This paper establishes performance guarantees for online algorithms that
schedule stochastic, nonpreemptive jobs on unrelated machines to minimize the
expected total weighted completion time. Prior work on unrelated machine
scheduling with stochastic jobs was restricted to the offline case, and
required linear or convex programming relaxations for the assignment of jobs to
machines. The algorithms introduced in this paper are purely combinatorial. The
performance bounds are of the same order of magnitude as those of earlier work,
and depend linearly on an upper bound on the squared coefficient of variation
of the jobs' processing times. Specifically for deterministic processing times,
without and with release times, the competitive ratios are 4 and 7.216,
respectively. As to the technical contribution, the paper shows how dual
fitting techniques can be used for stochastic and nonpreemptive scheduling
problems.Comment: Preliminary version appeared in IPCO 201

Gupta, Varun

Moseley, Benjamin

Uetz, Marc

Xie, Qiaomin

English

arXiv

This corrigendum fixes an incorrect claim in the paper Gupta et al. [Gupta V, Moseley B, Uetz M, Xie Q (2020) Greed works—online algorithms for unrelated machine stochastic scheduling. Math. Oper. Res. 45(2):497–516.], which led us to claim a performance guarantee of 6 for a greedy algorithm for deterministic online scheduling with release times on unrelated machines. The result is based on an upper bound on the increase of the objective function value when adding an additional job j to a machine i (Gupta et al., lemma 6). It was pointed out by Sven Jäger from Technische Universität Berlin that this upper bound may fail to hold. We here present a modified greedy algorithm and analysis, which leads to a performance guarantee of 7.216 instead. Correspondingly, also the claimed performance guarantee of (6 + 3Δ)h(Δ) in theorem 4 of Gupta et al. for the stochastic online problem has to be corrected. We obtain a performance bound (7:216 + 3:608Δ)h(Δ).</p

