Information Processing Letters 110 (2010) 325–330

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Information Processing Letters

www.elsevier.com/locate/ipl

An optimal semi-online algorithm for a single machine schedulingproblem with bounded processing time

Jiping Tao ∗, Zhijun Chao, Yugeng Xi, Ye Tao

Department of Automation, Shanghai Jiao Tong University, 800 Dong Chuan Road, Shanghai, China

a r t i c l e i n f o a b s t r a c t

Article history:Received 11 March 2009Received in revised form 16 October 2009Accepted 24 February 2010Available online 26 February 2010Communicated by W.-L. Hsu

Keywords:Analysis of algorithmsSemi-online algorithmsSingle machine schedulingTotal weighted completion timeCompetitive ratio

The single machine semi-online scheduling problem is considered with the assumptionthat the ratio of the longest processing time to the shortest one is not greater than γ . Thegoal is to minimize the total weighted completion time. We design a semi-online algorithm

and prove that it has a competitive ratio of 1 +√

4γ 2+1−12γ , which is also shown to be the

best performance achieved by any deterministic semi-online algorithm for the problem.© 2010 Elsevier B.V. All rights reserved.

1. Introduction

We consider the following single machine online sched-uling problem. A sequence of jobs arrive over time. Foreach job J j , its processing time p j and its weight w j can-not be known until its release date r j . The objective is tominimize the total weighted completion time. The problemcan be denoted by 1|r j|∑ w j C j in terms of the three-fieldnotation for scheduling problems [1]. An online algorithmdetermines the starting time for each job with the con-straint that jobs must be processed without interruption.If the algorithm makes use of some additional knowledgeor assumptions about the problem, we call it a semi-onlinealgorithm. The quality of an online or semi-online algo-rithm is often assessed by its competitive performance. Anonline or semi-online algorithm is called ρ-competitive if,for any instance, the objective function value of the sched-ule obtained from this algorithm is no worse than ρ timesthe objective value of the optimal offline schedule [2]. The

* Corresponding author.E-mail addresses: jipingtao@gmail.com (J. Tao), zhijun.chao@gmail.com

(Z. Chao), ygxi@sjtu.edu.cn (Y. Xi), tao.ye.tina@gmail.com (Y. Tao).

0020-0190/$ – see front matter © 2010 Elsevier B.V. All rights reserved.doi:10.1016/j.ipl.2010.02.013

competitive ratio is defined as the infimum of all the ρssuch that the algorithm is ρ-competitive.

Numerous results have been obtained for the corre-sponding problem where there is no limitation on process-ing times. For the case of identical weights, i.e., 1|r j |∑ C j ,Hoogeveen and Vestjens [3] prove that any online algo-rithm has a competitive ratio of at least 2. They slightlymodify the shortest processing time (SPT) rule and con-struct an online algorithm named by the delayed SPT(D-SPT) rule, which is proven to be 2-competitive. Theiranalysis is established by exploiting the structural rela-tion between a so-called “pseudo-schedule” and the op-timal preemptive schedule. For the same problem, Philipset al. [4] also develop an online algorithm with the samecompetitive ratio of 2. As for the case of nontrivial weights,i.e., 1|r j |∑ w j C j , Hall et al. [5] first present an online al-gorithm whose competitive performance is upper boundedby 3 + ε for any ε > 0. The upper bound is improved to1 + √

2 in [6] via the linear programming relaxation ofa time-index formulation and the α-point schedule. It isAnderson and Potts [7] who take an important step to-ward the optimality in the year of 2004 and develop theoptimal deterministic online algorithm for the problem.They extend the D-SPT rule in [3] and present the de-

326 J. Tao et al. / Information Processing Letters 110 (2010) 325–330

layed SWPT (D-SWPT) rule. They establish an elaboratecompetitive analysis via introducing two auxiliary schedul-ing problems called by “doubled problem” and “extendedproblem”. The analysis method does not need to make ex-plicit use of lower bounds on the optimal solution value.As a new proof technique, it has potential applications toother problems.

