Ideal Two-Machine Schedules of Jobs with Unit-Execution-Time Operations: An Extended Abstract Edward G. Coffman, Jr. Department of Electrical Engineering, Columbia University, New York NY 10027, U.S.A.
[email protected] Vadim G. Timkovsky CGI Group Inc., 30 Wellington St. W., Suite 300, Toronto Ontario M5L 1G1, Canada and Department of Computing and Software, McMaster University, Hamilton Ontario L8S 4L7, Canada
[email protected] Abstract Ideal schedules minimize both maximum completion time and total completion time of jobs. This paper presents polynomial-time algorithms finding ideal nonpreemptive and preemptive two-machine schedules of jobs with arbitrary release dates and precedence constraints and unit-execution-time operations on identical parallel machines, in a flow shop and an open shop. Heretofore, the complexity status of these problems has been unknown. 1.
Introduction
We study the problem of scheduling jobs Jj, j=1, 2, ..., n, with release dates rj on machines Mi , i=1, 2, ..., m, under precedence constraints. For identical parallel machines, each job Jj has only one operation Oj which can be run for an execution time pj on any of the machines. In a flow shop or an open shop, each job Jj has m operations Oij which must be run for execution times pij on Mi . The operations of each job should be run in the order O1j O2j ... Omj in flow shops and in any order in open shops. In preemptive scheduling on parallel machines a currently running operation can be transferred to another machine without an interruption, and it can be interrupted and resumed later on the same or another machine. In preemptive shop scheduling interruptions are allowed, but not transfers. There is no restriction on the number of times an operation can be preempted. Preemptions of fractionality 1/m mean that the lengths of nonpreempted parts of operations are multiples of 1/m. In nonpreemptive scheduling operations cannot be preempted. Ideal schedules minimize both maximum completion time and total completion time of jobs. The problem of interest here is finding in the machine environments mentioned above nonpreemptive and preemptive ideal two-machine schedules of jobs with arbitrary release dates and precedence constraints and unit-execution-time operations. For problem notation we use the α|β|γ notation [14]. α=1, P, P2, Q2, F2, O2 denotes a single machine, an arbitrary number of identical parallel machines, two identical parallel machines, two uniform parallel machines (which may have different speeds), a two-machine flow shop or a two-machine open shop. β may include pmtn, prec, intree, outtree, chains, rj, pj=m, pj=p, pj=1 or pij=1 which denote preemptions allowed, arbitrary or intree-, outtree- or chain-like precedence constraints, arbitrary release dates, execution times equal m, equal execution times, unit execution times on parallel machines or in shops. γ=Cmax, ΣCj, CmaxΣCj, Lmax, ΣUj, ΣwjCj, where Cj is completion time of Jj, denotes maximum completion time, total completion time, the
Pareto criterion with components Cmax and ΣCj, maximum lateness, maxj Lj, where Lj=Cj-dj, and number of late jobs for given due dates dj of jobs, where Uj=1 if Lj>0 or Uj=0 otherwise, and total weighted completion time for given weights wj of jobs. All numerical parameters are assumed to be integers.
2.
History
Earliest results on two-machine scheduling of jobs with arbitrary precedence constraints and unit-execution-time operations were devoted to finding polynomial-time algorithms for
P2|prec,pj=1|Cmax [7] and P2|pmtn,prec,pj =1| Cmax [16]. Extensions and new polynomial-time algorithms were obtained later for
P2|prec,rj,pj =1|Lmax [12],
F2|prec,rj,pij=1|Lmax [6],
Q2|pmtn,prec,rj|Lmax [13],
O2|prec,rj, pij=1|Lmax [5] and O2|nowait,prec, pij=1|Lmax [20]. Note that P2|prec,rj,pj=2|Lmax and O2|nowait,prec,rj,pij=1|Lmax are isomorphic [5] and remain open. Related problems with criteria more general than Lmax are already NP-hard because even 1|chains,pj=1|ΣUj [15], F2|chains,pij=1|ΣUj [2], P2|pmtn,rj|ΣUj [8],
O2|chains,pij=1|ΣUj [20] and O2|nowait,chains,pij=1|ΣUj [20] are NP-hard. For the criterion ΣCj it has been observed that the Coffman-Graham (CG) algorithm, originally designed for P2|prec,pj=1|Cmax [7], solves
P2|prec,pj=1|ΣCj [11],
F2|prec,pij=1|ΣCj [2] and O2|nowait,prec,pij=1|ΣCj [20]
in polynomial time as well. Note that the problem on parallel machines and the no-wait open-shop problem here are isomorphic [5]. The result for the flow shop was inspired by the proof [17, 10] that the CG algorithm also solves the bump number problem which is isomorphic to F2|prec,pij=1|Cmax. Although the observation that the CG algorithm solves P2|prec,pj=1|ΣCj has been well known for a long time [14], we are not aware of any published proof.
