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Approximability of scheduling problems with resource
consuming jobs
P´eter Gy¨orgyi · Tam´as Kis
Received: date / Accepted: date
Abstract The paper presents new approximability results for single machine
scheduling problems with jobs requiring some non-renewable resources (like raw
materials, energy, or money) beside the machine. Each resource has an initial stock
and additional supplies over time. A feasible schedule specifies a starting time for
each job such that no two jobs overlap in time, and when a job is started, enough
resources are available to cover its requirements. The goal is to find a feasible
schedule of minimum makespan. This problem is strongly NP-hard.
Recently, the authors of this paper have proposed a PTAS for the special case
with a single non-renewable resource and with a constant number of supply dates,
as well as an FPTAS for the special case with two supply dates and one resource
only. In this paper we prove APX-hardness of the problem when the number of
resources is part of the input, and new polynomial time approximation schemes
are devised for some variants, including (1) job release dates, and more than one,
but constant number of resources and resource supply dates, and (2) only one
resource, arbitrary number of supply dates and job release dates, but with resource
requirements proportional to job processing times.
Keywords Single machine scheduling, non-renewable resources, approximation
schemes, vertex cover problem
1 Introduction
In this paper we study scheduling problems with resource consuming jobs. In these
problems there are non-renewable resources (like raw materials, energy, or money)
consumed by the jobs. Each non-renewable resource has an initial stock, which
is replenished at a-priori known moments of time and in known quantities. We
will consider only single-machine problems, i.e., all the jobs have to be sequenced
on the same machine. The sole optimization objective will be the schedule length
P. Gy¨orgyi · T. Kis
Institute for Computer Science and Control, H1111 Budapest, Kende str. 13–17, Hungary
Tel.: +36 1 2796156; Fax: +36 1 4667503
E-mail: gyorgyi.peter@sztaki.mta.hu, kis.tamas@sztaki.mta.hu
P. Gy¨orgyi
E¨otvos Lor´and University, P´azm´any P´eter S´et´any 1/C, Budapest, Hungary, H1117
2 P´eter Gy¨orgyi, Tam´as Kis
(makespan). Our analysis relies on connections with variants of the knapsack prob-
lem, and with the vertex cover problem in graphs.
More formally, there is a single machine, a finite set of jobs J , and a finite set of
non-renewable resources R consumed by the jobs. The machine can perform only
one job at a time, and preemption is not allowed. Each job J
j
has a processing
time p
j
∈ Z
+
, a release date r
j
, and resource requirements a
ij
∈ Z
+
from the
resources i ∈ R. The resources are supplied in q different moments in time, 0 =
u
1
< u
2
< . . . < u
q
; the vector
˜
b
`
∈ Z
|R|
+
represents the quantities supplied at
u
`
. A schedule σ specifies the starting time S
j
of each job and it is feasible if (i)
the jobs do not overlap in time, (ii) S
j
≥ r
j
for all j ∈ J , and if (iii) at any time
point t the total material supply from every resource is at least the total request of
those jobs starting not later than t, i.e.,
P
(` : u
`
≤t)
˜
b
`i
≥
P
(j : S
j
≤t)
a
ij
, ∀i ∈ R.
The objective is to minimize the makespan, i.e., the completion time of the job
finished last.
Assumption 1
P
q
`=1
˜
b
`i
=
P
j∈J
a
ij
, ∀i ∈ R, holds without loss of generality.
Assumption 1 implies that there must exist a feasible solution (if every job
starts not before u
q
, the last supply date) and at least one job must start not
before u
q
(thus the optimal makespan C
∗
max
is greater than u
q
).
1.1 Previous work
Scheduling problems with resource consuming jobs were introduced by Carlier
(1984), and Carlier and Rinnooy Kan (1982). Further results can be found in
e.g., Slowinski (1984), Toker et al. (1991), Neumann and Schwindt (2002), La-
borie (2003), Grigoriev et al. (2005), Briskorn et al. (2010, 2013), Gafarov et al.
