On the Local Dominance Properties in Single Machine Scheduling
Problems
Natalia Jorquera-Bravo
1,2
and
´
Oscar C. V
´
asquez
1,2 a
1
University of Santiago of Chile (USACH), Faculty of Engineering, Program for the Development of Sustainable
Production Systems (PDSPS), Santiago, Chile
2
University of Santiago of Chile (USACH), Faculty of Engineering, Department of Industrial Engineering, Santiago, Chile
Keywords:
Single Machine Scheduling Problem, Local Dominance Properties, Search Space, Computational Complexity.
Abstract:
We consider a non-preemptive single machine scheduling problem for a non-negative penalty function f . For
this problem every job j has a priority weight w
j
and a processing time p
j
, and the goal is to find an order on
the given jobs that minimizes
∑
w
j
f (C j), where C
j
is the completion time of job j. This paper explores the
local dominance properties in this problem, which provide a powerful theoretical tool to better describe the
structure of optimal solutions by identifying rules that at least one optimal solution must satisfy, reducing the
search space from n! to n!/3
n/3
schedules and providing insights to show the computational complexity status
for problem with a convex penalty from a general framework, such as the problem of minimizing the sum of
weighted mean squared deviation of the completion times with respect to a common due date and jobs with
arbitrary weights.
1 INTRODUCTION
In this paper we consider a non-preemptive single-
machine scheduling problem, which has several oper-
ations represented by a set J of n jobs, where each job
j has a processing time p
j
∈ N and a priority weight
w
j
∈Q
+
where the objective is to find a schedule σ of
a set J of jobs that minimizes
∑
j∈J
w
j
f (C
j
), where f
is a given penalty non-negative function and C
j
is the
completion time of each job j ∈J with a value greater
than or equal to
∑
σ
i
≤σ
j
p
i
, where σ
j
is the order of job
j ∈ J in the schedule σ.
We focus on the dominance properties, which pro-
vide a powerful theoretical tool to better describe the
structure of optimal solutions by identifying rules that
at least one optimal solution must satisfy. This in-
formation can be used to enhance different exhaus-
tive algorithms to find an optimal solution by reduc-
ing the search space of n! different schedules and
pruning early ineffective partial solutions in several
problems with convex, concave, and piecewise linear
penalty function (V
´
asquez, 2015; Bansal et al., 2017),
even when the penalty functions are non-monotone
increasing (Pereira and V
´
asquez, 2017; D
´
ıaz-N
´
u
˜
nez
et al., 2018; D
´
ıaz-N
´
u
˜
nez et al., 2019; Falq et al., 2021;
Falq et al., 2022), highlighting the left-shifted prop-
a
https://orcid.org/0000-0002-1393-4692
erty of any optimal schedule. This property means
that all executions happen without idle time between
times t
0
and t
n
, with 0 ≤t
0
and t
n
= t
0
+
∑
j∈J
p
j
.
In this setting, we consider an instance I contain-
ing two jobs i, j ∈ J and distinguish two kinds of
properties.
• We say that jobs i, j ∈J satisfy local precedence at
time t - denoted i ≺
`(t)
j - if whenever in a sched-
ule σ job j starts at time t and is followed imme-
diately by job i then the schedule σ is not optimal.
• We say that jobs i, j ∈J satisfy global precedence
in the time interval [a,b] - denoted by i ≺
g[a,b]
j —
if whenever in a schedule σ we have a ≤C
j
−p
j
≤
C
i
− p
i
− p
j
≤ b, then σ is sub-optimal, no matter
if i, j are adjacent or not.
We use the notation i ≺
g
j as a shorthand for
i ≺
g[a,∞]
j. In addition i ≺
`[a,b]
j means i ≺
`(t)
j for
all t ∈ [a,b].
For convenience, we denote F(S) the objective
value from the jobs of schedule S and define the fol-
lowing function on the domain t ∈ [0,∞)
ν
i j
(t) :=

1 −
w
j
w
i

f (t + p
i
+ p
j
) +
w
j
w
i
f (t + p
j
).
Let S
1
and S
2
schedules for I of the form S
1
= Ai jB
and S
2
= A jiB, for some sets of jobs A and B. Let t
be the sum of processing time of all jobs in A. Thus,
208
Jorquera-Bravo, N. and Vásquez, Ó.
On the Local Dominance Properties in Single Machine Scheduling Problems.
DOI: 10.5220/0010871600003117
In Proceedings of the 11th International Conference on Operations Research and Enterprise Systems (ICORES 2022), pages 208-213
ISBN: 978-989-758-548-7; ISSN: 2184-4372
Copyright
c
 2022 by SCITEPRESS – Science and Technology Publications, Lda. All rights reserved

we have that i ≺
`(t)
j is equivalent to
0 <F(S
2
) −F(S
1
)
=w
j
f (t + p
j
) + w
i
f (t + p
i
+ p
j
)
−w
i
f (t + p
i
) −w
j
f (t + p
i
+ p
j
)
=w
i