3.
Results
In this paper we give a proof that the CG algorithm solves P2|prec,pj=1|ΣCj and consider an extension of the CG algorithm that solves in polynomial time P2|pmtn,prec,pj=1|CmaxΣCj. Note that the Muntz-Coffman (MC) algorithm, originally designed for P2|pmtn,prec,pj=1|Cmax [16], does not find ideal schedules, i.e., there are instances of P2|pmtn,prec,pj=1|ΣCj for which the MC algorithm does not find an optimal schedule. We also present extensions of the CG algorithm that solve in polynomial time
P2|prec,rj,pj=1|CmaxΣCj, F2|prec,rj,pij=1|CmaxΣCj, P2|pmtn,prec,rj,pj=1|CmaxΣCj, O2|prec,rj,pij=1|CmaxΣCj and O2|nowait,prec,rj,pij=1|CmaxΣCj,
problems whose complexity status has been unknown up to now. As we show, the flow-shop problem and the open-shop problem can be solved in polynomial time by slight elaborations of the nonpreemptive and preemptive algorithms for identical parallel machines. Note that the preemptive problem here cannot be solved by a nonpreemptive algorithm because preemptions in P2|pmtn,pj=1|Cmax and P2|pmtn,intree,pj=1|ΣCj can be advantageous [1]. On the other hand, P|pmtn,outtree,rj,pj=1|CmaxΣCj and P|outtree,rj,pj=1|CmaxΣCj have a common solution that can be found in polynomial time by the Brucker-Hurink-Knust algorithm [3]. Two-machine problems with criteria more general than ΣCj are NP-hard because
P2|chains,pj=1|ΣwjCj [20], P2|pmtn,chains,pj=1|ΣwjCj [20], F2|chains,pij=1|ΣwjCj [19], O2|chains,pij=1|ΣwjCj [20] and O2|nowait,chains,pij=1|ΣwjCj [20] are NP-hard. Note that preemptions in the preemptive problem here are redundant because they are redundant even in P|pmtn,chains|ΣwjCj [9].
Conclusions and Open Problems 1.
The results of this paper close the list of problems with the unknown complexity status in the
α|β|γ classification where
α belongs to {P2, F2, O2} and β includes pj=1 or pij=1
2.
3. 4.
5.
6.
7.
8.
with the exception of O2|nowait,prec,rj,pij=1|Lmax which is isomorphic to P2|prec,rj,pj=2|Lmax. The problem remains open even for chain-like precedence constraints. The many-machine problems without precedence constraints, O|nowait,rj,pij=1|Lmax and P|rj,pj=m|Lmax, are equivalent [5] and can be solved in polynomial time by the Simons algorithm [18] for P|rj,pj=p|Lmax. The case α=Q2 is not well studied. To the result we mentioned at the beginning of Section 2 we can only add that Q2|chains,pj=1|Cmax can be solved in polynomial time [4]. This paper shows that P2|pmtn,prec,rj,pj=1|CmaxΣCj has a solution of fractionality ½. Hence, the recognition version of the problem is obviously in NP. However, it is unknown whether the recognition version of P3|pmtn,prec,pj=1|Cmax, P3|pmtn,prec,pj=1|ΣCj or their extensions belongs to NP. The fractionality conjecture [16] that P|pmtn,prec|Cmax has a solution of fractionality 1/m remains unverified even for jobs with unit-execution-time operations. In application to any preemptive problem in the α|β|γ classification we refer to it as the extended fractionality conjecture. We also introduce the NP-preemption hypothesis that the recognition version of any preemptive problem in the α|β|γ classification belongs to NP. It is obviously true if the extended fractionality conjecture is true. The following fact gives an idea to what extent the extended fractionality conjecture is stronger than the NP-preemption hypothesis. Let the recognition version of a preemptive problem A in the α|β|γ classification belong to NP, and let q be a polynomial upper bounding the size of A. Then A has a solution of fractionality 1/q! [21]. However, it is unknown whether the extended fractionality conjecture is true for all preemptive problems whose recognition versions belong to NP. If the NP-preemption hypothesis is true, then any preemptive problem on identical parallel machines in the α|β|γ classification polynomially reduces to a nonpreemptive problem with unit-execution-time operations [21, 2, 20].
