We consider parallel machine scheduling problems where the jobs are subject to precedence constraints, and the processing times of jobs are governed by independent probability distributions. The objective is to minimize the weighted sum of job completion times &;j wjC_j in expectation, where wj&; 0. Building upon an LP-relaxation by Möhring, Schulz, and Uetz (J.ACM 46 (1999), pp.924-942) and an idle time charging scheme by Chekuri, Motwani, Natarajan, and Stein (SIAM J. Comp., to appear) we derive the first approximation algorithms for this model

Skutella, Martin

Uetz, Marc

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  PCUPPPITPQUIIG WEIVGE OPUS DCI SCHEDULING PRECEDENCE-CONSTRAINED JOBS WITH STOCHASTIC PROCESSING TIMES ON PARALLEL MACHINES by MARTIN SKUTELLA Marc UETZ No. 691/2000     FACHBEREICH 3   MATHEMATIKScheduling Precedence-Constrained Jobs with Stochastic Processing ‘Times on Parallel Machines Martin Skutella* Marc Uetz* (Extended Abstract) Abstract We consider parallel machine scheduling problems where the jobs are subject to precedence constraints, and the processing times of jobs are gov- erned by independent probability distributions. The objective is to mini- mize the weighted sum of job completion times }), w; Cj in expectation, where w; > 0. Building upon an LP-relaxation by Méhring, Schulz, and Uetz (J. ACM 46 (1999), pp. 924-942) and an idle time charging scheme by Chekuri, Motwani, Natarajan, and Stein (SIAM J. Comp., to appear) we derive the first approximation algorithms for this model. 1 Preliminaries Denote by V = {1,...,n} a set of jobs which must be scheduled on m parallel machines. Precedence constraints are given by an acyclic digraph G = (V, A). In the stochastic model, a job processing time p; is known only upon completion of the job, however, the distribution of the corresponding random variable p; is given beforehand. Let p = (pi,..-,pn), and denote by p = (p1,.--,Dn) a particular realization of the processing times. A scheduling policy consists of an online process of decisions which must not anticipate future information; we refer to [2] for details. A given policy eventually yields a feasible m-machine schedule for each realization p. Let S;(p) and C;(p) denote the start and completion times of job j for a given realization p, and let S;(p) and Cj(p) denote the associated random variables. 2 List scheduling with idle time charging Given a priority list LZ, Graham’s classical list scheduling algorithm schedules the first available job(s) from the list whenever a machine falls idle. Hence, jobs may be scheduled ‘out of order’ w.r.t. the given list Z. For the makespan * Technische Universitat Berlin, Fachbereich Mathematik, Sekr. MA 6-1, Strafe des 17. Juni 136, 10623 Berlin, Germany. Email:{skutella,uetz}@math.tu-berlin.de. The second author is supported by the German- Israeli Foundation for Scientific Research and Development (GIF) under grant I 246-304.02/97.objective, Graham’s list scheduling achieves a performance ratio of 2 — 1/m, which is also true for the stochastic setting. The weighted completion time ob- jective, however, turns out to be more difficult to approximate. In particular, no approximation result is known for the parallel machine setting with precedence- constraints and stochastic processing times. In order to derive approximation bounds for the weighted completion time objective in deterministic scheduling, a refined list scheduling algorithm has been suggested by Chekuri et al. [1]. The idea is to extend Graham’s list scheduling in such a way that a job may be scheduled out of order (w.r.t. the given list L) only if ‘enough’ idle time has accu- mulated. The analysis of the algorithm relies on a charging scheme for idle time. We show that an appropriate adaption of the list scheduling algorithm from [1], based on an optimal solution to a generalized LP-relaxation from [3], leads to constant worst-case performance guarantees also for the stochastic model. The algorithm proceeds over time, starting at time ¢ = 0, until all jobs have been planed. As usual, a job is called available at time ¢ if all predecessors have already been completed by ¢, and for a given realization p, the