It is important to notice that all the above results areachieved without any restriction on the processing timesof jobs. We also note that the lower bound 2 in [3] is ob-tained by an instance where an infinite number of jobswith processing time 0 are released immediately after themachine starts processing an infinitely long job. However,such extreme instances may never happen in real schedul-ing problems. Therefore, it is not necessary to considerthat the processing time in an instance can cover such alarge range. A natural assumption is to limit the processingtime in a bounded interval. Under the assumption, Kamin-sky and Simchi-Levi [8] prove that the shortest processingtime (SPT) rule is asymptotically optimal for the case ofidentical weights, where “asymptotically optimal” meansthat the relative error from the optimum approaches zeroas the number of jobs tends to infinity. For the case ofnontrivial weights, if weights are also bounded, Chou [9]proves that the shortest weighted processing time (SWPT)rule is asymptotically optimal. The boundedness assump-tion about the processing time is also adopted for parallelmachine scheduling problems [10,11].

Based on the above discussion, we know that D-SWPTis an optimal algorithm when the processing time is un-bounded [7]. We also know that the SWPT rule is asymp-totically optimal when the processing time and weightsare both bounded [9]. However, nothing is known aboutthe semi-online problem with bounded processing timesand finite number of jobs. It is just what we are goingto investigate in this work. We denote the problem by1|r j,

pmaxpmin

� γ |∑ w j C j , where γ is a given constant whichgives an upper bound on the ratio of the longest to theshortest processing time. We believe that it is possibleto design a better semi-online algorithm for the modifiedproblem compared with the case without a restriction onprocessing times. The intuitive explanation is that it needsnot to consider such an instance where jobs with an ex-tremely short processing time arrive immediately after themachine determines to schedule a long job. In this work,we construct a deterministic semi-online algorithm for theproblem 1|r j,

pmaxpmin

� γ |∑ w j C j by modifying the D-SWPTrule presented in [7] and prove that it is optimal with the

competitive ratio of 1 +√

4γ 2+1−12γ . The competitive analy-

sis method is based on the idea of instance transformation,which is first introduced by us in [12]. The method tries tosearch for the worst-case instance in the instance space.It starts from an arbitrary instance and modifies the in-stance towards the possible structure of the worst-caseinstance with respect to the given algorithm. The modifica-tion guarantees that the performance ratio of the modifiedinstance does not decrease. Eventually, it comes up with arelatively simple instance with a special structure, whoseperformance ratio can be directly analyzed.

The paper is organized as follows. In Section 2, a lowerbound is established by constructing two special instances.The semi-online algorithm is presented in Section 3 and itsperformance ratio is analyzed in Section 4. Conclusion andremarks are given in Section 5.

2. A lower bound on the competitive ratio for1|r j,

pmaxpmin

��� γ |∑ wi Ci

In this section we give a lower bound on the compet-itive ratio of any semi-online algorithm for 1|r j,

pmaxpmin

�γ |∑ wi Ci . We construct two special instances for whichno semi-online algorithm can guarantee an outcome strictlyless than a given value.

Theorem 1. For the problem 1|r j,pmaxpmin

� γ |∑ w j C j , anysemi-online algorithm has a competitive ratio of at least 1 +√

4γ 2+1−12γ .

Proof. Given an instance I , denote the objective value ofthe semi-online schedule by ALG(I), and the optimal ob-jective function value by OPT(I). Consider the followingsituation. The first job J1 arrives at time 0 with p1 = 1 andw1 = 1. Suppose that the semi-online algorithm decides toschedule J1 at time S . Depending on S , the adversary hastwo choices, i.e., either to release no jobs any more, or torelease the second job J2 at time S +ε with p2 = 1/γ andw2, where ε is an infinitely small positive number. In thefirst case the optimal schedule obviously processes J1 attime 0, thus we get

ALG(I)

OPT(I)= S + 1. (1)

In the second case, the optimal schedule performs noworse than the one which first schedules J2 at time S + εthen followed by J1. So we get another performance ratioas

ALG(I)

OPT(I)� S + 1 + w2(S + 1 + 1/γ )

S + ε + 1/γ + 1 + w2(S + ε + 1/γ ). (2)

When ε tends to 0 and w2 tends to positive infinity, theabove ratio tends to 1 + 1

S+1/γ .Combining (1) and (2), we can obtain that the compet-

itive ratio is not less than the maximal one of the tworatios, i.e.,

ρ � max

{S + 1,1 + 1

S + 1/γ

}. (3)

