New and Improved Algorithms for Minsum Shop Scheduling*

Transcription

1 871 New and Improved Algorithms for Minsum Shop Scheduling* Maurice Queyranne t Maxim Sviridenko ~ Abstract We consider a general class of multiprocessor shop scheduling problems with a reinsure objective, and present approximation methods based on linear programming relaxations in the operation completion times. These LP relaxations use new classes of valid inequalities for multistage jobs. We first consider open shop problems with total weighted job completion time objective. For the nonpreemptive problem O[[ ~ wj Cj, we introduce "LP-based precedence constraints" and derive a 5.83-approximation algorithm. For its preemptive ver- sion, O[pmtn[ y~wjcj, we show that a simple job-based greedy algorithm, using directly the LP solution, yields a 3o approximation. We then consider a general class of multiprocessor shop scheduling problems, preemptive or nonpreemptive, with precedence constraints between operations, with job or operation release dates, and with a general reinsure objective. This class of objectives includes, among others, weighted sums of operations completion times, job completion times, stage mal~espans and the overall makespan. We combine the LP relaxation with two known techniques: (i) partitioning the set of operations using time intervals with geometrically increasing lengths; and (ii) an approximation algorithm for the makespan version of these problems without release dates. When the latter produces a schedule with makespan no larger than p times the "trivial lower bound" consisting of the largest of all stage average loads (or "congestion?) and job lengths (or "dilation"), we obtain a 2epapproximation, where 2 e < This leads in particular to a polylogarithmic approximation for the reinsure multiprocessor dag-shop problem, and a O(1) approximation for the reinsure, acyclic job shop problem with unit processing times. These performance guarantees also bound the gap between optimum preemptive and nonpreernptive schedules for these problems. *Research supported by a research grant from NSERC (the Natural Sciences and Research Council of Canada) to the first author. tfaculty of Commerce and Business Administration, University of British Columbia, Vancouver, B.C., Canada V6T 1Z2; Maurice. Queyranne@comnerce. ubc. ca SUniversity of Aarhus, BRICS, Aarhus, Denmark; sviri@brics, dk 1 Introduction. Approximation algorithms have been developed during the past three decades for many scheduling problems with minmax objective primarily the makespan. In contrast, scheduling problems with minsum objective-- typically, a weighted sum of completion times--have only recently received extensive attention. Many resuits for the makespan problems use a trivial lower bound which is the largest of the maximum stage average load, also known as "congestion" [15], and the maximum job length, or "dilation" (ibid.). On the other hand, two of the most successful tools for the reinsure problems have been the use of time intervals with geometrically increasing lengths (as in Hall et al. [10, 11]) and Chakrabarti et al. [2]), and linear programming (LP) relaxations (see Queyranne and Schulz [20], Shmoys [26] and Skutella [29] for extensive reviews and references). In this paper, we combine these three techniques--makespan approximation algorithms relative to the trivial lower bound, geometric time intervals, and LP relaxations--to obtain new or improved approximation algorithms for two broad classes of shop scheduling problems with reinsure objective. In Section 2, we consider preemptive and nonpreemptive open shops with a weighted sum of job completion times objective. We present an LP relaxation, from which we derive a 5.83-approximation algorithm for the nonpreemptive problem O]]~wjCj. This approximation factor is slightly weaker than the (5.78+e)- approximation factor obtained by Chakrabarti et al. [2] for the open shop problem Omlrj[)'~.wjC j with release dates but a fixed number m of machines. On the other hand, our algorithm, which uses a novel idea of "LP-based job precedence constraints" derived from the LP solution, is polynomial for any number of machines. We also present a simple 3-approximation algorithm for the preemptive problem O]pmtn I ~wjcj. In contrast with the other algorithms in this paper, it is a simple job-based greedy algorithm using directly the LP solution, without recourse to geometrically increasing time intervals. Its approximation factor is also slightly weaker than the ( e)-approximation factor for Om]rj, pmtn I ~ wjcj in Chakrabarti et al. [2], but again our algorithm is polynomial for any number of machines.