University of Twente Research Information

This article was downloaded by: [2001:67c:2564:ad00::d:1008] On: 10 March 2022, At: 01:06Publisher: Institute for Operations Research and the Management Sciences (INFORMS)INFORMS is located in Maryland, USAMathematics of Operations ResearchPublication details, including instructions for authors and subscription information:http://pubsonline.informs.orgCorrigendum: Greed Works—Online Algorithms forUnrelated Machine Stochastic SchedulingVarun Gupta , Benjamin Moseley , Marc Uetz , Qiaomin XieTo cite this article:Varun Gupta , Benjamin Moseley , Marc Uetz , Qiaomin Xie  (2021) Corrigendum: Greed Works—Online Algorithms forUnrelated Machine Stochastic Scheduling. Mathematics of Operations Research 46(3):1230-1234. https://doi.org/10.1287/moor.2021.1149Full terms and conditions of use: https://pubsonline.informs.org/Publications/Librarians-Portal/PubsOnLine-Terms-and-ConditionsThis article may be used only for the purposes of research, teaching, and/or private study. Commercial useor systematic downloading (by robots or other automatic processes) is prohibited without explicit Publisherapproval, unless otherwise noted. For more information, contact permissions@informs.org.The Publisher does not warrant or guarantee the article’s accuracy, completeness, merchantability, fitnessfor a particular purpose, or non-infringement. Descriptions of, or references to, products or publications, orinclusion of an advertisement in this article, neither constitutes nor implies a guarantee, endorsement, orsupport of claims made of that product, publication, or service.Copyright © 2021, INFORMSPlease scroll down for article—it is on subsequent pagesWith 12,500 members from nearly 90 countries, INFORMS is the largest international association of operations research (O.R.)and analytics professionals and students. INFORMS provides unique networking and learning opportunities for individualprofessionals, and organizations of all types and sizes, to better understand and use O.R. and analytics tools and methods totransform strategic visions and achieve better outcomes.For more information on INFORMS, its publications, membership, or meetings visit http://www.informs.orgCorrigendum: Greed Works—Online Algorithms for UnrelatedMachine Stochastic SchedulingVarun Gupta,a Benjamin Moseley,b Marc Uetz,c Qiaomin XiedaUniversity of Chicago, Chicago, Illinois 60637; bCarnegie Mellon University, Pittsburgh, Pennsylvania 15213; cUniversity of Twente,Enschede 7500AE, Netherlands; dCornell University, Ithaca, New York 14850Contact: varun.gupta@chicagobooth.edu, https://orcid.org/0000-0001-7373-1734 (VG); moseleyb@andrew.cmu.edu,https://orcid.org/0000-0001-8162-017X (BM); m.uetz@utwente.nl, https://orcid.org/0000-0003-4223-2435 (MU); qiaomin.xie@cornell.edu, https://orcid.org/0000-0003-2834-6866 (QX)Received: July 10, 2020Accepted: February 11, 2021Published Online in Articles in Advance:July 16, 2021MSC2000 Subject Classifications: Primary:90B36, 68M20; secondary: 90C27, 90C59OR/MS Subject Classification: Primary:analysis of algorithms: suboptimal algorithms;production/scheduling: stochastic, multiplemachine, sequencing: stochastic,approximations/heuristichttps://doi.org/10.1287/moor.2021.1149Copyright: © 2021 INFORMSAbstract. This corrigendum fixes an incorrect claim in the paper Gupta et al. [Gupta V,Moseley B, Uetz M, Xie Q (2020) Greed works—online algorithms for unrelated machinestochastic scheduling.Math. Oper. Res. 45(2):497–516.], which led us to claim a performanceguarantee of 6 for a greedy algorithm for deterministic online scheduling with release timeson unrelatedmachines. The result is based on an upper bound on the increase of the objectivefunction value when adding an additional job j to a machine i (Gupta et al., lemma 6). It waspointed out by Sven Jäger from Technische Universität Berlin that this upper bound may failto hold. We here present a modified greedy algorithm and analysis, which leads to a perfor-mance guarantee of 7.216 instead. Correspondingly, also the claimed performance guaranteeof (6+ 3Δ)h(Δ) in theorem 4 of Gupta et al. for the stochastic online problem has to be cor-rected.We obtain a performance bound (7:216+ 3:608Δ)h(Δ).In Gupta et al. [1], we claim several results for stochastic online scheduling on unrelated machines to minimizethe total weighted completion time∑j wjCj. For the version of the problem where jobs are released over timeand become known at individual release times rj, the analysis of the greedy algorithm as presented in Guptaet al. [1], section 6.1.1, is incorrect, as the upper bound