On the Performance of Smith’s Rule in
Single-Machine Scheduling with Nonlinear Cost
Wiebke H¨ohn
1,
and Tobias Jacobs
2,
1
Technische Universit¨at Berlin, Germany
hoehn@math.tu-berlin.de
2
NEC Laboratories Europe, Heidelberg, Germany
tobias.jacobs@neclab.eu
Abstract. We consider the problem of scheduling jobs on a single ma-
chine. Given some continuous cost function, we aim to compute a sched-
ule minimizing the weighted total cost, where the cost of each individual
job is determined by the cost function value at the job’s completion
time. This problem is closely related to scheduling a single machine with
nonuniform processing speed. We show that for piecewise linear cost
functions it is strongly NP-hard.
The main contribution of this article is a tight analysis of the approx-
imation factor of Smith’s rule under any particular convex or concave
cost function. More specifically, for these wide classes of cost functions
we reduce the task of determining a worst case problem instance to a
continuous optimization problem, which can be solved by standard alge-
braic or numerical methods. For polynomial cost functions with positive
coefficients it turns out that the tight approximation ratio can be cal-
culated as the root of a univariate polynomial. To overcome unrealistic
worst case instances, we also give tight bounds that are parameterized
by the minimum, maximum, and total processing time.
1 Introduction
We address the problem of scheduling jobs on a single machine so as to mini-
mize the weighted sum of completion costs. The input consists of a set of jobs
j =1,...,n,whereeachjobj has an individual weight w
j
≥ 0 and process-
ing time p
j
≥ 0, and the goal is to find a one-machine schedule minimizing
n
j=1
w
j
f(C
j
), where C
j
denotes the completion time of job j in the schedule.
The only assumption we make about the cost function f : R → R at this point
is that it is continuous and monotone. In the classic three-field notation [6], the
problem we consider reads as 1 ||
w
j
f(C
j
) . Note that the question of allowing
preemption does not play a role here, because the jobs do not have release times
and so the possibility of preemption never leads to a cheaper optimal schedule.
Supported by the German Research Foundation (DFG) as part of the priority pro-
gramme “SPP 1307: Algorithm Engineering”.
Work supported by a fellowship within the Postdoc-Programme of the German Aca-
demic Exchange Service (DAAD).
D. Fern´andez-Baca (Ed.): LATIN 2012, LNCS 7256, pp. 482–493, 2012.
c
Springer-Verlag Berlin Heidelberg 2012
On the Performance of Smith’s Rule 483
An important alternative interpretation of problem 1 ||
w
j
f(C
j
) is the sce-
nario of linear cost and nonuniform processor speed. Assume that the processor
speed at any time t is given by a nonnegative function g : R → R,andthe
processing times (or workloads) p
j
of the jobs are given with respect to a unit
speed processor. The total workload processed until time t is G(t):=
t
0
g(t)dt.
Conversely, if the total workload of job j and all jobs processed before it is t
,
then the cost of j in the schedule is G
−1
(t
). Therefore, the problem is equivalent
to 1 ||
w
j
G
−1
(C
j
) . Note that G
−1
is always monotone, and it is continuous
even if g is not. Moreover if g is increasing or decreasing then G
−1
is convex and
concave, respectively. The case of cost function f and processor speed function g
is equivalent to problem 1 ||
w
j
f(G
−1
(C
j
)) .
Related Work. The problem 1 ||
w
j
f(C
j
) with nonlinear cost function f has
been studied for half a century. For quadratic cost functions there is a long series
of articles on branch-and-bound schemes; see e.g., [12,9]. In a companion paper
we combine and improve the methods of these articles, and compare them in an
extensive computational study [7]. Further references can be found therein.
The problem of minimizing the total weighted flowtime on one or multiple
machines with or without preemption is a well studied problem, and efficient
approximation schemes are known for many variants [1,3]. In [2], Bansal and
Pruhs motivate the usage of monomial cost functions in the context of processor
scheduling, where jobs have nonuniform release dates. They show that even in
the case of uniform weights there is no n
o(1)
-competitive online algorithm, and
they analyze a number of scheduling strategies using resource augmentation.
A more general problem version, where each job has its individual cost func-
tion, has recently attracted attention. Bansal and Pruhs have given a geometric
interpretation that yields a O(log log nP )-approximation in the presence of re-
lease dates and preemption. In the special case of uniform release dates, their
method achieves the constant factor of 16. That factor has recently been im-
proved to 2 + via a primal-dual approach by Cheung and Shmoys [4].