References [1] Baptiste, Ph. and V. G. Timkovsky, V. G. (2001). On Preemption Redundancy in Scheduling Unit Processing Time Jobs on Two Parallel Machines, Operation Research Letters, 28, 205212. [2] Brucker, P. and S. Knust, S. (1999). Complexity Results for Single-Machine Problems with Positive Finish-Start Time-Lags, Computing, 63, 299-316 [3] Brucker, P., Hurink, J. and Knust, S. (2001). A Polynomial Algorithm for P|pj=1,rj,outtree|ΣCj, Technical Report, University of Osnabrueck. [4] Brucker, P., Hurink, J. and Kubiak, W. (1999). Scheduling Identical Jobs with Chain Precedence Constraints on Two Uniform Machines, Mathematical Methods of Operations Research, 49, 211-219. [5] Brucker, P., Jurisch, B. and Jurisch, M. (1993). Open Shop Problems with Unit Time Operations, ZOR - Methods and Models of Operations Research, 37, 59-73. [6] Bruno, J., Jones, J. W. and So, K. (1980). Deterministic Scheduling with Pipelined Processors, IEEE Transactions on Computers, C-29, 308-316. [7] Coffman, E. G., Jr. and Graham, R. L. (1972). Optimal Scheduling for Two-Processor System, Acta Informatica, 1, 200-213. [8] Du, J., Leung, J. Y.-T. and Wong, C. S. (1992). Minimizing the Number of Late Jobs with Release Time Constraints, Journal of Combinatorial Mathematics and Combinatorial Computing, 11, 97-107. [9] Du, J., Leung, J. Y.-T. and Young, G. H. (1991). Scheduling Chain-Structured Tasks to Minimize Makespan and Mean Flow Time, Information and Computation, 92, 219-236. [10] Finta, L. and Liu, Z. (1996). Single Machine Scheduling Subject to Precedence Delays, Discrete Applied Mathematics, 70, 247-266. [11] Garey, M. R. Unpublished. [12] Garey, M. R. and Johnson, D. S. (1977). Two-Processor Scheduling with Start-Times and Deadlines, SIAM Journal on Computing, 6, 416-426. [13] Lawler, E. L. (1982). Preemptive Scheduling of Precedence-Constrained Jobs on Parallel Machines, in: Deterministic and Stochastic Scheduling, M.A.H. Dempster, J.K. Lenstra and A.H.G. Rinnooy Kan, eds., Reidel, Dordrecht, 101-123. [14] Lawler, E. L. et al. (1993). Sequencing and Scheduling: Algorithms and Complexity, in: Handbook on Operations Research and Management Science, vol. 4, Logistics of Production and Inventory, S.C. Graves, A.H.G. Rinnooy Kan, and P. Zipkin, eds., Elsevier, 445-552. [15] Lenstra, J. K. and Rinnooy Kan, A.H.G. (1980). Complexity Results for Scheduling Chains on a Single Machine, European Journal of Operational Research, 4, 270-275. [16] Muntz, R. R. and Coffman, E. G., Jr. (1969). Optimal Preemptive Scheduling on TwoProcessor Systems, IEEE Transactions on Computers, C-18, 1014-1020. [17] Schaeffer, A. A. and B. Simons, B. (1988). Computing the Bump Number with Techniques from Two-Processor Scheduling, Order, 5, 131-141. [18] Simons, B. (1983). Multiprocessor Scheduling of Unit-Time Jobs with Arbitrary Release Times, SIAM Journal on Computing, 12, 294-299. [19] Tanaev, V. S., Sotskov, Yu. N. and Strusewich, V. A. (1994). Scheduling Theory: MultiStage Systems, Kluwer. [20] Timkovsky, V. G. (1998). Identical Parallel Machines vs. Unit-Time Shops and Preemptions vs. Chains in Scheduling Complexity, Technical Report, Department of Computing and Software, McMaster University, Ontario. [21] Timkovsky, V. G. (1998). Is a Unit-Time Job Shop Not Easier Than Identical Parallel Machines?, Discrete Applied Mathematics, 85, 149-162.