2 872 In Section 3, we consider a large class of multiprocessor shop scheduling problems with release dates on operations and a general reinsure objective. Minsum objectives includes, among others, such "composite objectives" as weighted sums of operations completion times, job completion times, stage makespans and the overall makespan, as may arise, for example, in multi-objective scheduling problems (e.g., [17]). Our LP relaxation includes a new type of inequalities, the ' "job capacity construints" whereby a job (whose operations cannot be simultancously processed) is considered as a "resource" with unit capacity. Our main result is the following: if there exists a polytime algorithm for a class of multiprocessor job shop problems which guarantees a makespan no larger than p times the trivial lower bound, then we obtain a polytime algorithm for the minsum version of the problem, with release dates, which guarantees an objective value no larger than 2 e p times the optimum, where e ~ is the base of the natural logarithms. This yields algorithms with the best known performance guarantees for various classes of job shop, flow shop and certain open shop problems with release dates and minsum objective. On the other hand, Hoogeveen et al. [12] have shown that the problems F I I ~ Cj and OII ~ Cj are MAX-SNP-hard, hence there is no hope for obtaining approximation guarantees arbitrarily close to 1 for these problems, unless 7) = Jq'7 ~. Using known makespan approximation algorithms, we obtain the new approximation bounds for weighted sum of operations (or job) completion times objectives given in Table 1 (at the end of this paper). In this table, /~ = maxj/~j denotes the largest number of operations per job. We use the notation dagj to indicate that the only precedence constraints axe between operations of a same job. A job shop is acyclic ff each job visits each stage at most once. For all but the two open shop results from Section 3 we use the reduction from the multiprocessor makespan problem J(P)[dagj[Cmax to the single processor makespan problem J] dagj ]Cma~ due to Shmoys et al. [28]. The polylogarithmic performance guarantee in the first row of Table 1 dominates 1 earlier O(m) results for: the classical job-shop problem Jl[ ~ Cj with mean flow time objective (Gonzalez and Sahni [8]); the multiprocessor flow-shop problem F(P)lri[~wjCj with weighted sum of job completion times objective (Schulz [241/; and the open shop problem O]1 ~ Cj with mean flow time objective (Achugbue and Chin [1]). 1Except for instances where /~ is very large, namely, where m = o((log(m/~)/loglog(m/~))2)), discussed under the name of %ery long jobs" in [21]. Since the LP relaxations are valid for the preemptive problems, the approximation bounds obtained for the nonpreemptive versions also bound the "power of preemption", i.e., the relative gap between optimum preemptive and nonpreemptive schedules. Detailed proofs of the results in this paper can be found in our working paper [211. Chandra Chekuri's Ph.D. dissertation [3] concludes with the following question: "That is, is there a polynomial time algorithm that uses as a subroutine a procedure for minimizing makespan and outputs an approximate schedule for minimizing average weighted completion time?" Our results in Section 3 provide a partial positive answer to this question, where we relaxed the requirement of "minimizing makespan" to that of "approximately minimizing makespan relative to the trivial lower bound." Providing a fully positive answer to Chekuri's original question remains an interesting challenge. This would, for example, lead to polynomial time approximation schemes for minsum job shop problems with a fixed number of machines (see [13]). 2 Open Shop Scheduling. In this section we consider open shop problems with a weighted sum of job completion times objective wjcj. We first present a linear programming relaxation which is valid for the preemptive version of the problem, and therefore also for its nonpreemptive version. We use the LP solution in two different ways and obtain a 5.83-approximation algorithm for the nonpreemptive problem OI]~'~wjC j, and a 3-approximation algorithm for the preemptive problem Olpmtn I ~ wjcj. In an open shop problem, we are given a set 7 ) = (P1,..., Pro) of machines and a set J = (J1,..., Jn) of jobs. Each job Jj E ff consists of a set Oj of operations, where operation o E Oj must be processed without interruption on machine re(o) for Po time units, and may not start earlier than date 0. Each machine can process at most one job at any given time, and no two operations on a same job may be performed simultaneously. The open shop problems we consider here are more general than usual, in that each job may include any number of operations (possibly none) on every machine, thus allowing a limited form of "partial preemption" of that job on that machine. Let Cj denote the completion time of job Jj, that is, the latest completion time of an operation of that job in a schedule. We seek a feasible schedule of all the job so as to minimize the weighted sum ~j wjcj of the job completion times, where wj are given nonnegative job weights.

