Lemma 15.
Fix any q ≥ Q − M + 1 and ≤ j ≤ q . Then, V qj = η q + ( η j − η j − ) · G qj − η j − · F qj + η j − . Proof. As g ≥ Q − M + 1, it holds that R q = 0, and thus (24) reduces to V qj = q − X ℓ = j η ℓ · ( F q,ℓ +1 − F qℓ ) . As in the proof of Lemma 14, to further evaluate V qj , it is useful to trace values F qj , F q,j +1 , . . . ,F qq (cf. Figure 3), where the increases of these values contribute to V qj . We also note thatfor q ≥ Q − M + 1, the possible increases are from 0 to 1 (between F q,q − M − and F q,q − M )and from 1 to ξ q (between F q,Q − M and F q,Q − M +1 ). We consider three cases. j ≤ q − M −
1. Then, F qj = 0 and V qj = η Q − M · ( F qq − F q,q − ) + η q − M − · ( F q,q − M − F q,q − M − )= η Q − M · ( ξ q −
1) + η q − M − = η q + η q − M − − η j − + η j − − η j − · F qj . The lemma follows as ( η j − η j − ) · G qj = η q − M − − η j − (see (26)). j ∈ { q − M, . . . , Q − M } . Then F qj = 1, and V qj = η Q − M · ( F qq − F q,q − )= η Q − M · ( ξ q − η q − η j − · F qj + η j − . The lemma follows as ( η j − η j − ) · G qj = 0 (see (26)). j ∈ { q − M, . . . , Q − M } . Then F qj = ξ q = 1 + δ q − Q + M , and V qj = 0= η q + η q + j − Q + M − − η q − η j − · (1 + δ q − Q + M ) + η j − = η q + η q + j − Q + M − − η q − η j − · F qj + η j − The lemma follows as ( η j − η j − ) · G qj = η q + j − Q + M − − η q (see (26)). ◀ Showing Inequality (22) . We show that (22) holds with equality. Using Lemma 15 and (25),we obtain V qj + η j − · H qj − η j · G qj = η q + ( η j − η j − ) · G qj − η j − · F qj + η j − − ( η j − η j − ) · G qj − η j − + η j − · F qj = η q . Showing Inequality (23) . Using Lemma 15, (25), and the definition of C q , we obtain V qj + η j − · F qj − C q = η q + ( η j − η j − ) · G qj − η j − · F qj + η j − + η j − · F qj − η q − η q − M − = ( η j − η j − ) · G qj + η j − − η q − M − ≥ . where the last inequality follows by (27). B Omitted Proofs ▶ Lemma 4.
Fix any input I . There exists value T I , such that for any τ ≥ T I , S τ completesall jobs of I and is an optimal (cost-minimal) solution for I . Proof.
Let
Opt be a cost-optimal schedule for I and let t be its length. Let w be the weightof the lightest job from I . We fix T I = max { t, ( val t ( Opt ) + 1) /w } . Now, we pick any τ ≥ T I , and investigate properties of S τ .As τ ≥ T I ≥ t , the schedule of Opt can be trivially extended to a τ -schedule A thatdoes nothing in its suffix of length τ − t . Both A and Opt complete all jobs, and thus val τ ( A ) = val t ( Opt ). Moreover, as S τ minimizes function val τ , val τ ( S τ ) ≤ val τ ( A ) = val t ( Opt ) < T I · w ≤ τ · w , and thus S τ completes all jobs (as otherwise val τ would includea penalty of at least τ · w ). As S τ and Opt complete all jobs, cost S τ ( I ) = val τ ( S τ ) ≤ val t ( Opt ) = cost
Opt ( I ), i.e., S τ is an optimal solution for I . ◀▶ Lemma 9.
Fix any γ , any input I for γ -resettable scheduling, and parameters of Disc :integer M and β ∈ (0 , /M ] . Then, E [ cost Disc ( I )] / cost Opt ( I ) ≤ P ∗ γ, I ,M,β /M , where P ∗ γ, I ,M,β is the value of the optimal solution to P γ, I ,M,β . Proof.