The semi-online algorithm can choose S so as to minimizethe above expression. Obviously the best choice for S is

S =√

4γ 2 + 1 − 1

2γ. (4)

This implies the desired lower bound

ρ � 1 +√

4γ 2 + 1 − 1

2γ. � (5)

J. Tao et al. / Information Processing Letters 110 (2010) 325–330 327

3. The βD-SWPT semi-online algorithm

It is well known that the SWPT rule is optimal when alljobs have identical release dates [13]. When jobs arrive dy-namically over time, the key point for an online algorithmis to determine whether to keep the machine idle or to im-mediately schedule the available jobs in the nondecreasingorder of p j/w j . Inspired by the D-SWPT rule [7] for theproblem without any additional limitation on processingtimes, we directly extend it and construct a semi-online al-gorithm for the problem 1|r j,

pmaxpmin

� γ |∑ Ci . We call thisalgorithm β-delayed SWPT (βD-SWPT) rule.

Denote the current decision time by t . The βD-SWPTrule can be described as follows. Whenever the machineis idle and some jobs are available, choose a job with thesmallest value of the ratio p j/w j among all the arrivedand unscheduled jobs. When ties occur, choose the onewith the smallest index. Say that J i is chosen, if pi � βt ,schedule J i ; otherwise, wait until the above inequality issatisfied or a new job arrives, where β is a positive pa-rameter to be designed.

In the extreme case where γ tends to infinity, the D-SWPT algorithm [7] is the best deterministic online al-gorithm, which is equivalent to the βD-SWPT rule withβ = 1. Therefore, intuitively speaking, when γ is a finitevalue, β should be greater than 1 such that less waitingtime would be inserted in the schedule and a better com-petitive performance could be obtained compared with theD-SWPT algorithm. In the following section, we will give acompetitive analysis of the βD-SWPT algorithm and prove

that√

4γ 2+1+12γ is the best choice for β .

4. Competitive analysis of the βD-SWPT algorithm

Competitive analysis is not a trivial routine for mostscheduling problems. The difficulty lies in two aspects. Thefirst one is that the instance space for a given problemis not a very structured one. The number of jobs and therelease dates and processing times of jobs can be as ar-bitrary as possible. The difficulty is how to handle thearbitrariness and explore the worst-case instance. Anotherdifficulty is that no realizable optimal offline algorithm isavailable in most cases due to the NP-hardness character-istic. An appropriate lower bound on the optimal objectivevalue has to be constructed in order to calculate the per-formance ratio. These two difficulties make most of thecompetitive analysis methods typically problem dependent.

In this work, we develop a straightforward analysismethod based on instance transformation. The method notonly can find the worst-case instance in the space of in-stances, but also does not require to know what the opti-mal schedule exactly looks like. The basic procedure is tobegin from an arbitrary instance I , then construct a newinstance I ′ by modifying the data in I , such that I ′ notonly has a more simple and special structure, but also sat-isfies

ALG(I)

OPT(I)� ALG(I ′)

OPT(I ′)or

ALG(I)

OPT(I)� max

{ALG(I ′)OPT(I ′)

, F (β)

},

where F (β) is a function of β , which is an upper bound onthe performance ratio of another instance. In other words,

we maybe construct two new instances by the transfor-mation, one of which possesses a very simple and specialstructure such that its performance ratio can be directlyobtained or upper bounded. Execute the above procedurestep by step. The finally achieved instance may have asimple structure such that its performance ratio can be di-rectly analyzed. In this way, we can obtain an upper boundon the competitive ratio.

First we introduce an important lemma. It can be de-rived from simple analysis and the detailed proof is at-tached in Appendix A. The lemma will be repeatedly usedin the competitive analysis.

Lemma 2. Let f (x) and g(x) be two positive functions definedin the interval [u, v], moreover f (x) is convex and g(x) is con-cave. Then f (x)/g(x) reaches its maximum at either endpointof the interval, i.e., f (x)

g(x) � max{ f (u)g(u)

,f (v)g(v)

} ∀x ∈ [u, v].

It can be easily shown that the above lemma still holdswhen the interval is open at some endpoint on conditionthat the limit of f (x)

g(x) exists at the corresponding endpoint.For any instance I , denote the optimal non-preemptive

schedule by σ ∗(I), and the schedule produced by the βD-SWPT rule by σ(I).