(2011), Gy¨orgyi and Kis (2014, 2015), Morsy and Pesch (2015). In particular, in
Grigoriev et al. (2005) and Gafarov et al. (2011) the complexity of several variants
was studied and some constant ratio approximation algorithms were developed in
Grigoriev et al. (2005). Briskorn et al. (2010), Briskorn et al. (2013) and Morsy and
Pesch (2015) examined scheduling problems where there is an initial inventory, and
no more supplies, but some of the jobs produce resources, while other jobs consume
the resources. In Briskorn et al. (2010) and Briskorn et al. (2013) problems with
the objective of minimizing the inventory levels were studied. Morsy and Pesch
(2015) designed approximation algorithms to minimize the total weighted comple-
tion time. In Gy¨orgyi and Kis (2014) a PTAS for scheduling resource consuming
jobs with a single non-renewable resource and a constant number of supply dates
was developed, and also an FPTAS was devised for the special case with q = 2
supply dates and one non-renewable resource only. In Gy¨orgyi and Kis (2015)
it was shown, among other results, that there is no FPTAS for the problem of
scheduling jobs on a single machine with two non-renewable resources and q = 2
supply dates, unless P = NP , which is in strong contrast with the existence of an
FPTAS for the special case with one non-renewable resource only (Gy¨orgyi and
Kis, 2014). In Gy¨orgyi and Kis (2015) and Gy¨orgyi and Kis (2014), variants of the
knapsack problem are solved as a subproblem using combinatorial techniques like
enumeration of feasible packings. However, an important and very fruitful algo-
rithmic technique for solving packing type problems is linear programming based
rounding, see e.g., Fleischer et al. (2011), that will be used in this paper as well.
Approximability of scheduling problems with resource consuming jobs 3
While there are some algorithmic results for the more general resource con-
strained project scheduling problem (RCPSP) with non-renewable resources, see
e.g., Neumann and Schwindt (2002), Laborie (2003), but, to our best knowledge,
the hardness of approximation has only been studied for RCPSP without any
non-renewable resources, see Gafarov et al. (2014).
1.2 Results of the paper
Our positive and negative results are presented in the following two subsections.
1.2.1 Non-approximability results
If the number of non-renewable resources is constant and the number of supply
dates is 2, then the problem 1|rm = const., q = 2|C
max
admits a PTAS (Gy¨orgyi
and Kis, 2015). In contrast, if the number of resources is part of the input, we can
prove the following result.
Theorem 1 Unless P = NP , there is some constant ε > 0 such that it is NP-
hard to approximate the problem 1|rm, q = 2|C
max
better than 1 + ε if the number
of resources is part of the input.
Since the problem 1|rm|C
max
admits a 2-approximation algorithm (Grigoriev
et al., 2005), and 1|rm, q = 2|C
max
is just a special case, we can deduce the
following:
Corollary 1 1|rm, q = 2|C
max
is APX-complete.
It is also known that if the number of resources is constant and at least 2, then
there is no FPTAS for 1|rm = const., q = 2|C
max
.
1.2.2 Approximation schemes
Our new approximation schemes can solve more general problems than 1|rm =
1, q = const|C
max
, for which a PTAS has been developed in Gy¨orgyi and Kis
(2014). On the one hand, we allow more than one, but a constant number of
resources, and on the other hand, we consider job release dates as well.
Theorem 2 There is a PTAS for the problem 1|rm = const., q = const., #{r
j
:
r
j
< u
q
} = const.|C
max
.
The condition #{r
j
: r
j
< u
q
} = const. reads that the number of distinct
job release dates before u
q
is bounded by a constant. The new PTAS inherits
some of the components from the earlier result, like scheduling small and big
jobs separately, but in this paper we use linear programming based rounding to
schedule the small jobs, and in the analysis we prove only that the rounding is just
good enough to get the desired approximation for the original scheduling problem,
instead of the stronger result proved in Gy¨orgyi and Kis (2014) showing that the
scheduling of the small jobs is a good approximation for a subproblem similar to
the multiple knapsack problem. The following result dispenses with the condition
on the number of distinct job release dates.
4 P´eter Gy¨orgyi, Tam´as Kis
Theorem 3 There is a PTAS for the problem 1|rm = const., q = const.|C
max
.
The proof of this results uses a rounding argument, and relies on the PTAS
developed for proving Theorem 2. Finally, we can get rid of the constant bound
on the number of resource supplies at the expense of considering one resource only
and restricting the resource requirements to be proportional to the job processing
times, i.e., there exists a positive constant λ such that a
j
= λp
j
for all j ∈ J .
The constant λ of course depends on the problem instance. This assumption may
be quite reasonable in some practical applications. Since we can get an equivalent
problem by dividing all the supplies, and all the resource requirements of a prob-
lem instance by the (instance specific) constant λ, from now on we consider the
case a
j
= p
j
only. Notice that in the above transformation, the
˜
b
`
may become
fractional after dividing by λ. However, this does not create any difficulty for the
approximation algorithms proposed below.
Theorem 4 There is a PTAS for the problem 1|rm = 1, a
j
= p
j
|C
max
.
One can generalize this result by enabling job specific release dates as well.
Theorem 5 There is a PTAS for the problem 1|rm = 1, p
j
= a
j
, r
j
|C
max
.
In Table 1 we summarize the results of the paper, and for the sake of com-
pleteness, we also mention previous results for this class of problems.