By scaling all variables by the same value, P γ, I ,M,β is equivalent to the (non-linear)optimization program P ′ γ, I ,M,β , whose objective is to maximize ( P Qq =0 P qj =0 η q · w F qj + g F qj ) / max Q − M +1 ≤ q ≤ Q P qj =0 g qj , subject to constraints (8)–(11). In particular, the optimalvalues of these programs, P ∗ γ, I ,M,β and P ′∗ γ, I ,M,β are equal.Next, we set the values of variables w F qj , w S qj , g F qj and g S qj on the basis of input I , andparameters M and β . (Note that the variables depend on these parameters, but not on therandom choices of Disc .) We now show that they satisfy the constraints of P ′∗ γ, I ,M,β and werelate E [ cost Disc ( I )] / cost Opt ( I ) to P ∗ γ, I ,M,β .By Lemma 5 and the relations w F q = P qj =0 w F qj and w S q = P qj =0 w S qj , the variables satisfy(8) and (9). Next, Lemma 6 implies (10). Inequalities (12), (13) and (11) follow directlyby the definition of costs and weights. Finally, by Lemma 7 and Lemma 8, for any q ∈{ Q − M +1 , . . . , Q } , it holds that E [ cost Disc ( I )] / cost Opt ( I ) = (1 /M ) · ( P Qq =0 P qj =0 η q · w F qj + g F qj ) / ( P qj =0 g qj ), and thus E [ cost Disc ( I )] / cost Opt ( I ) ≤ P ′∗ γ, I ,M,β /M = P ∗ γ, I ,M,β /M . ◀ . Bienkowski, A. Kraska and H.-H. Liu 21 ▶ Theorem 11.
For any γ , there are γ -resettable scheduling problems, such that for any ω ∈ [ − , , the competitive ratio of Mimic ( γ, ω ) is at least γ . Proof.
We fix a small ε > α = 2 + γ . The input I contains two jobs: the firstone of weight ε that arrives at time 1, and second one of weight 1 that arrives at time α ω + ε . We assume that there exists a schedule S that serves the first job at the time ofits arrival and a schedule S that serves both jobs at the times of their arrivals. Therefore, cost Opt ( I ) = ε · · ( α ω + ε ) = α ω + 2 · ε .For analyzing the cost of Mimic , note that at at time 1,
Mimic observes the first job andlearns the value of min( I ) = 1. This is the sole purpose of the first job: setting min( I ) = 1makes the algorithm miss the opportunity to serve the second job early. At time τ = α ω , Mimic executes the τ -schedule S ′ , which is schedule S prolonged trivially to length τ .Next, at time τ = α ω , Mimic executes the τ -schedule S ′ , which is schedule S prolongedtrivially to length τ . This way it completes the second job at time τ + ( α ω + ε ), and thus cost Mimic ( I ) ≥ τ + α ω + ε = α ω · (1 + α ) + ε . By taking appropriately small ε >
0, theratio between cost
Mimic ( I ) and cost Opt ( I ) becomes arbitrarily close to 1 + α = 3 + γ . ◀▶ Theorem 12.
For any γ , there are γ -resettable scheduling problems, such that the compet-itive ratio of a randomized algorithm that runs Mimic ( γ, ω ) with a random ω ∈ ( − , is atleast γ ) / ln(2 + γ ) . Proof.
Let α = 2 + γ . The input I contains a single job of weight 1 arriving at time 1. Wealso assume that for any τ ≥
1, there exists a τ -schedule S τ that completes this job at time 1.Clearly, cost Opt ( I ) = 1 · Mimic observes the only job of I and learns that min( I ) = 1. Its sets τ = α ω and at time τ it executes schedule S τ , thus completing the job at time τ + 1. Therefore, cost Mimic ( I ) = R − τ + 1 dω = R − α ω + 1 dω = 1 + ( α − / ln α = 1 + (1 + γ ) / ln(2 + γ ).This implies the desired lower bound. ◀ C Flaw in the Randomized Lower Bound for DARP
The authors of [23] claim a lower bound of 3 for randomized k -DARP (for any k ≥ m and with a realnumber v ∈ [0 , k -DARP problem is at least L m,v = 3 m − km − km + v m +12 − m · (3 + (4 km + 4 km + 6 m + 6) · v ) − − m − km − km + v m +12 − m · (3 + (2 km + 2 km + 4 m + 4) · v )and they claim that there exists v , such that L m,v = 3 when m tends to infinity. However, forany fixed k and any v (also being a function of m ), by dividing numerator and denominatorby m , we obtain thatlim m →∞ L m,v = − k + v m +12 − m · k · v − k + v m +12 − m · k · v = 2 ..