Operations Research Letters 26 (2000) 107–109
www.elsevier.com/locate/orms
A comment on scheduling on uniform machines under chain-type
precedence constraints
Gerhard J. Woeginger
1Institut fur Mathematik B, TU Graz, Steyrergasse 30, A-8010 Graz, Austria
Received 1 August 1999; received in revised form 1 October 1999
Abstract
In a recent paper, Chekuri and Bender (Proceedings of IPCO’98, pp. 383–393) derive (among other results) a polynomial-time 6-approximation algorithm for makespan minimization on uniform parallel machines subject to chain-type precedence constraints. In this short note, we combine some straightforward observations and thereby derive an extremely simple 2-approximation algorithm for this problem. c 2000 Elsevier Science B.V. All rights reserved.
Keywords:Approximation algorithm; Worst-case analysis; Scheduling; Uniform machines; Precedence constraints; Makespan
1. Introduction
Consider m uniformly related parallel machines
M1; : : : ; Mmwhere machineMi(16i6m) has a speed
factorsi¿1. There arenjobsJ1; : : : ; Jn where jobJj
has a processing requirement of pj; the processing
of jobJj on machineMi takespj=si time units. The
processing of the jobs is constrained by a precedence relation. Preemption of the jobs is not allowed. The goal is to nd a feasible schedule that minimizes the makespan, i.e., the maximum job completion time. In the standard three-eld scheduling notation (see e.g. [4]), this problem is denoted by Q|prec|Cmax.
The special case where the precedence constraints are
chain-type(i.e., where every job has at most one
di-1Supported by the START program Y43-MAT of the Austrian Ministry of Science.
E-mail address: gwoegi@opt.math.tu-graz.ac.at (G.J. Woe-ginger)
rect predecessor and at most one direct successor) is denoted byQ|chain|Cmax.
Apolynomial-time approximationalgorithm is an algorithm that returns near-optimal solutions in poly-nomial time. If the cost of these near-optimal solu-tions is always at most a factor ofabove the optimal cost, then the algorithm is called a-approximation
algorithm, and the valueis called aworst-case per-formance guarantee. In 1997, Chudak and Shmoys [2] developed a polynomial-time approximation algo-rithm forQ|prec|Cmaxwith worst-case performance
guarantee O(logm). In 1998, Chekuri and Bender [1] gave another polynomial-time approximation algorithm with the same order of worst-case perfor-mance. Chekuri and Bender [1] also proved that for the special caseQ|chain|Cmax, their algorithm is a
6-approximation algorithm.
Contribution of this note: In this short note, we present a very simple 2-approximation algorithm for
108 G.J. Woeginger / Operations Research Letters 26 (2000) 107–109 Q|chain|Cmax. This approximation algorithm is
de-scribed in Section 2. Section 3 presents a tool that is used in the analysis of the approximation algorithm.
2. The main result
Throughout this section, we will work with in-stances of three closely related scheduling problems. These instances are denoted byIchain; Inon, andIpmtn,
and they are specied as follows.
• InstanceIchainis an instance of problem Q|chain|Cmax
as dened above.
• InstanceInonis an instance of problemQ k C
max,
the variantwithoutany precedence constraints. The machines in Inon are exactly the same as in Ichain
and also have the same speeds, but the jobs inInon
are dierent from the jobs inIchain. For every chain
Cof jobs inIchain, there is a corresponding single
new jobJ(C) inInon. The processing requirement of J(C) equals the sum of all processing requirements
of the jobs inC.
• Finally, instance Ipmtn is an instance of problem Q|pmtn|Cmax, the variant with preemption and
without precedence constraints. The machines in
Ipmtnare the same as inIchainandInon, and its jobs are the same as inInon. Hence, instanceIpmtnmay be considered as the preemptive relaxation of in-stanceInon.
ByCchain
max ,Cmaxnon, andC
pmtn
max , we denote the optimal
ob-jective value of instanceIchain,Inon, andIpmtn,
respec-tively. Observe that every feasible solution forInoncan
be transformed into a feasible solution forIchainwith the same makespan: Simply replace every jobJ(C) by
its corresponding chain C. Moreover, every feasible
solution forIchain can be transformed into a feasible
solution forIpmtnwith the same makespan: Simply
re-place every chainCby appropriate preempted pieces
of the corresponding jobJ(C). This yields
Cmaxpmtn6Cmaxchain6Cmaxnon: (1) In Section 3, we will show that Cnon
max62C
pmtn
max and
that for Inon a non-preemptive schedule non with
makespan at most 2Cmaxpmtn can be computed in
poly-nomial time. Every jobJ(C) in non is replaced by
its corresponding chainC to get a feasible schedule
chain forIchain with makespan at most 2Cpmtn
max . With
(1), this yields the following theorem.
Theorem 2.1. There is a polynomial-time2- approxi-mation algorithm forQ|chain|Cmax.
A general result of Shmoys et al. [6] states that this type of result carries over to the case of release dates while losing only a factor of 2 in the performance guarantee. Hence, there exists a polynomial-time 4-approximation algorithm forQ|rj; chain|Cmax.