#Supplies #Resources Release PTAS FPTAS
q rm dates r
j
2 1 no yes (Gy¨orgyi and Kis, 2014) yes (Gy¨orgyi and Kis, 2014, 2015)
2 1 yes yes (Sect. 5) ?
2 const. ≥ 2 no yes (Gy¨orgyi and Kis, 2014) no
a
(Gy¨orgyi and Kis, 2015)
2 const. ≥ 2 yes yes (Sect. 5) no
a
(Gy¨orgyi and Kis, 2015)
2 arbitrary yes/no no
a
(Sect. 3) no
a
(Sect. 3)
const. ≥ 3 1 yes/no yes (Sect. 5) ?
const. ≥ 3 const. ≥ 2 yes/no yes (Sect. 5) no
a
(Gy¨orgyi and Kis, 2015)
arbitrary 1 no yes
b
(Sect. 6) no
a
(Grigoriev et al., 2005)
arbitrary 1 yes yes
b
(Sect. 7) no
a
(Grigoriev et al., 2005)
a
if P 6= N P
b
under the condition a
j
= p
j
Table 1 Known approximability results for scheduling problems with resource consuming
jobs. In the column of release dates ”yes / no” means that the result is valid in both cases.
The question mark ”?” indicates that we are not aware of any definitive answer.
1.3 Structure of the paper
In Section 2 we provide a problem formulation in terms of a mathematical program
which will be used throughout the paper. In Sections 3, 4, 5, 6, and 7 we prove
Theorems 1, 2, 3, 4 and 5, respectively. Some final remarks and open questions
are collected in Section 8.
Approximability of scheduling problems with resource consuming jobs 5
1.4 Terminology and definitions
An optimization problem Π consists of a set of instances, where each instance has
a set of feasible solutions, and each solution has a cost. In a minimization problem
a feasible solution of minimum cost is sought, while in a maximization problem one
of maximum cost. The value of the best (or optimal) solution of instance x of Π
is denoted by opt(x). An ε-approximation algorithm for an optimization problem
Π delivers in polynomial time for each instance of Π a solution whose objective
function value is at most (1 + ε) times the optimum value in case of minimization
problems, and at least (1−ε) times the optimum in case of maximization problems.
For an optimization problem Π, a family of approximation algorithms {A
ε
}
ε>0
,
where each A
ε
is an ε-approximation algorithm for Π is called a Polynomial Time
Approximation Scheme (PTAS) for Π. If, in addition, each A
ε
in the family is of
polynomial time in 1/ε as well, then {A
ε
}
ε>0
, is called a Fully Polynomial Time
Approximation Scheme (FPTAS) for Π. The class PTAS / class FPTAS consists
of those optimization problems which admit a polynomial time approximation
scheme / fully polynomial time approximation scheme. The above definitions are
mainly from the book of Garey and Johnson (1979).
The class APX consists of those optimization problems that can be approx-
imated within some constant factor in polynomial time in the size of the input.
Clearly, the class PTAS is a subset of the class APX. The class APX-complete
comprises those problems from APX which do not belong to PTAS, unless P=NP.
Let Π
1
and Π
2
be two optimization problems. We say that Π
1
L-reduces to Π
2
if
there exist two polynomial time algorithms f and g, and two constants α, β > 0,
such that for every instance x of Π
1
:
i) opt
Π
2
(f(x)) ≤ α · opt
Π
1
(x),
ii) for any solution of f(x) with cost c
2
, g provides a solution of x with cost c
1
such that |c
1
− opt
Π
1
(x)| ≤ β · |c
2
− opt
Π
2
(f(x))|.
The two most important properties of L-reductions are that they compose, and
if Π
1
L-reduces to Π
2
, and there is an ε-approximation algorithm for Π
2
, then,
through the reduction, we get an αβε-approximation algorithm for Π
1
. See the
original paper by Papadimitriou and Yannakakis (1991) for more details.
2 Problem formulation
We can model our scheduling problem by means of a mathematical program. To
this end, firstly we construct a set of time points T consisting of all the distinct
values from the set of time moments u
`
, ` = 1, . . . , q, (when some non-renewable
resource is supplied), and the set of release dates of the jobs r
j
, j ∈ J . Suppose
T has τ elements, denoted by v
1
through v
τ
, with v
1
= 0. We define the values
b
`i
:=
P
k : u
k
≤v
`
˜
b
ki
for i ∈ R, that is, b
`i
equals the total amount supplied from
resource i up to time point v
`
.
We introduce τ · |J | binary decision variables x
j`
, (j ∈ J , ` = 1, . . . , τ ) such
that x
j`
= 1 if and only if job j is assigned to the time point v
`
, which means that
the requirements of job j must be satisfied by the resource supplies up to time