3 Open Shops: A Linear Progran3Lming 1;relaxation. For every job Jj E ff let Plj denote the total processing time of all operations of job Jj on machine Pi. Note that pij = 0 if there is no such operation, i.e., if job Jj does not use machine Pi- Let 5 = )'~m=l Pij denote the length of job Ji. For every subset A C_ ff of the job set, define 1 (z / 1 Pij" \SEA / SEA Consider the following linear program: (2.1) min ~'~wjcj (2.2) S.t. Cj ~_ ~ Jj E J; (2.3) Z pijcj ~_ fi(a) A C_ N, Pi e P jea where N -- {1,...,n}. Note that this LP includes a single variable Cj per job, and thus ignores the details of scheduling the individual operations on the corresponding machines. The objective (2.1) is to minimize the weighted sum of job completion times, while the job length constraints (2.2) reflect the fact that no two operations of job Jr can be processed simultaneously. Constraints (2.3) are relaxed versions of the machine capacity constraints due to Wolsey [30] and Queyranne [18, 19]. They prevent too much work from being completed too early on each machine, and are valid for both preemptive and nonpreemptive schedules. Indeed, consider a machine Pi and a feasible (preemptive or nonpreemptive) schedule. Let Cij denote the latest completion time of an operation of job Jj on this machine, with Cij = 0 if there is no such operation. Note that all the operations of a job Jj on this machine may be viewed as a single "job" processed preemptively for a total of pij time units on this single machine. Since all these "jobs" form a feasible preemptive schedule on machine Pi, their completion times Cij must satisfy the single machine capacity constraints E pijcij )_ fi(a) for all A C N, jea see [19] or [20]. Since all Pii >_ 0 and Cij < Cj, the validity of constraints (2.3) follows. Therefore the linear program (2.1)--(2.3) defines a relaxation of both the preemptive and nonpreemptive problems O[pmtn I ~ wjcj and O[1 ~ wjcj. This LP relaxation has one variable per job, but an exponential number of constraints (2.3). Since the separation problem for these constraints can be solved in polynomial time (Queyranne [19]) it follows that the LP (2.1)--(2.3) can itself be solved in polynomial time [9]. Let C LP denote an optimal solution to LP (2.1)--(2.3). 2.2 Nonpreemptive Open Shop Scheduling: A 5.83-Approximation Algorithm. Given the LP solution C LP, we define LP-based job precedence constraints as follows: Jj "~LP Jk if and only if 7C Lp < Ck LP, where yr2 ~ We will use these LP-based job precedences Jj "~LP J~ in our approximation algorithm to prevent starting job Jt on machine Pi until every job Jj with "sufficiently early LP completion time" (as defined by the condition 7C LP < C LP) has completed on this machine. (Remark that they do not prevent starting job Jk on another machine Pv, provided of course that every such other job Jj has completed on that machine Pv.) We schedule the operations using "machine-based scheduling" consistent with the LP-based job precedences. (A related idea is used by Chekuri et al. [4] for the parallel machine problems Plrjl~w~Cj and Plr~, precl ~ w~cj.) Namely, we start with all jobs that have no predecessor in "~LP being available for each machine. We let the availability date of each machine Pi be Ai = 0. We repeat the following until all job completion times have been determined. Let Pi be a machine with the earliest availability date t - Ai = minh Ah. If there is a job Jj which is available for machine Pi and not in process on some other machine at date t then do the following: (1) process Jj on Pi in the time interval [t, t+pij]; (2) update Ai = t+pii; (3) declare job Jj not available any more for machine Pi; and (4) declare available for machine Pi every job Jk such that Jj "~LP Jk and each of its predecessors Ju "~LP Jk has been entirely processed on machine Pi. (Remark that a job Jk may be available for a machine even when some operations of some of its predecessors are not complete on some other machines.) Otherwise (that is, if there is no job which, at date t, is available for machine Pi and is not in process on some other machine) update Ai to the next smallest machine availability date min{ah : Ah > t}: machine Pi will be idle during the time interval It, Ai]. We obtain the following job-by-job bound: LEMMA 2.1. Let C LP be an optimal solution to (2.1)- (2.3). Then the algorithm described in the preceding paragraph defines a feasible nonpreemptive schedule with job completion times Cff satisfying, for every job Jj, c7 <_ + From this job-by-job bound, we obtain a approximation for the nonpreemptive open shop problem with a weighted sum of job completion times objective:

4 874 THEOREM 2.1. There exists a polynomial time (3 + 2~/2) -approximation algorithm for the nonpreemptire open shop problem Oll ~ w~cj. 2.3 Preemptive Open Shop Scheduling: A 3- Approximation Algorithm. In contrast with the nonpreemptive open shop algorithm above, we use a job-based greedy algorithm to preemptively schedule the jobs in order of their LP completion times C LP. Namely, assume w.l.o.g, that the jobs are indexed in nondecreasing order of their LP completion times, that is1 that C LP < C LP ~... ~ CLn P. We consider the jobs Jj one by one, in this order j = 1,2,...,n, and all operations of job Jj are scheduled as follows, without modifying the schedule of jobs J1,...,J j-1. Given the partial schedule of these jobs J1,---, Jj-1, job Jj is preemptively assigned to the earliest available time slots, so long as its corresponding operations are not complete. Ties between machines and/or operations available at a same time are broken arbitrarily. When all work for job J~ has been scheduled, defming its completion time C~, we turn to job Jj+l. The schedule is complete when the last job Jn is scheduled. We obtain the following job-by-job bound: LEMMA 2.2. Let C LP be an optimal solution to (2.1)- (2.3). Then the algorithm described in this section defines a feasible preemptive schedule with job completion times C H satisfying, for every job Jj, cy <3c? From this job-by-job bound, we obtain a 3- approximation for the preemptive open shop problem with a weighted sum of job completion times objective: THEOREM 2.2. There exists a polynomial time 3- approximation algorithm for the preemptive open shop problem Olpmtnl ~ w~cj. 3 Minsum Multiprocessor Job Shop Scheduling. An instance Z of the minsum multiprocessor job shop problem J(P)lrhj, prec[ ~ wijcij is defined as follows: 7 ) = (P1,.--, Pro) is the set of stages; stage Pi consists of mi identical parallel machines (or processors) (mi > 1). 7" = (J1,..., Jn) is the set of jobs; job Jj consists of a set of/~j operations 01j,..., Oujj, where Ohj must be processed without interruption on one machine in stage re(h, j) for Phi time units, where Phi is the given processing time of operation Obj. No two operations on a same job may be performed simultaneously. Let Oi be the set of all operations Ohj which must be processed on stage Pi, i.e., such that m(h,j) = i, and let (.9 = UiOi denote the set of all operations. There are two additional types of constraints on the operations: -~ denotes the precedence constraints between operations: Ogj -~ Oh~ means that operation Ogj must be completed before operation Ohk can start; operation Ogj is a predecessor of Ohk and Ohk is a successor of Ogj; we assume that -~ defines a (strict) partial order, i.e., an acyclic digraph; rhj >_ 0 is the release date of operation Ohj; w.l.o.g., we may assume that the release dates are compatible with the precedence constraints, that is, rgj +pgj < rhk whenever Ogj -~ Oak. Finally, we are also given a family/4 C 2 of subsets of operations and nonnegative subset weights wu >_ 0 for all U E L/. For any schedule with operation completion times Chj (Ohj e 0), let Cmax(U) = max{chj : Ohj E U} denote the completion time of subset U E U. The minsum objective is the weighted sum ~veuwvcmax(u) of subset completion times. This allows us to model, among others, "composite objectives" including weighted sums of operation completion times (U = {Ohj}), job completion times (U = {Olj,..., O~jj}), stage makespans (U -- Oi) and overall makespan (U = O). The problem is to schedule all operations on machines in the prescribed stages, according to the precedence and release date constraints, and so as to minimize the minsum objective ~-~u~u wucma~(u). We are interested in both the preemptive and nonpreemptive versions of this job shop problem. We assume that all Phj and rhj axe nonnegative integers. Zero processing times (Phi ) and release dates (rhj = 0) are allowed, but we assume w.l.o.g. that rhj q-phi _ 1 for all Ohj, since all operations with zero processing time and release date may be performed at time zero without increasing the objective value or affecting the rest of the schedule. We define the average load of stage 7)i as Li = (1/mi) ~(h,j)eo, Phi and the length of job Jj as gj = )-~h Phj. The trivial lower bound for the makespan problem is max{max/li, maxj gj}. 3.1 Multiprocessor Job Shops: A Linear Programming Relaxation. In contrast with the open shop LP of Section 2.1, the linear programming relaxation below uses one variable, the completion time Chj,