earliest point in time when job 7 is available in the schedule constructed by Algorithm 1 is denoted by r;(p). Algorithm 1 (DELAy LIsT [1]). while there are unscheduled jobs do: let t be the earliest point in time when a machine falls idle, or the next tentative decision time (see case (c)), whatever occurs first; let j be the first unscheduled job and i the first unscheduled and available job (if any) in the list L; (a) if j ts available, then start j at time t and charge all uncharged idle time in the interval [r;(p),t] to J; (b) else, if there is at least 6 Elp;] uncharged idle time in the interval [r;(p), t], then start i at time t and charge all uncharged idle time in [ri(p), t] to i; (c) else define the next tentative decision time as the first point in time when (b) applies to i. 3 Analysis We analyze the outcome of Algorithm 1 point-wise, that is, for every realization p of processing times; the analysis is job-by-job. Like in [1], denote by B; and A; the set of jobs that come before and after job j in the list L, respectively (by convention, B; also includes j). For a given realization p, let O;(p) C Aj be the set of jobs in A; that Algorithm 1 starts before job j. The basic idea is to partition the time interval [0,C;(p)] into two disjoint categories: time intervals f, where a chain j1,j2,---,jn = j of predecessors of j is in process (as in Graham’s analysis); the total length of this chain is denoted by ¢;(p). The total processing in the remaining time intervals [r;(p),S(p)], & = j1,---, jn, which are denoted by J2, can be partitioned into three categories: processing of jobs in Bj, processing of jobs in O;(p), and idle time. It follows from the analysis in [1] that any job k is charged no more than GE|p,| idle time. Moreover, there is no uncharged idle time in [r;(p), S;,(p)], and the idle time in [rz(p), S,(p)] is charged only to jobs in B,. This holds in particular for k = j1,...,j,. Since Bj, C++: C B;, = Bj, the total amount of idle time in Iz is bounded from above by B Dic p, = [pi]. Hence, we obtain for every realization p of processingtimes Ci(r) <G() +4 (¥ (+ 8Efpil) + Dv). (1) t€B; 1€0;(p) Before we take expectations in (1), we concentrate on the term )/jco,(p) Pi- First, we require: Lemma 1. E [Zico ip) P| =E [Dicos 0) Efpil| . Proof. We can write Dico; (p) Pi equivalently as ies; 6;(p) p;, where 6;(p) is a binary random variable which is 1 if and only if i € O;(p). Linearity of ex- pectation yields E( Vico; (p) pil = ied; E|6;(p) p;]. But 6;(p) is stochastically independent of the processing time p; — when job i is started, it is already decided whether i € O;(p). In particular, this decision is independent of the actual processing time of job i (processing times are independent). Hence, Dies, E[6i(P) pi] = Vica, Pr € Oj) Epi] = E[Lico,(p) Elpil)- O Next, as in [1], it can be shown that the amount of idle time in [0, S;(p)] charged to jobs in A; is bounded by (m — 1)é;(p). If a job 7 is scheduled out of order w.r.t. j (that is, 1 € O;(p)), then 6E[p;| idle time is charged to 2. Hence, we obtain 8)’ ico, (p) lpi] < (m — 1) £;(p). Taking expectations in (1), Lemma 1 now yields E[C;(p)] < (1+ 33) E[G@)] +42 YO Epi. (2) 1€B; 4 Linear programming relaxation To obtain a priority list L as input for Algorithm 1, Chekuri et al. [1] use a single machine relaxation. This approach does not help in the stochastic setting, since the single machine problem does not necessarily provide a lower bound for the parallel machine problem (see [3] for an example). Instead, we use an LP- relaxation which extends the one proposed in [3] by adding inequalities which represent the precedence constraints. Define f : 2” > IR by FW) = se (( Biel) + ¥ Hp) — tea Y Hp), WC. Here, A > 0 is an upper bound on Var'\p,]/E[p;|* for all j, where Var[p;] = Elp;] — Elp;]° is the variance of p;. It follows from [3] that the inequalities jew Elp;| E[Cj@)] = f(W) are valid for all W C V and any scheduling policy. Hence, the following is a valid LP-relaxation for the problem at hand min > Wj cy jEV 8.t. >» Elps)Cy? = f(W), W CY, (3) JEW CYP > Cy? + Elpy], (i,j) € A. (4)We assume that job 1 is artificial with p, = 0, predecessor of all other jobs, and fixed at time 0, then (4) yields C}? > Elp,] for all jobs j. Since inequalities (3) can be separated in time O(nlogn), see [3], this LP-relaxation can be solved in polynomial time. From an optimum solution to the LP-relaxation, we define a priority list L according to non-decreasing ‘LP completion times’ C}?’