1 −
w
j
w
i

f (t + p
i
+ p
j
)

+
w
j
w
i
f (t + p
j
) − f (t + p
i
)
=w
i
(ν
i j
(t) − f (t + p
i
)),
and then, the following equivalence hold
i ≺
`(t)
j ≡ 0 < ν
i j
(t) − f (t + p
i
).
1.1 Our contribution
We explore the local dominance properties in single
machine scheduling problems. We show that the total
number of solutions that satisfy the local dominance
properties has an upper bound defined by n!/3
n/3
,
which is a dramatic improvement over the n! differ-
ent schedules of the search space. In addition, we
study the NP-hardness for convex penalty in a gen-
eral framework and address particularly the problem
of minimizing the sum of weighted mean squared de-
viation of the completion times with respect to a com-
mon due date, whose computational complexity status
is still open (V
´
asquez, 2014), providing some insights
to show the computational complexity status based on
dominance properties.
2 SEARCH SPACE AND LOCAL
DOMINANCE PROPERTIES
Given an algorithm that uses some search tree proce-
dure to solve the scheduling problem, we consider a
node of the search tree specified by a partial sched-
ule S
0
. Let i, j be two jobs not in S
0
, and let t
be t
n
−
∑
k∈S
p
k
. Then the descendants of this node
include the partial right-to-left schedules i + S
0
and
j + S
0
. Now except in some degenerate cases (e.g.
identical jobs i and j) for comparable jobs exactly one
of i ≺
`(t)
j, j ≺
`(t)
i holds. This implies that exactly
one of the sub-descendants partial right-to-left sched-
ules j + i + S
0
and i + j + S
0
exists in the search tree.
By considering the left-shifted property of any opti-
mal schedule, note that here we used the fact that the
partial schedule was extended from right to left, and
the local precedence relations between i and j were
done for the same time point, which would not have
been the case, if the schedule were constructed from
left to right.
The effect of this observation is that the number of
nodes in the second level of the tree is upper bounded
by

n
2

. Thus, by multiplying these numbers for all
even levels, the total number of leaves of the tree is
upper bounded by n!/
√
2
n
. However, we improve this
upper bound.
In order to prove the new upper bound, we intro-
duce the following lemma.
Lemma 1. Consider three jobs i, j, k ∈J with w
i
> w
j
and i ≺
`(t+p
k
)
j. The following expression holds
ν
jk
(t + p
i
) <
w
k
w
i

w
j
w
k
+ w
2
i
−2w
i
w
k
w
i
w
k
f (t + p
k
+ p
i
+ p
j
)

−
w
k
w
i

w
j
−w
i
w
i
f (t + p
k
+ p
j
) − f (t + p
k
+ p
i
)

.
Proof. By case assumption i ≺
`(t+p
k
)
j, we have
0 <ν
i j
(t + p
k
) − f (t + p
k
+ p
i
)
=

1 −
w
j
w
i

f (t + p
k
+ p
i
+ p
j
)
+
w
j
w
i
f (t + p
k
+ p
j
) − f (t + p
k
+ p
i
)
=
w
j
w
i
w
k
(w
i
−w
j
)
w
k
(w
i
−w
j
)
2
w
2
i
w
j
!
f (t + p
k
+ p
i
+ p
j
)
!
+
w
j
w
i
w
k
(w
i
−w
j
)
w
k
(w
i
−w
j
)
w
2
i
f (t + p
k
+ p
j
)
!
−
w
j
w
i
w
k
(w
i
−w
j
)

w
k
(w
i
−w
j
)
w
j
w
i
f (t + p
k
+ p
i
)

.
The last equality follows from the assumption w
i
>
w
j
. Thus we have
0 <

w
k
(w
i
−w
j
)
2
w
2
i
w
j

f (t + p
k
+ p
i
+ p
j
)
+
w
k
(w
i
−w
j
)
w
2
i
f (t + p
k
+ p
j
)
−
w
k
(w
i
−w
j
)
w
j
w
i
f (t + p
k
+ p
i
) (1)
We use the following equality
w
k
(w
i
−w
j
)
2
w
2
i
w
j
=
w
2
k
w
2
i
−2w
2
k
w
i
w
j
+ w
2
k
w
2
j
+ w
2
i
w
j
w
k
−w
2
i
w
j
w
k
w
2
i
w
j
w
k
!
=

w
k
w
j
−
2w
k
w
i
w
2
i
+
w
k
w
j
w
2
i
+
w
2
i
w
2
i
−
w
j
w
j

=

w
j
w
k
+ w
2
i
−2w
i
w
k
w
2
i
−

1 −
w
k
w
j

, (2)
On the Local Dominance Properties in Single Machine Scheduling Problems
209