For 1 ||
w
j
f(C
j
) with arbitrary concave f , Stiller and Wiese [11] show that
Smith’s rule (see below for a definition) guarantees an approximation factor of
(
√
3+1)/2. The result is tight in the sense that for a certain cost function f there
is a problem instance where this factor is reached by Smith’s rule. Epstein et al.
provide an approximation algorithm for the problem variant with release dates
by generalizing their results on scheduling unreliable machines [5]. Their method
generates a schedule which has approximation guarantee 4 + ε for any cost
function. Both the algorithm by Epstein et al. as well as Smith’s rule analyzed
by Stiller and Wiese yield schedules that are universal in the sense of being
generated without knowledge of the cost function.
Our Contribution. The computational complexity of problem 1 ||
w
i
f(C
i
)
is a long standing open question [9,11]. In Section 4 we give a first result in
that direction by showing that for piecewise linear and monotone cost functions
the problem is NP-hard in the strong sense. The instances we reduce to can be
interpreted as a processor that alternates between two different speeds. Such
484 W. H¨ohn and T. Jacobs
Table 1. The first table shows the tight approximation factor of Smith’s rule for
various cost functions. The factors for polynomials hold under the assumption of non-
negative coefficients. In the second table, examples of the parameterized analysis are
shown.
cost function approx. factor
square root 1.07
degree 2 polynomials 1.31
degree 3 polynomials 1.76
degree 4 polynomials 2.31
degree 5 polynomials 2.93
degree 6 polynomials 3.60
degree 10 polynomials 6.58
degree 20 polynomials 15.04
exponential ∞
cost fct. p
min
p
max
P approx. factor
x
2
1 20 500 1.028
x
2
1 20 1000 1.014
x
2
1 20 5000 1.003
x
2
1 100 500 1.136
x
2
1 100 1000 1.071
x
2
1 100 5000 1.015
x
3
1 100 1000 1.149
x
5
1 100 1000 1.296
x
10
1 100 1000 1.630
scenarios are likely to occur in practice, e.g., when some extra computational
power becomes available at nighttime.
Our main result is a tight analysis of the approximation factor of Smith’s
rule [10] also known as WSPT (Weighted-Shortest-Processing-Time-First). This
well known strategy first computes the WSPT ratio q
j
:= w
j
/p
j
for each job
and then sorts the jobs by descending q
j
, which is optimal in the linear cost
case. In Section 2, we show that for all convex and all concave cost functions
tight bounds for the approximation factor can be obtained as the solution of
a continuous optimization problem with at most two degrees of freedom. In
the case of cost functions that are polynomials with positive coefficients, it will
turn out that the approximation factor can be calculated simply by determining
the root of a univariate polynomial. An overview of approximation factors with
respect to a number of cost functions are depicted in Table 1, showing that
WSPT achieves the best known approximation factor for cost functions that
are polynomials of degree up to three. Regarding universal scheduling methods,
WSPT provides the best known approximation factor for up to degree six.
The worst case approximation factors are established by extreme instances
that consist of one large job and an infinite number of infinitesimally small jobs.
In order to analyze the performance of WSPT for realistic instances, we intro-
duce three parameters that restrict the problem instances under consideration.
These parameters are the minimum, maximum, and total job length p
min
,p
max
and P , respectively. In Section 3 we show how to obtain tight bounds for the
approximation ratio of Smith’s rule under any parameter configuration. Some
examples of this analysis are given in Table 1.
2 Tight Analysis of Smith’s Rule
In this section we analyze the worst case approximation factor obtained by
Smith’s rule in the case of any convex or concave cost function. The following
simple observation will be used a number of times.
On the Performance of Smith’s Rule 485
Observation 1. Problem 1 ||
w
j
f(C
j
) is invariant to weight scaling, i.e.,
if I is a problem instance and I
is obtained from I by multiplying all job weights
with a constant c, then the cost of any schedule for I
is c times the cost of the
same schedule for I.
We denote by WSPT(I) the schedule computed for instance I by Smith’s rule,
and by OPT(I) an optimal schedule for I. Slightly abusing notation, the cost of
these schedules will also be denoted by WSPT(I)andOPT(I).