3. An auxiliary result
In this section we show that for any instance of
Q k Cmaxwithmmachines, the optimal
non-preemp-tive makespan is at most a factor of 2−1=mabove the optimal preemptive makespan. Although the schedul-ing literature contains many results on makespan minimization on uniformly related machines, we were not able to nd this statement in the literature and we were not able to deduce it from known results. For example, Gonzalez et al. [3] prove that the largest processing time heuristic has worst-case performance guarantee 2−2=(m+ 1) for Q k Cmax. In another
paper in this area, Lenstra et al. [5] show that for
R k Cmax (makespan minimization on unrelated
ma-chines), the optimal non-preemptive makespan is at most a factor of 2 above the objective value of a cer-tain linear programming relaxation. Both statements are fairly close to our statement. However, the proofs in [3,5] do not compare the optimal non-preemptive makespan against the optimal preemptive makespan, but against stronger lower bounds.
Now consider an instance of Q k Cmax with n
jobs J1; : : : ; Jn and with m uniformly related
par-allel machines M1; : : : ; Mm. The largest processing
time(LPT) algorithmassigns the jobs to machines in order of non-increasing processing requirements
p1¿· · ·¿pn. Every job is assigned to the machine
on which its completion time will be earliest. Ties are broken by assigning the job to the machine with smallest index. We claim that the makespan of the schedule produced by LPT is at most (2−1=m)Cmaxpmtn,
where Cmaxpmtn is the optimal preemptive makespan.
G.J. Woeginger / Operations Research Letters 26 (2000) 107–109 109
counter examples with the minimum number of ma-chines consider one with the minimum number of jobs. Then n¿m holds (otherwise, neither the pre-emptive optimum nor the LPT algorithm will use all machines, and removing the slowest machine would yield a smaller counter example). Moreover, LPT will assign the rstn−1 jobs with completion times6(2−1=m)Cmaxpmtn, and only the assignment of
Jn brings the LPT makespan above (2−1=m)Cmaxpmtn
(otherwise, we could remove job Jn and produce a counterexample with a smaller number of jobs).
For 16i6m, letLidenote the total processing time
assigned to machine Mi by LPT at the moment just
before the last jobJnis assigned. ByL=pn+Pmi=1Li
we denote the total processing requirement of all jobs. Since the assignment ofJn brings the LPT makespan
above (2−1=m)Cmaxpmtn, we get for all 16i6m the
inequality
(2−1=m)Cmaxpmtn¡
1
si(Li+pn): (2)
Multiplying (2) bysiand adding up these inequalities over alliyields (2−1=m)CmaxpmtnPmi=1si¡
Pm i=1Li+
mpn, which is equivalent to
(2−1=m)Cmaxpmtn m X
i=1
si¡ L+ (m−1)pn: (3)
It is straightforward to see that L6CmaxpmtnPmi=1si.
Plugging this inequality into the left-hand side of (3) yields (1−1=m)L ¡(m−1)pnand hence,L ¡ mpn.
On the other hand, there aren¿mjobs with process-ing times ¿pn, and therefore L¿mpn must hold.
This contradiction completes the argument.
Theorem 3.1. For any instance of Q k Cmax with
m machines; the optimal non-preemptive makespan Cnon
max is at most a factor of 2 −1=m above the
optimal preemptive makespan Cmaxpmtn. Moreover; a
non-preemptive schedule with makespan at most
(2−1=m)Cmaxpmtncan be computed in polynomial time.
The factor 2−1=min the statement of Theorem 3.1 cannot be improved: Considermmachines with speeds
s1=s2=· · ·=sm−1= 1 andsm=
1
2, andn=mjobs
with processing requirementspj ≡ 1−1=2m. Then
Cmaxpmtn= 1 andCmaxnon= 2−1=m. Finally, we remark that
for the case ofmidenticalmachines (i.e., for the case wheresi≡1) an improvement of the statement indeed is possible: In this case, the optimal non-preemptive makespan is at most a factor of 2−2=(m+ 1) above the optimal preemptive makespan. The instance with
n=m+1 jobs with processing timespj ≡1−1=(m+1) illustrates that the bound 2−2=(m+ 1) is best possible for identical machines.
References
[1] C. Chekuri, M.A. Bender, An ecient approximation algorithm for minimizing makespan on uniformly related machines, Proceedings of the 6th Conference on Integer Programming and Combinatorial Optimization (IPCO’98), Springer Lecture Notes in Computer Science, Vol. 1412, Springer, Berlin, 1988, pp. 383–393.
[2] F.A. Chudak, D.B. Shmoys, Approximation algorithms for precedence-constrained scheduling problems on parallel machines that run at dierent speeds, Proceedings of the eighth Annual ACM-SIAM Symposium on Discrete Algorithms (SODA’97), pp. 581–590. Journal version in J. Algorithms 30 (1999) 323–343.
[3] T. Gonzalez, O.H. Ibarra, S. Sahni, Bounds for LPT schedules on uniform processors, SIAM J. Comput. 6 (1977) 155–166. [4] E.L. Lawler, J.K. Lenstra, A.H.G. Rinnooy Kan, D.B. Shmoys, Sequencing and scheduling: Algorithms and complexity, in: S.C. Graves, A.H.G. Rinnooy Kan, P.H. Zipkin (Eds.), Logistics of Production and Inventory, Handbooks in Operations Research and Management Science, Vol. 4, North-Holland, Amsterdam, 1993, pp. 445–522. [5] J.K. Lenstra, D.B. Shmoys, E. Tardos, Approximation
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