σ(I) is composed of some blocks. In any block jobs areprocessed continuously, and idle time only exists betweenblocks. The idle time occurs either due to the waiting strat-egy of βD-SWPT, or because all the arrived jobs have beenfinished before a new batch of jobs arrive. The jobs sched-uled before an idle period do not influence the schedulingdecision of the jobs scheduled after that idle period, andvice versa [3]. Therefore, the instance I can be split intoseveral smaller instances each of which consists of jobs inone block of σ(I). Moreover the independently constructedschedules by βD-SWPT for these smaller instances keepthe starting times of all the jobs the same as in σ(I). Thusat least one of these smaller instances has a performanceratio no less than that of I . Therefore, from the worst-casepoint of view, we only need to focus on such instance forwhich the schedule produced by βD-SWPT is composed ofa single block. Denote any one of these smaller instancesby I1. Denote the first release date by r0, and the starttime of the block by S . Denote the shortest and longestprocessing time among jobs in I1 by pmin and pmax.

Next we will develop two lemmas to show that the in-stance I1 can be modified step by step such that the newinstance after modification has a more simple and specialstructure.

Lemma 3. A new instance, denoted by I2 , can be constructedby modifying I1 , such that there is no job released after S in I2 ,moreover

ALG(I1)

OPT(I1)� max

{ALG(I2)

OPT(I2),1 + βγ

β + γ

}. (6)

Proof. Consider the last release date after time S . For easeof exposition, denote the release date by rL . Denote thejob being processed at rL in σ(I1) by J i . It starts at time(rL − p̃i), where p̃i is the fragment of J i finished before rL .

328 J. Tao et al. / Information Processing Letters 110 (2010) 325–330

Since rL is the last release date, all the remained jobs areprocessed according to SWPT rule after J i is finished. Wedivide the jobs released at rL into three sets as follows:

Q 1 ={

J j

∣∣∣ p j

w j� pi

wi, r j = rL

}, (7)

Q 2 ={

J j

∣∣∣ 0 <p j

w j<

pi

wi, r j = rL

}, (8)

Q 3 ={

J j

∣∣∣ p j

w j→ 0 (w j → ∞), r j = rL

}. (9)

We should point out that the weights of all the jobs in Q 2are finite values. Modify the weight w j of each job J j inQ 2 to δw j where δ is a parameter to be chosen in the nextstep. Let

δ = max

{p j

w j

∣∣∣ J j ∈ Q 2

}/ pi

wi. (10)

Denote the intermediate instance after this adjustmentby I ′1. Jobs are continuously processed in σ(I ′1) in the sameorder as in σ(I1) for any δ ∈ [δ,+∞) since the modi-fication does not change the SWPT order of these jobs.So ALG(I ′1) is a monotonically increasing linear functionwith respect to δ. Consider how OPT(I ′1) changes with re-spect to δ. For a given active feasible schedule, if the jobskeeps the processing order unchanged when δ changes in[δ,+∞), then the objective value of this feasible sched-ule is a monotonically increasing linear function of δ. Sincethe optimal schedule σ ∗(I ′1) is the one with the mini-mal objective value among all the active feasible sched-ules, OPT(I ′1) is the minimum of multiple linear functions,so it is a piecewise linear function of δ with the slopenot increasing when δ increases. Thus OPT(I ′1) is a con-cave function with respect to δ. According to Lemma 2, wecan obtain an instance with a worse performance ratio bychoosing δ as either δ or +∞. If δ is chosen as δ, thereis at least one job belonging to Q 2 whose weighted pro-cessing time is modified to pi/wi according to the equality(10). Whereas if δ tends to infinity, the weights of all thejobs in Q 2 are modified to +∞. Then update Q 1, Q 2 andQ 3 according to (7)–(9). If there remain jobs in Q 2, modifytheir weights by repeating the above procedure.

After finite steps, all the jobs in Q 2 can be removedinto either Q 1 or Q 3. For ease of exposition, we still useI1 to denote the final instance obtained from the abovemodification.

