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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 1
Institut 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
c 2000 Elsevier Science B.V. All rights reserved.
simple 2-approximation algorithm for this problem.
Keywords: Approximation algorithm; Worst-case analysis; Scheduling; Uniform machines; Precedence constraints; Makespan

1. Introduction
Consider m uniformly related parallel machines
M1 ; : : : ; Mm where machine Mi (16i6m) has a speed
factor si ¿1. There are n jobs J1 ; : : : ; Jn where job Jj

has a processing requirement of pj ; the processing
of job Jj on machine Mi takes pj =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 di1

Supported by the START program Y43-MAT of the Austrian
Ministry of Science.
E-mail address: gwoegi@opt.math.tu-graz.ac.at (G.J. Woeginger)

rect predecessor and at most one direct successor) is
denoted by Q | chain | Cmax .
A polynomial-time approximation algorithm is an
algorithm that returns near-optimal solutions in polynomial time. If the cost of these near-optimal solutions is always at most a factor of  above the optimal
cost, then the algorithm is called a -approximation

algorithm, and the value  is called a worst-case performance guarantee. In 1997, Chudak and Shmoys
[2] developed a polynomial-time approximation algorithm for Q | prec | Cmax with worst-case performance
guarantee O(log m). In 1998, Chekuri and Bender
[1] gave another polynomial-time approximation
algorithm with the same order of worst-case performance. Chekuri and Bender [1] also proved that for
the special case Q | 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

c 2000 Elsevier Science B.V. All rights reserved.
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PII: S 0 1 6 7 - 6 3 7 7 ( 9 9 ) 0 0 0 7 6 - 0

108

G.J. Woeginger / Operations Research Letters 26 (2000) 107–109

Q | chain | Cmax . This approximation algorithm is described in Section 2. Section 3 presents a tool that is
used in the analysis of the approximation algorithm.

its corresponding chain C to get a feasible schedule
pmtn
chain for I chain with makespan at most 2Cmax
. With
(1), this yields the following theorem.

2. The main result

Theorem 2.1. There is a polynomial-time 2-approximation algorithm for Q | chain | Cmax .

Throughout this section, we will work with instances of three closely related scheduling problems.
These instances are denoted by I chain ; I non , and I pmtn ,
and they are specied as follows.

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 for Q | rj ; chain | Cmax .

• Instance I chain is an instance of problem
Q | chain | Cmax

3. An auxiliary result

as dened above.
• Instance I non is an instance of problem Q k Cmax ,
the variant without any precedence constraints. The
machines in I non are exactly the same as in I chain
and also have the same speeds, but the jobs in I non
are dierent from the jobs in I chain . For every chain
C of jobs in I chain , there is a corresponding single
new job J (C) in I non . The processing requirement of
J (C) equals the sum of all processing requirements
of the jobs in C.
• Finally, instance I pmtn is an instance of problem
Q | pmtn | Cmax , the variant with preemption and
without precedence constraints. The machines in
I pmtn are the same as in I chain and I non , and its jobs

are the same as in I non . Hence, instance I pmtn may
be considered as the preemptive relaxation of instance I non .
pmtn
chain
non
, Cmax
, and Cmax
, we denote the optimal obBy Cmax
jective value of instance I chain , I non , and I pmtn , respectively. Observe that every feasible solution for I non can
be transformed into a feasible solution for I chain with
the same makespan: Simply replace every job J (C) by
its corresponding chain C. Moreover, every feasible
solution for I chain can be transformed into a feasible
solution for I pmtn with the same makespan: Simply replace every chain C by appropriate preempted pieces
of the corresponding job J (C). This yields
pmtn
chain
non
6Cmax
6Cmax

:
Cmax

(1)
pmtn
non
62Cmax
Cmax
non

and
In Section 3, we will show that
with
that for I non a non-preemptive schedule 
pmtn
can be computed in polymakespan at most 2Cmax
nomial time. Every job J (C) in non is replaced by

In this section we show that for any instance of
Q k Cmax with m machines, the optimal non-preemptive makespan is at most a factor of 2 − 1=m above the

optimal preemptive makespan. Although the scheduling 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 machines), the optimal non-preemptive makespan is at
most a factor of 2 above the objective value of a certain 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 parallel machines M1 ; : : : ; Mm . The largest processing
time (LPT) algorithm assigns 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
pmtn
,
schedule produced by LPT is at most (2 − 1=m)Cmax
pmtn
where Cmax is the optimal preemptive makespan.
Suppose otherwise. Consider a counter example with
the minimum number of machines, and among all

G.J. Woeginger / Operations Research Letters 26 (2000) 107–109

counter examples with the minimum number of machines consider one with the minimum number of
jobs. Then n¿m holds (otherwise, neither the preemptive 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 rst n − 1 jobs with completion
pmtn
, and only the assignment of
times 6(2 − 1=m)Cmax

pmtn
Jn brings the LPT makespan above (2 − 1=m)Cmax
(otherwise, we could remove job Jn and produce a
counterexample with a smaller number of jobs).
For 16i6m, let Li denote the total processing time
assigned to machine Mi by LPT at the moment
Pm just
before the last job Jn is assigned. By L = pn + i=1 Li
we denote the total processing requirement of all jobs.
Since the assignment of Jn brings the LPT makespan
pmtn
above (2 − 1=m)Cmax
, we get for all 16i6m the
inequality
pmtn
¡
(2 − 1=m)Cmax

1
(Li + pn ):

si

(2)

Multiplying (2) by si and addingP
up these inequalities
Pm
m
pmtn
over all i yields (2 − 1=m)Cmax
i=1 si ¡
i=1 Li +
mpn , which is equivalent to
pmtn
(2 − 1=m)Cmax

m
X

si ¡ L + (m − 1)pn :

(3)

i=1

pmtn Pm
It is straightforward to see that L6Cmax
i=1 si .
Plugging this inequality into the left-hand side of (3)
yields (1 − 1=m)L ¡ (m − 1)pn and hence, L ¡ mpn .
On the other hand, there are n¿m jobs with processing 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
non
is at most a factor of 2 − 1=m above the
Cmax
pmtn
. Moreover; a
optimal preemptive makespan Cmax
non-preemptive schedule with makespan at most
pmtn
can be computed in polynomial time.
(2 − 1=m)Cmax

109

The factor 2 − 1=m in the statement of Theorem 3.1
cannot be improved: Consider m machines with speeds
s1 = s2 = · · · = sm−1 = 1 and sm = 12 , and n = m jobs
with processing requirements pj ≡ 1 − 1=2m. Then
pmtn
non
Cmax
= 1 and Cmax
= 2 − 1=m. Finally, we remark that
for the case of m identical machines (i.e., for the case
where si ≡ 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 times pj ≡ 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.
 Tardos, Approximation
[5] J.K. Lenstra, D.B. Shmoys, E.
algorithms for scheduling unrelated parallel machines,
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[6] D.B. Shmoys, J. Wein, D.P. Williamson, Scheduling parallel
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