5 875 for every operation Ohj E O, as well as one variable Ctr for every subset U E/~. (3.4) min Z wvcu UE/4 (3.5) s.t. Cu >_ Chj U e LI, Ohj E U; (3.6) Chj >_ rhj + Phi h e Mj, Jj e fl; (3.7) Chk >_ Cgj + Phk 09j -< Oak; (3.8) Z > S m)(a) Ohj E A A C_ O~, P~ e P; (3.9) Z phjchj > gj(b) B C Mj, Jj e fl; heb where Mj -- {1,..., p j}, and 1 (z / 1 2 \Oh~eA ] Oh~A 1 (z3 1 z 2 gj(s) = ~ Phi + ~ Paj. \heb / heb The objective function is formulated by (3.4) and (3.5). Constraints (3.6) and (3.7) are the release date and precedence constraints, respectively. Constraints (3.8) form (a relaxation of) the stage capacity constraints. They are parallel machine extensions of the single machine inequalities of Wolsey and Queyranne; see [23] or [20] for a proof of their validity. Constraints (3.9) are a new feature of this work. They may be interpreted as "job capacity constraints", and their validity follows from the requirement that no two operations of the same job may be processed simultaneously. They have the same mathematical structure as the single-machine capacity constraints (3.8), except that they correspond to subsets of operations on a same job. A physical interpretation is that every job may be considered as a "resource" with unit capacity. Note that these job capacity constraints are implied by the release date and precedence constraints (3.6) and (3.7) when a job consists of a chain (totally ordered set) of operations, such as in flow shop and "classical" job shop problems. The LP-relaxation (3.4)-(3.9) has one variable per operation, hut an exponential number of constraints (3.8) and (3.9). Since the separation problem for these constraints can be solved in polynomial time, it follows that this LP can be solved in polynomial time. More precisely, we obtain: PROPOSITION 3.1. Consider the reinsure multiprocessor shop scheduling problem defined above and let T = maxad rn~ + Y~h,j Phj. Then one can find in polynomial time an optimal solution C LP to the LPrelaxation (3.4)-(3.9) such that 1 < C LP < T and Y~'~treu wucma~(u) < OPT, where OPT denotes the optimal objective value of the preemptive scheduling problem. 3.2 A Minsum Job Shop Approximation Algorithm. In the first step of our algorithm we find an optimal solution C LP to the linear program (3.4)-(3.9) satisfying the properties in Proposition 3.1. Let e denote the base of the natural logarithms, and lgt - rlnt] +1. We use a uniform random variable X on [0, 1] to determine a random base point ~ = exp(-x). Thus, at the end of the first step, ~ is a fixed number satisfying 1/e<~< 1. In the second step, we define operation subsets O (k) - {Ohj : j3exp(k) _< clf < ~exp(k + 1)), i.e., O (k) is the set of all operations with LP completion times clf in interval [~ exp(k), ~ exp(k + 1)), for k = 0,..., lgt. Note that the nonempty subsets O (k) form a partition of the set O of all operations and that their number, at most 1 + lg T, is polynomially bounded. For each nonempty O (k), we define an instance Z (k) of the makespan job shop problem restricted to the operations in O(k) and ignoring the release date constraints (but not the precedence constraints between operations). Let L~ k) " Z {Phi: Ohj E 0 (k) and m(h,j) = i} ITt i denote the average load of stage Pi in instance Z (k), and let ~k). ~'~{paj : Oaj e O (k) } denote the length of job Jj in Z (k). Let C (k) denote the completion time vector of the schedule, starting at time 0 and obtained by applying to 2: (k) a makespan approximation algorithm with the property that where ]Z(k)[ is the size of instance 2:(k) and p is a nondecreasing function such that p 0Z(k)]) > 1. To simplify notation, we use "p" as an abbreviation for "p (IZI)" hereafter. In the third step of our algorithm, we concatenate these partial schedules C (k) into the heuristic schedule C H by letting C~ = ~exp(k + 2)p + Ch(~ ) for all operations Ohj, and keeping the same assignment of operations to the parallel machines at every stage. We obtain the following expected operation-byoperation bound:

6 876 LEMMA 3.1. Let C LP be an optimal solution to the LP relaxation (3.4)-(3.9) of the minsum multiprocessor job shop problem. The algorithm described in this section defines a feasible nonpreemptive schedule with expected f ~ satisfying, for every operation Oh1 G 0 E [cg] < 2e, C$f. The randomized algorithm above can be derandomized as in Roundy [22] and Goemans and gleinberg [6]. We obtain a polynomial time algorithm with approximation guarantee 2 e p < 5.44 p: THEOREM 3.1. Consider a class of instances of the multiprocessor job shop problem J(P)lpreclCmax with makespan objective, for which there exists a polynomial time algorithm producing a feasible schedule with makespan no larger than p times the trivial lower bound (largest of all stage average loads and job lengths), where p may be a nondecreasing function of the size of the instance. Then there exists a polynomial time approximation algorithm for the reinsure job shop problem J ( P)lrhl, precl ~tr wvcmax(u) with performance guarantee 2 e p. We now briefly compare our approximation algorithm with relevant parts of the work in Hail et ai. [11] and Chakrabarti et ai. [2]. A 7-approximation algorithm for the precedence constrained single-stage problem Plrj, prec I ~ wjcj in [11] uses an LP relaxation similar to those herein, combined with power-of-two time intervals and Graham's list scheduling algorithm within each interval. The approximation guarantee is improved to 5.33 in [2] by the use of a randomized base period as in our Section 3. The job capacity constraints used in our LP relaxation are key to the extension of this approach to (multi-stage) shop scheduling problems. Using different techniques, [2] also presents, for a fixed number of machines, a ( )-approximation factor for the job shop problem Jmlrjl ~ wjcj with release dates but a fixed number of machines. Acknowledgements: The authors want to thank Andreas Schulz (MIT) for drawing their attention to relevant parts in the work of Hail et ai. [11] and of Chakrabarti et ai. [2], for suggesting the randomized base intervai method used in Section 3.2, and for extensive discussions and suggestions on an earlier version of this work. They also thank Chandra Chekuri (Bell Laboratories) for pointing out the question raised in his Ph.D. dissertation and quoted herein. References [1] J.O. Achugbue and F.Y. Chin, "Scheduling the open shop to minimize mean flow time." SIAM Journal on Computing 11 (1982) [2] S. Chakrabarti, C.A. Phillips, A.S. Schulz, D.B. Shmoys, C. Stein and J.Wein, "Improved Scheduling Algorithms for Minsum Criteria." In: F. Meyer auf der Heide and B. Monien, eds, Automata, Languages and Programming (Proceedings of the 23rd International Colloquium ICALP'96, 1996), Lecture Notes in Computer Science 1099, Springer, [3] C. Chekuri, Approximation Algorithms for Scheduling Problems. Ph.D. Dissertation, Dept. of Computer Science, Stanford University, August [4] C. Chekuri, R. Motwani, B. Natarajan, and C. Stein, "Approximation Techniques for Average Completion Time Scheduling." Proceedings of the 8th Symposium on Discrete Algorithms (SODA'97, 1997) [5] U. Feige and C. Scheideler, "Improved bounds for acyclic job shop scheduling." In: Proceedings of the 30th Annual ACM Symposium on Theory of Computing (STOC'98, 1998), ACM Press, New York, NY, [6] M. Goemans and J. Kleinberg, "An improved approximation ratio for the minimum latency problem." In: Proceedings of the 7th ACM-SIAM Symposium on Discrete Algorithms (SODA'96, 1996) [7] L. A. Goldberg, M. Paterson, A. Srinivasan, and E. Sweedyk, "Better approximation guarantees for jobshop scheduling." Dept. of Computer Science, University of Warwick, Coventry, UK. Preliminary version in: Proceedings of the 8th Symposium on Discrete Algorithms (1997) [8] T. Gonzalez and S. Sahni, "Fiowshop and jobshop schedules: Complexity and approximation." Operations Research 26 (1978) [9] M. GrStschel, L. Lovdsz, and A. Schrijver, Geometric Algorithms and Combinatorial Optimization. Springer, [10] L.A. Hall, D.B. Shmoys, and J. Wein, "Scheduling to Minimize Average Completion Time: Off-line and Online Algorithms", Proceedings of the 7th Symposium on Discrete Algorithms (1996) [11] L.A. Hall, A.S. Schulz, D.B. Shmoys, and J. Wein, "Scheduling to Minimize Average Completion Time: Off-Line and On-Line Approximation Algorithms", Mathematics of Operations Research 22 (1997) [12] H. Hoogeveen, P. Schuurman, and G. Woeginger, "Nonapproximability results for scheduling problems with reinsure criteria." In: R.E. Bixby, E.A. Boyd, and R.Z. Rios-Mercado, eds, Integer Programming and Combinatorial Optimization (IPCO-VI Proceedings, 1998), Lecture Notes in Computer Science 1412, Springer, [13] K. Jansen, R. Solis-Oba and M. Sviridenko, "Makespan Minimization in Job Shops: a Polynomial Time Approximation Scheme." In: Proceedings of the 31th

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