given in lemma 6 in Gupta et al. [1], section 6.1.2, mayfail to hold. To understand what the issue is, recall that the greedy algorithm works by modifying release times rj torij when job j is assigned to a machine i, where rij :max{rj, c · pij}, and c ≥ 0 is a parameter that is optimized later inthe analysis. The reason why lemma 6 in Gupta et al. [1] may fail to hold is potential low-priority jobs with respectto a given job j—that is, jobs h with wh=pih < wj=pij—that could be released in the open time interval (rj, rij). Suchjobs could delay the start of job j beyond the upper bound claimed in Gupta et al. [1], lemma 6.This problem can be fixed by simply accounting for the potential additional delay that such jobs could im-pose on job j. This can be done because we know at most one low-priority job could be in process upon timerij and impose such additional delay. We therefore suggest a small modification of the greedy algorithm,namely, in the way jobs are assigned to machines in step 3 of the algorithm: Instead of letting this assignmentdepend on the actual increase in the objective function value, we let it depend on an upper bound on the in-crease in the objective function value. This upper bound also includes the potential delay that could becaused by one low-priority job. Technically speaking, in contrast to Gupta et al. [1], we now define cost j→ i( )as an upper bound on the increase of the objective function value when assigning job j to machine i and as-sign a job to any of the machines minimizing this quantity. This leads to several necessary changes in thesubsequent analysis, including a different choice for several of the parameters used in the analysis (e.g.,parameter c is chosen to be 2=3 instead of 1). The rest of this corrigendum gives all necessary changes for theanalysis.Greedy Algorithm (Online Time Model for Deterministic Processing Times) Consider any fixed job j that is releasedat time t  rj with processing times pij on machines i  1, : : : ,m. Then, we proceed as follows.1. Modified release times: On machine i, the release time of job j is modified to rij :max{rj, c · pij}; we will opti-mize parameter c ≥ 0 later.2. LetUi(t) denote the set of jobs that have been assigned to machine i at time t and that have not been started yet(excluding the fixed job j).1230MATHEMATICS OF OPERATIONS RESEARCHVol. 46, No. 3, August 2021, pp. 1230–1234ISSN 0364-765X (print), ISSN 1526-5471 (online)http://pubsonline.informs.org/journal/moor3. To decide on the assignment of job j to a machine, we define cost j→ i( ) as an upper bound on the additionalcost of job j, when included into a hypothetical greedy weighted shortest processing time (WSPT) schedule of jobsUi(rj) onmachine i:cost j→ i( ) : wj 1+ 1c( )rij + pij +∑k∈Ui(rj), wkpik≥wjpijpik⎛⎜⎜⎜⎜⎜⎜⎝⎞⎟⎟⎟⎟⎟⎟⎠+ ∑k∈Ui(rj), wkpik<wjpijwkpij: (1)The reason to work with this upper bound is potential jobs that could be released in the interval (rj, rij). One ofthese could delay the earliest possible start time of job j beyond rij. In defining cost j→ i( ), we account for themaximum additional delay that such jobs could impose on j.4. Among all machines i ∈ {1, : : : ,m}, assign job j to a machine m( j) that minimizes cost j→ i( ), ties brokenarbitrarily.5. On each machine i, schedule jobs following the greedy weighted shortest processing time rule with modifiedrelease times rij. That is, as soon as a machine falls idle at time t, we schedule among all unscheduled jobs k as-signed to machine iwith rik ≤ t, any job jwith maximal ratio wk=pik.The following lemma replaces lemma 6 of Gupta et al. [1]:Lemma 6. If ALG denotes the objective function value of the above greedy algorithm, and if m(j) is the machine to whichjob j got assigned, thenALG ≤∑j∈Jcost j→m j( )( ):Proof. Denote by Xi(t) the remaining processing time of a job that is in process on machine i at time t, withXi(t)  0 if no such job exists. Consider a fixed job j’s contribution to the objective ∑jwjCj. When job j is releasedat time rj, it is assigned to a machine that minimizes cost j→ i( ).We bound the contribution of this job j to∑j wjCj as follows: First, we estimate the contribution of job j itself: jcan be started no earlier than time rij, and at time rij, the machine might be blocked for another Xi(rij) time unitsby some job h. Note that such job h could even get released later than rj, namely, in time interval (rj, rij). Indepen-dent of this, job j’s start can be further delayed by “high-priority” jobs k from Ui(rj), where high priority meansthat wk=pik ≥ wj=pij. Second, the fact that job j got assigned to machine i could delay the “low-priority” jobs fromUi(rj), where low priority means that wk=pik < wj=pij.Altogether, the increase of∑jwjCj, caused by job j being assigned to machine i, is at mostwj rij +Xi(rij) +∑k ∈Ui(rj), wkpik ≥wjpijpik + pij( )+ ∑k ∈Ui(rj), wkpik <wjpijwkpij ≤ cost j→ i( ):To see why the last inequality is true, let h be the potential job in process at time rij, thenXi(rij) ≤ pih ≤ rihc ≤rijc:The claim of the lemma now follows by summing over all jobs j ∈ J, and because of the following observation: Intime interval (rj, rij), an even higher priority job k could get released, and such a job k can cause j’s start to be de-layed even further. But the delay that these jobs will impose on j will be accounted for in the term cost k→ i( ). Theset of all low-priority jobs that could get released in interval (rj, rij) can cause j’s start to be delayed by at most Xi(rij).Dual Lower BoundThe analysis in Gupta et al. [1] proceeds by defining a linear programming (LP) relaxation for the deterministicversion with processing times E[Pij], which is termed (Pr), and finding a lower bound on the optimal solutionvalue zPr of this LP relaxation by any feasible solution to its dual linear program, which is:max zDr  ∑j∈Jαj −∑i∈M∑s∈Z≥0βi,ss:t:αjE Pij[ ] ≤ βi,s +wj s+ 12E Pij[ ]+ 12( )for all i ∈M, j ∈ J, s ∈ Z≥rj ,βi,s ≥ 0 for all i ∈M, s ∈ Z≥0:(Dr)Gupta et al.: Online Algorithms for Unrelated Machine Stochastic SchedulingMathematics of Operations Research, 2021, vol. 46, no. 3, pp. 1230–1234, © 2021 INFORMS 1231Presaging the speed-scaling analysis, defineαj : cost j→m j( )( )βi,s :∑k:m k( )i; rk≤s;Ck≥swk, (2)and their analogues in the modified instance where both the release times and processing times are scaled by afactor f ≥ 1,αfj αjf,βfi,s  βi,f s:(3)In words, αj is the previously defined upper bound on the contribution of job j to the objective, and βi,s is the totalweight of jobs assigned to machine i that were released by time s, but are yet unfinished. The values α fj and βfi,sare the corresponding values for an identical instance where time is scaled by factor f . The following lemma re-places lemma 7 of Gupta et al. [1]:Lemma 7. With α f and β f as defined in (3), the values (α f=a,β f=b) are a feasible solution for the dual (Dr), given thata ≥ 1, f ≥ 1, af ≥ 2(2+ c), 1=c ≤ f (a− 1), and af ≥ b. Specifically, one feasible solution is obtained when c  2=3,a  32=23, b  16=3, and speed f  23=6.Proof. Denoting E[Pij] by pij, the dual constraints require that, for all jobs j and machines i, and for all times s ≥ rj,αjpij≤ βi,s +wjs+ 12pij+wj · 12 : (4)Fixing job j and machine i, and plugging in the values α fj =a and βfi,s=b, we need to showαfja · pij ≤βfi,sb+wj s+12pij+wj · 12 , (5)for all s ≥ rj. Equivalently, noting that α f  α=f , we have to show thatαjpij≤ af · βfi,sb+wj s+12pij· af +wj · af2 : (6)Because β fi,s  βi,f s, and replacing s+ 1=2 by s, it therefore suffices to showαjpij≤ af · βi,f sb+wj spij · af +wj ·af2(7)for all s ≥ rj. Because of our choice of αj as minimizer of cost j→ i( ), we have for all machines iαjpij≤ wjpij· 1+ 1c( )rij + pij +∑k∈Ui(rj), wkpik≥wjpijpik⎛⎜⎜⎜⎜⎜⎜⎝⎞⎟⎟⎟⎟⎟⎟⎠+ ∑k∈Ui(rj), wkpik<wjpijwk : (8)Hence, it suffices to show that the right-hand side in (8) is upper-bounded by the right-hand side in (7). To thatend, we even show that a slightly stronger inequality is true: Note that βi,f s is the total weight of jobs k assignedto machine i and unfinished at time f s, but with rk ≤ f s. As long as f ≥ 1, and because rj ≤ s, we have rj ≤ f s.Hence, βi,f s ≥∑k:m(k)  i,rk ≤ rj ,Ck≥f swk ≥∑k∈Ui(rj),Ck≥f swk. Therefore, it suffices to show that the right-hand side of (8)is bounded from above byafb· ∑k∈Ui(rj),Ck≥f swk +wj spij · af +wj ·af2 afb· ∑k∈Ui(rj),Ck≥f swk +wjpij · f s− rj( )( )+wjpij· f s a− 1( ) + rj( )+wj · af2 :Gupta et al.: Online Algorithms for Unrelated Machine Stochastic Scheduling1232 Mathematics of Operations Research, 2021, vol. 46, no. 3, pp. 1230–1234, © 2021 INFORMSMultiplying everything with pij, we therefore need to argue that the following inequality is true:wj 1+ 1c( )rij + pij +∑k∈Ui(rj), wkpik≥wjpijpik⎛⎜⎜⎜⎜⎜⎜⎝⎞⎟⎟⎟⎟⎟⎟⎠+ ∑k∈Ui(rj), wkpik<wjpijwkpij≤ afb· ∑k∈Ui(rj):Ck≥f swkpij +wj · f s− rj( )( )+wj · f s a− 1( ) + rj( )+wjpij · af2 :Let us rewrite this more conveniently aswj · 1+ 1c( )rij + pij( )︸︷︷︸I+ ∑k∈Ui(rj), wkpik≥wjpijwjpik +∑k∈Ui(rj), wkpik<wjpijwkpij︸︷︷︸II≤ afb· ∑k∈Ui(rj):Ck≥f swkpij + wj · f s− rj( )︸︷︷︸II∗+ wj · f s a− 1( ) + rj( )+ pij · af2( )︸︷︷︸I∗: (9)The following observations and conditions are sufficient for the above inequality to be true:1. I ≤ I∗: Distinguish two cases. When rij  rj, we have I  (1+ 1=c)rij + pij  rj + rj=c+ pij. Moreover, since s ≥ rj,I∗  ( f s(a− 1) + rj) + pij · af=2 ≥ rj + f (a− 1)rj + pij · af=2. Therefore, we get that I ≤ I∗ under the conditions that1=c ≤ f (a− 1), and af ≥ 2. On the other hand, when rij  cpij, we get I  (2+ c)pij, and we get that I ≤ I∗ under thecondition that 2(2+ c) ≤ af , whenever a ≥ 1. Summarizing, we get that I ≤ I∗ for both cases, conditioned on 1=c ≤f (a− 1) and 2(2+ c) ≤ af .2. II ≤ II∗: We have by definition ofUi(rj) thatwj f s− rj( ) ≥ wj ∑k∈Ui(rj),Ck<f spik:Therefore, under the condition that af=b ≥ 1, we get that II ≤ II∗, because thenII∗ − II ≥ ∑k∈Ui(rj), wkpik≥wjpij,Ck≥f swkpij −wjpik( ) +∑k∈Ui(rj), wkpik<wjpij,Ck<f swjpik −wkpik( ) ≥ 0:One can now check that parameter values c  2=3, a  32=23,b  16=3, and f  23=6 fulfill all necessary requirements.Main ResultsFinally, the following theorem replaces theorem 3 of Gupta et al. [1]. Note that the competitive ratio 7.216 is stillan improvement over the best prior algorithm that was known to be 8-competitive (Hall et al. [2]).Theorem 3. The greedy algorithm for the deterministic online scheduling problem with release times has competitive ratio7.216 for minimizing the total weighted completion times∑jwjCj on unrelated machines. That is, ALG ≤ 7:216OPT.Proof. By definition of α in (2) and by Lemma 6, we haveALG ≤∑jαj:Moreover, because∑iβi,s is the total weight of the jobs that have been released by time s, but are yet unfinished,∑i,s βi,s ∑jwj (Cj − rj) ≤∑j wjCj. Therefore, we haveALG ≥∑i, sβi,s:Consider now a modified problem instance where both the release times and the processing times are scaled bya factor f as follows:r fj :rjfand p fij :pijf,Gupta et al.: Online Algorithms for Unrelated Machine Stochastic SchedulingMathematics of Operations Research, 2021, vol. 46, no. 3, pp. 1230–1234, © 2021 INFORMS 1233so that we also haver fij rijf:Then, α fj as defined in (3) is the analogous upper bound on the increase in total weighted completion time due tothe presence of job j in the modified instance, and β fi,s as defined in (3) is the weight of the released, but yet unfin-ished, jobs on machine i at time s in the modified instance. Here, we assume without loss of generality that all jobsizes and release times are integer multiples of f , which can be achieved by scaling. Also, let us denote by ALG fthe value achieved by the greedy algorithm for the modified instance, and note that ALG=f  ALG f ≤∑jα fj , andALG=f  ALG f ≥∑i,sβ fi,s.By Lemma 7, which gives a lower bound on the optimal solution value OPT via a feasible solution for (Dr) ofthe form (α f=a,β f=b) for constants (a,b), we getOPT ≥∑jαfja−∑i, sβfi,sb≥ ALGf1a− 1b( ),orALG ≤ f ·1=a− 1=bOPT:Now, because parameters c  2=3, a  32=23,b  16=3, and speed f  23=6 are feasible choices for Lemma 7, weget that ALG ≤ (7+ 11=51) ·OPT < 7:216 ·OPT.The above rational parameters have been obtained by using a basic nonlinear solver, combined with binarysearch on one of the parameters. As a matter of fact, optimizing over the four parameters a,b, f , c using SCIP as asolver for nonlinear constrained optimization yields as optimal solution f=(1=a− 1=b) ≈ 7:2151018 < 7+ 11=51,however, with nonrational solution values. For the sake of simplicity and for feasibility of scaling arguments, wedecided to go with the rational solution as presented above.The proof of theorem 4 of Gupta et al. [1] is unchanged; the new competitive ratio of Theorem 3 implies the fol-lowing modification to the main result of Gupta et al. [1]:Theorem 4. The greedy algorithm has a performance guarantee of (7:216+ 3:608Δ)h(Δ) for online scheduling of stochasticjobs with release times on unrelated machines to minimize the expectation of the total weighted completion timesE∑jwjCj[ ]. That is, ALG ≤ (7:216+ 3:608Δ)h(Δ)OPT.AcknowledgmentsThe authors are grateful to Sven Jäger from TU Berlin for pointing out the flaw in lemma 6 in the original version of thepaper (Gupta et al. [1]). The authors also thank the reviewers of this corrigendum for their careful reading and remarksand for suggesting another bug fix in the proof of Theorem 3.References[1] Gupta V, Moseley B, Uetz M, Xie Q (2020) Greed works—online algorithms for unrelated machine stochastic scheduling. Math. Oper. Res.45(2):497–516.[2] Hall LA, Schulz AS, Shmoys DB, Wein J (1997) Scheduling to minimize average completion time: Off-line and on-line approximation algo-rithms. Math. Oper. Res. 22(3):513–544.Gupta et al.: Online Algorithms for Unrelated Machine Stochastic Scheduling1234 Mathematics of Operations Research, 2021, vol. 46, no. 3, pp. 1230–1234, © 2021 INFORMS