Theorem 1. Let f be a convex cost function. Then the tight approximation ratio
of Smith’s rule can be calculated as
sup
WSPT(I)
OPT(I)
=max
q
0
f(t)dt + p · f(q + p)
p ·f (p)+
p+q
p
f(t)dt
| p ≥ 0,q ≥ 0
. (1)
When f is concave, the tight ratio is
sup
WSPT(I)
OPT(I)
=max
p ·f (p)+
p+q
p
f(t)dt
q
0
f(t)dt + p · f(q + p)
| p ≥ 0,q ≥ 0
. (2)
These equalities hold regardless of the tie breaking strategy used by Smith’s rule.
In what follows, we prove a number of lemmas which successively narrow the
space of instances we need to consider when searching for a worst case problem
instance for Smith’s rule. Determining the worst case solution in the final in-
stance space will then be shown to be equivalent to the continuous optimization
problem described in the above theorem.
Very similar to the analysis of Stiller and Wiese [11], we first show that it
is sufficient to consider instances with constant WSPT ratio, and that a most
expensive schedule is obtained by inverting the optimal job order. Thereafter,
again as Stiller and Wiese, we restrict to instances with several small jobs and
one large job. However, their proof of this property is based on a modification of
the cost function which makes it invalid for our problem setting. The remainder
of our proof follows a completely different line of argumentation.
The following observation can be shown by continuity considerations, see the
full version of this paper for a formal proof or [8] for an explanation of the general
principle behind this argumentation.
Observation 2. If the cost function f is continuous, then the approximation
factor sup{WSPT(I)/OPT(I)} is independent of the tie breaking policy em-
ployed by WSPT.
As a consequence, we can assume that WSPT always breaks ties in the worst
possible way. In terms of an adversary model, we can assume that the adversary
not only chooses the problem instance, but also the way WSPT breaks ties.
Thenextlemmashowsthatwecanrestrict our attention to problem instances
where Smith’s ratio is the same for all jobs. By Observation 2, we can further
486 W. H¨ohn and T. Jacobs
f
w
j
· f (C
j
)
line job
p
j
= w
j
C
j
time
cost
(a) A job j’s cost is represented by a rect-
angle. Line jobs are a collection of in-
finitesimally small jobs. Their total cost
is given by the area under the graph of f.
f
i
Δ
INC
L
j
Δ
INC
M
C
INC
j
C
INC
i
time
f(C
INC
i
)
f(C
INC
j
)
p
j
cost
(b) Jobs i and j in schedule INC(I)from
Lemma 3. The marked areas represent
the change in cost when merging the
jobs or when making job j alinejob.
Fig. 1. Geometric interpretation of a schedule for instances with w
j
= p
j
for all jobs j
assume that WSPT schedules the jobs in the worst possible order while OPT
schedules them in the best possible order. When the cost function is convex or
concave, there is a very simple characterization of these special orders, as shown
in Lemma 2. Due to the analogy to Stiller and Wiese [11], the proofs of these
two lemmas are omitted in this extended abstract.
Lemma 1. For the worst case ratio of Smith’s rule we can assume that the
WSPT ratio q
j
is 1 for all jobs j. More formally,
sup
WSPT(I)
OPT(I)
=sup
WSPT(I)
OPT(I)
| w
j
= p
j
for each job j ∈ I
.
Lemma 2. If the cost function f is convex, then
sup
WSPT(I)
OPT(I)
=sup
INC(I)
DEC(I)
| w
j
= p
j
for each job j ∈ I
where INC(I) and DEC(I) denotes the cost of the schedule where the jobs in I are
processed in order of their increasing and decreasing processing time, respectively.
If f is concave, then sup{WSPT(I)/OPT(I)} is obtained analogously with the
reciprocal of INC(I)/DEC(I).
At this point we introduce a geometric interpretation of our scheduling problem.
In this interpretation each job j is represented by a rectangle having width w
j
and height f(C
j
). As we can restrict our attention to unit ratio jobs, the width
equals p
j
. Hence, by arranging the rectangles along the x-axis in the order in
which the corresponding jobs appear in some schedule S, each rectangle ends
at the x-axis at its completion time in S. When drawing the graph of the cost
function f into the same graphic, all upper right corners of the rectangles lie on
this graph. The total cost of S results as the area of all rectangles. Note that
the area below the graph of f, i.e.,
w
j
0
f(x)dx is a lower bound on the cost
any schedule. An example is depicted in Figure 1(a).