. Using Algorithm 1 with input L, we obtain: Theorem 1. LP-based list scheduling using algorithm DELAY LIST is an a- approzimation, where a=1+ 45 + (1+ 6)(1 +max{1, "tA}). Proof. Use (2) and [3, Lemma 4.2] in order to obtain an upper bound on 1/m Diez, Elpi] in terms of the LP-completion time C}*. The fact that E[¢;(p)] is a lower bound on the expected completion time of job j for any scheduling policy then yields the desired result. O Using 8 = 1/V2, this yields a constant worst-case performance bound of a < 5.83 if A < 1, which is the case, e.g., for exponentially distributed processing times, or more generally for so-called NBUE distributions (that is, the expected remaining processing time of a job never exceeds its total expected processing time). 5 Final Remarks The presented results can be slightly improved by a more involved analysis, which also allows to recover the single machine results mentioned in [3]. With minor modifications our results also carry over to problems with release dates. Finally, improved results can be obtained for in-tree precedence constraints using similar ideas as in [1, Sec. 4.4]. References [1] C. Chekuri, R. Motwani, B. Natarajan, and C. Stein, Approzimation tech- niques for average completion time scheduling, to appear in SIAM J. Comp. [2] R. H. Mohring, F. J. Radermacher, and G. Weiss, Stochastic scheduling problems I: General strategies, ZOR — Zeitschrift fiir Oper. Res. 28 (1984), 193-260. [3] R. H. Méhring, A. S. Schulz, and M. Uetz, Approsimation in stochastic scheduling: The power of LP-based priority policies, J. ACM 46 (1999), 924-942.Reports from the group “Combinatorial Optimization and Graph Algorithms” of the Department of Mathematics, TU Berlin 691/2000 Martin Skutella and Marc Uetz: Scheduling Precedence Constrained Jobs with Stochastic Processing Times on Parallel Machines 689/2000 Rolf H. Méhring, Martin Skutella, and Frederik Stork: Scheduling with AND/OR precedence constraints 685/2000 Martin Skutella: Approximating the single source unsplittable min-cost flow problem 684/2000 Han Hoogeveen, Martin Skutella, and Gerhard J. Woeginger: Preemptive scheduling with rejection 683/2000 Martin Skutella: Convex quadratic and semidefinite programming relax- ations in Scheduling 682/2000 Rolf H. Méhring and Marc Uetz: Scheduling Scarce Resources in Chemical Engineering 681/2000 Rolf H. Mohring: Scheduling under Uncertainty: Optimizing Against a Randomizing Adversary 680/2000 Rolf H. Méhring, Andreas S. Schulz, Frederik Stork, and Marc Uetz: Solv- ing Project Scheduling Problems by Minimum Cut Computations (Journal ver- sion for the previous Reports 620 and 661) 674/2000 Esther M. Arkin, Michael A. Bender, Erik D. Demaine, Séndor P. Fekete, Joseph S. B. Mitchell, and Saurabh Sethia: Optimal Covering Tours with Turn Costs 669/2000 Michael Naatz: A Note On a Question of C. D. Savage 667/2000 Sdndor P. Fekete and Henk Meijer: On Geometric Maximum Weight Cliques 666/2000 Sdéndor P. Fekete, Joseph S. B. Mitchell, and Karin Weinbrecht: On the Continuous Weber and k-Median Problems 664/2000 Rolf H. Méhring, Andreas S. Schulz, Frederik Stork, and Marc Uetz: A Note on Scheduling Problems with Irregular Starting Time Costs 661/2000 Frederik Stork and Mare Uetz: Resource-Constrained Project Scheduling: From a Lagrangian Relaxation to Competitive Solutions 658/1999 Olaf Jahn, Rolf H. Méhring, and Andreas S. Schulz: Optimal Routing of Traffic Flows with Length Restrictions in Networks with Congestion 655/1999 Michel X. Goemans and Martin Skutella: Cooperative facility location games 654/1999 Michel X. Goemans, Maurice Queyranne, Andreas S. Schulz, Martin Skutella, and Yaoguang Wang: Single Machine Scheduling with Release Dates 653/1999 Andreas S. Schulz and Martin Skutella: Scheduling unrelated machines by randomized rounding646/1999 Rolf H. Méhring, Martin Skutella, and Frederik Stork: Forcing Relations for AND/OR Precedence