Corrigendum: Greed Works—Online Algorithms for Unrelated Machine Stochastic Scheduling

This paper establishes performance guarantees for online algorithms that schedule stochastic, nonpreemptive jobs on unrelated machines to minimize the expected total weighted completion time. Prior work on unrelated machine scheduling with stochastic jobs was restricted to the offline case and required linear or convex programming relaxations for the assignment of jobs to machines. The algorithms introduced in this paper are purely combinatorial. The performance bounds are of the same order of magnitude as those of earlier work and depend linearly on an upper bound on the squared coefficient of variation of the jobs' processing times. Specifically for deterministic processing times, without and with release times, the competitive ratios are 4 and 6, respectively. As to the technical contribution, this paper shows how dual fitting techniques can be used for stochastic and nonpreemptive scheduling problems.</p

Greed works-online algorithms for unrelated machine stochastic scheduling

This paper establishes performance guarantees for online algorithms that schedule stochastic, nonpreemptive jobs on unrelated machines to minimize the expected total weighted completion time. Prior work on unrelated machine scheduling with stochastic jobs was restricted to the offline case and required linear or convex programming relaxations for the assignment of jobs to machines. The algorithms introduced in this paper are purely combinatorial. The performance bounds are of the same order of magnitude as those of earlier work and depend linearly on an upper bound on the squared coefficient of variation of the jobs' processing times. Specifically for deterministic processing times, without and with release times, the competitive ratios are 4 and 6, respectively. As to the technical contribution, this paper shows how dual fitting techniques can be used for stochastic and nonpreemptive scheduling problems

Greed Works -- Online Algorithms For Unrelated Machine Stochastic Scheduling

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