If Q 3 is not empty, note that each job in Q 3 has aninfinite weight. Thus we can directly analyze the perfor-mance ratio of the instance by only considering these jobsin Q 3 since other jobs can be omitted with respect to theinfinite weighted completion times of jobs in Q 3. All thejobs in Q 3 are continuously processed immediately afterJ i is finished in σ(I1). For any job J j in Q 3, denote itscompletion time in σ(I1) by C j . The optimal schedule con-tinuously process these jobs in Q 3 from the time rL in thesame order as in σ(I1). Denote the corresponding optimalcompletion time of J j by C∗

j . Then we have

C j = C∗j + pi − p̃i . (11)

So in the sense of limit we can obtain

ALG(I1)

OPT(I1)=

∑J j∈Q 3

w jC j∑J j∈Q 3

w jC∗j

(12)

� maxJ j∈Q 3

C j

C∗j

(13)

� 1 + maxJ j∈Q 3

pi

C∗j

(14)

� 1 + pi

rL + pmin(15)

� 1 + pi

pi/β + pmin(16)

� 1 + βγ

β + γ. (17)

Inequality (14) comes from (11). Inequality (15) is truesince C∗

j is the corresponding completion time when jobsin Q 3 are processed from the time rL . Inequality (16) re-sults from the fact that J i starts at time (rL − p̃i) in σ(I1)

according to the βD-SWPT rule, which implies pi � β(rL −p̃i). Inequality (17) comes from the relation pi/pmin � γ .

For the jobs in Q 1, we can keep the processing orderunchanged and reduce their release dates to that of J i . Theadjustment does not decrease the performance ratio sinceit does not increase the optimal objective value and keepsthe βD-SWPT schedule unchanged because these jobs can-not be chosen before J i is finished according to the βD-SWPT rule.

Thus through the above transformation procedure, Q 1,Q 2 and Q 3 all become empty, i.e, there is no job releasedat rL . Therefore, rL is not a true “release date” any more,and the original next-to-last release date becomes the lastone in the new instance. Then we can repeat the abovetransformation procedure. Finally all the original releasedates after S are all “deleted”, moreover the inequality (6)is always satisfied. �Lemma 4. For the instance I2 constructed from Lemma 3, an-other new instance, denoted by I3 , can be constructed such thatthere is no job released at S, moreover

ALG(I2)

OPT(I2)� ALG(I3)

OPT(I3). (18)

Proof. In the case of r0 = S , we assume there is no jobreleased at S in order to avoid confusion. For the case ofr0 = S , the proof is almost the same as that for Lemma 3except that (7)–(9) are replaced by the following equali-ties:

Q 1 ={

J j

∣∣∣ p j

w j� min

ri<S

pi

wi, r j = S, r j = r0

},

Q 2 ={

J j

∣∣∣ 0 <p j

w j< min

ri<S

pi

wi, r j = S, r j = r0

},

Q 3 ={

J j

∣∣∣ p j

w j→ 0 (w j → ∞), r j = S, r j = r0

},

and (10) are replaced by

δ =max{ p j

w j| J j ∈ Q 2}

minr <Spi

.

i wi

J. Tao et al. / Information Processing Letters 110 (2010) 325–330 329

Fig. 1. The βD-SWPT schedule and the optimal schedule for I3.

Similar to the transformation procedure in Lemma 3, Q 2becomes empty. The jobs in Q 3 can also be deleted withthe exception that the upper bound becomes

ALG(I2)

OPT(I2)�

ALG(I ′2)OPT(I ′2)

. (19)

The reason is that jobs in Q 3 are processed from S inσ(I2), so the equality (11) is replaced by C j = C∗

j . As forthe jobs in Q 1, we can similarly reduce their release dateto r0. The detailed proof is omitted. �

Through the modification in Lemmas 3 and 4, I3 onlyincludes jobs released before S . For these jobs not releasedat r0, we can keep the processing order unchanged andreduce their release dates to r0. The adjustment does notdecrease the performance ratio since it does not increaseOPT(I3) and keeps σ(I3) and ALG(I3) unchanged becausethese jobs are not selected before the time S according tothe βD-SWPT rule. Index all the jobs from 1 to n in theorder in which they are processed in σ(I3). Then we have

p1

w1� p2

w2� · · · � pn

wn. (20)

Both σ(I3) and σ ∗(I3) are illustrated in Fig. 1. We can an-alyze the performance ratio of I3 by directly comparing jobcompletion times in σ(I3) and σ ∗(I3).