Constraints 640/1999 Foto Afrati, Evripidis Bampis, Chandra Chekuri, David Karger, Claire Kenyon, Sanjeev Khanna, Ioannis Milis, Maurice Queyranne, Martin Skutella, Cliff Stein, and Mazim Sviridenko: Approximation Schemes for Minimizing Av- erage Weighted Completion Time with Release Dates 639/1999 Andreas S. Schulz and Martin Skutella: The Power of a-Points in Preemp- tive Single Machine Scheduling 634/1999 Karsten Weihe, Ulrik Brandes, Annegret Liebers, Matthias Muller— Hannemann, Dorothea Wagner and Thomas Willhalm: Empirical Design of Geometric Algorithms 633/1999 Matthias Miiller-Hannemann and Karsten Weihe: On the Discrete Core of Quadrilateral Mesh Refinement 632/1999 Matthias Miller-Hannemann: Shelling Hexahedral Complexes for Mesh Generation in CAD 631/1999 Matthias Miiller-Hannemann and Alerander Schwartz: Implementing Weighted 6-Matching Algorithms: Insights from a Computational Study 629/1999 Martin Skutella: Convex Quadratic Programming Relaxations for Network Scheduling Problems 628/1999 Martin Skutella and Gerhard J. Woeginger: A PTAS for minimizing the total weighted completion time on identical parallel machines 627/1998 Jens Gustedt: Specifying Characteristics of Digital Filters with FilterPro 620/1998 Rolf H. Méhring, Andreas S. Schulz, Frederik Stork, and Marc Uetz: Re- source Constrained Project Scheduling: Computing Lower Bounds by Solving Minimum Cut Problems 619/1998 Rolf H. Méhring, Martin Oellrich, and Andreas S. Schulz: Efficient Algo- rithms for the Minimum-Cost Embedding of Reliable Virtual Private Networks into Telecommunication Networks 618/1998 Friedrich Hisenbrand and Andreas S. Schulz: Bounds on the Chvatal Rank of Polytopes in the 0/1-Cube 617/1998 Andreas S. Schulz and Robert Weismantel: An Oracle-Polynomial Time Augmentation Algorithm for Integer Proramming 616/1998 Alerander Bockmayr, Friedrich Eisenbrand, Mark Hartmann, and Andreas S. Schulz: On the Chvadtal Rank of Polytopes in the 0/1 Cube 615/1998 Ekkehard Kohler and Matthias Kriesell: Edge-Dominating Trails in AT- free Graphs 613/1998 Frederik Stork: A branch and bound algorithm for minimizing expected makespan in stochastic project networks with resource constraints 612/1998 Rolf H. Méohring and Frederik Stork: Linear preselective policies for stochastic project scheduling 611/1998 Rolf H. Méhring and Markus W. Schaffter: Scheduling series-parallel or- ders subject to 0/1-communication delays 609/1998 Arfst Ludwig, Rolf H. Mohring, and Frederik Stork: A computational study on bounding the makespan distribution in stochastic project networks 605/1998 Friedrich Eisenbrand: A Note on the Membership Problem for the Ele- mentary Closure of a Polyhedron596/1998 Andreas Fest, Rolf H. Méhring, Frederik Stork, and Marc Uetz: Resource Constrained Project Scheduling with Time Windows: A Branching Scheme Based on Dynamic Release Dates 595/1998 Rolf H. Méhring Andreas S. Schulz, and Mare Uetz: Approximation in Stochastic Scheduling: The Power of LP-based Priority Policies 591/1998 Matthias Miiller-Hannemann and Alerander Schwartz: Implementing Weighted b-Matching Algorithms: Towards a Flexible Software Design 590/1998 Stefan Felsner and Jens Gustedt and Michel Morvan: Interval Reductions and Extensions of Orders: Bijections to Chains in Lattices 584/1998 Alix Munier, Maurice Queyranne, and Andreas S. Schulz: Approxima- tion Bounds for a General Class of Precedence Constrained Parallel Machine Scheduling Problems 577/1998 Martin Skutella: Semidefinite Relaxations for Parallel Machine Scheduling Reports may be requested from: Hannelore Vogt- Moller Fachbereich Mathematik, MA 6-1 TU Berlin StraBe des 17. Juni 136 D-10623 Berlin — Germany e-mail: moellerQmath.TU-Berlin.DE Reports are also available in various formats from http: //www.math.tu-berlin.de/coga/publications/techreports/ and via anonymous ftp as ftp://ftp.math.tu-berlin.de/pub/Preprints/combi/Report-number-year .ps

Scheduling precedence-constrained jobs with stochastic processing times on parallel machines

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Scheduling precedence-constrained jobs with stochastic processing times on parallel machines

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