Lemma 5. For instance I3 ,

ALG(I3)

OPT(I3)� 1 + 1

β.

Proof. When r0 = S , σ(I3) is just the optimal schedule, soALG(I3)OPT(I3)

= 1.Next we consider the case of r0 < S . J1 does not start

before S according to the βD-SWPT rule, so

p1 = β S. (21)

For job J j , denote its completion time in σ(I3) and σ ∗(I3)

respectively by C j and C∗j . We have

C j = C∗j + S, j = 1, . . . ,n.

Hence we can obtain

ALG(I3)

OPT(I3)=

∑nj=1 w j(C∗

j + S)∑nj=1 w jC∗

j

(22)

� 1 + maxS

C∗ (23)

j

= 1 + S

p1(24)

= 1 + 1

β, (25)

where equality (24) is due to the fact that each job is fin-ished after J1 in σ ∗(I3), and the last relation comes fromthe equality (21). �

Combining the above three lemmas with Theorem 1, wecan immediately obtain the competitive ratio of the βD-SWPT algorithm.

Theorem 6. For the problem 1|r j,pmaxpmin

� γ |∑ w j C j , the βD-

SWPT algorithm with β =√

4γ 2+1+12γ has the competitive ratio

of 1 +√

4γ 2+1−12γ .

Proof. Applying Lemmas 3, 4 and 5, we can obtain thecompetitive ratio ρ of βD-SWPT as

ρ � max

{1 + 1

β,1 + βγ

β + γ

}.

The above inequality holds for any β > 0. Let 1 + 1β

=1 + βγ

β+γ so as to minimize the right-hand side, i.e., β =√4γ 2+1+1

2γ , we have

ρ � 1 +√

4γ 2 + 1 − 1

2γ. (26)

Combining it with Theorem 1, we immediately have

ρ = 1 +√

4γ 2 + 1 − 1

2γ. � (27)

Remark 7. The result includes the previous conclusionabout the best deterministic online algorithms for1|r j |∑ w j C j as a special case, since βD-SWPT has thecompetitive ratio of 2 when γ tends to infinity. When γis a finite value, βD-SWPT can achieve a competitive ratioless than 2. Especially, when γ = 1, i.e., each job has thesame processing time, it is a semi-online algorithm withthe competitive ratio of (1 + √

5)/2. To the best of ourknowledge, this is the first online competitive result aboutthe problem 1|p j = p, r j |∑ w j C j .

5. Concluding remarks

In this work, we consider a single machine semi-onlinescheduling problem of minimizing the total weighted com-pletion time with the assumption that the ratio of thelongest processing time to the shortest one in any instanceis not greater than a constant γ . We design a semi-onlinealgorithm and prove that it is optimal with the competitive

ratio of 1 +√

4γ 2+1−12γ .

We develop a novel method to analyze the competi-tive performance of the presented semi-online algorithm.The method is aiming to derive an upper bound on the

330 J. Tao et al. / Information Processing Letters 110 (2010) 325–330

competitive ratio. It exploits the possible structure of theworst-case instance with respect to the given semi-onlinealgorithm. The basic procedure is to begin from an arbi-trary instance and to modify it, such that the modifiedinstance shows a more special structure of which we cantake advantage to analyze the performance ratio. The anal-ysis method is deserved to be extended to other online orsemi-online algorithms in our further work.

Acknowledgement

This work is supported by the National Science Foun-dation of China (No. 60934007) and Specialized ResearchFund for the Doctoral Program of Higher Education(No. 20070248004).

The authors would like to thank Junjie Zhou for usefuldiscussion.

Appendix A. Proof of Lemma 2

For any x in [u, v], there exists a ∈ [0,1] such that x =au + (1 −a)v . Since f (x) is convex and g(x) is concave, wehave

f (x) � af (u) + (1 − a) f (v),

g(x) � ag(u) + (1 − a)g(v).

With the positivity of all functions, we can obtain

f (x)

g(x)� af (u) + (1 − a) f (v)

ag(u) + (1 − a)g(v)

� max

{f (u)

g(u),

f (v)

g(v)

}. �

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