replacing it in expression (1). Then, we obtain
0 <

w
j
w
k
+ w
2
i
−2w
i
w
k
w
2
i

f (t + p
k
+ p
i
+ p
j
)
−

1 −
w
k
w
j

f (t + p
k
+ p
i
+ p
j
)
+
w
k
(w
i
−w
j
)
w
2
i
f (t + p
k
+ p
j
)
−
w
k
(w
i
−w
j
)
w
j
w
i
f (t + p
k
+ p
i
)
=

w
j
w
k
+ w
2
i
−2w
i
w
k
w
2
i

f (t + p
k
+ p
i
+ p
j
)
−

1 −
w
k
w
j

f (t + p
k
+ p
i
+ p
j
)
+
w
k
(w
i
−w
j
)
w
2
i
f (t + p
k
+ p
j
) −
w
k
w
j
f (t + p
k
+ p
i
)
+
w
k
w
i
f (t + p
k
+ p
i
) (3)
We reorder the terms of expression (3) and have

1 −
w
k
w
j

f (t + p
k
+ p
i
+ p
j
) +
w
k
w
j
f (t + p
k
+ p
i
)
=ν
jk
(t + p
i
)
<

w
j
w
k
+ w
2
i
−2w
i
w
k
w
2
i

f (t + p
k
+ p
i
+ p
j
)
+
w
k
(w
i
−w
j
)
w
2
i
f (t + p
k
+ p
j
) +
w
k
w
i
f (t + p
k
+ p
i
)
=
w
k
w
i

w
j
w
k
+ w
2
i
−2w
i
w
k
w
i
w
k
f (t + p
k
+ p
i
+ p
j
)

−
w
k
w
i

w
j
−w
i
w
i
f (t + p
k
+ p
j
) − f (t + p
k
+ p
i
)

,
concluding the proof.
We now prove that the number of sequences that
respect the local dominance property among three
jobs is only two.
Theorem 1. Fix the sequences with three compara-
ble jobs i, j and k in some consecutive order, where t
1
and t
1
−(p
i
+ p
j
+ p
k
) are the completion time and
starting time, respectively. The number of sequences
that respect the local precedence among them is two.
Proof. Without loss of generality, we adopt t
1
=
p
i
+ p
j
+ p
k
. We know that the total se-
quences induces by three comparable jobs is 3! =
6. We denote S
1
,S
2
,S
3
,S
4
,S
5
and S
6
the sequences
i jk,ik j, jki, jik,ki j and k ji, respectively.
Note that the jobs a and b are comparable, and
therefore one of a ≺
`(t)
b, b ≺
`(t)
a holds.
Let → and ⇒ be the exclusion relations between
two sequences with the same pair of jobs ordered in
inverse form at the beginning and the end, respec-
tively. These two exclusions capture the fact that a
single order is possible between two comparable jobs
at the same point in time.
By using the exclusion relations, we have:
S
1
⇒ S
2
→ S
5
⇒ S
6
→ S
3
⇒ S
4
→ S
1
Therefore, the sequences that respect the exclu-
sion relations can be in two sets:
A = {S
1
,S
3
,S
5
} or B = {S
2
,S
4
,S
6
}
In addition, we distinguish six priority weight
orders among the jobs defined as follows:
o
1
: {w
i
> w
j
> w
k
} o
2
: {w
k
> w
i
> w
j
}
o
3
: {w
k
> w
j
> w
i
} o
4
: {w
j
> w
i
> w
k
}
o
5
: {w
i
> w
k
> w
j
} o
6
: {w
j
> w
k
> w
i
}.
Now, we prove by contradiction that only 2 of 3
sequences of each set respect the local precedence
among the jobs. First, we consider the set A and w
i
>
w
j
, which covers the order priority weights o
1
,o
2
and
o
5
Formally, suppose the sequence S
1
,S
3
and S
5
with
w
i
> w
j
respect the local precedence among the jobs.
We observe the job pairs with the same local prece-
dence for two particular sequences (in bold font) and
have:
1. S
1
= ijk ∧S
5
= jki imply j ≺
`(t)
k for t = p
i
,0
2. S
1
= ijk ∧S
3
= kij imply i ≺
`(t)
j for t = p
k
,0
3. S
3
= ki j ∧S
5
= jki imply k ≺
`(t)
i for t = p
j
,0
In particular,
f (p
i
+ p
j
) <ν
jk
(p
i
) :=

1 −
w
k
w
j

f (p
j
+ p
i
+ p
k
)
+
w
k
w
j
f (p
i
+ p
k
) (4)
f (p
k
+ p
i
) <ν
i j
(p
k
) :=

1 −
w
j
w
i

f (p
j
+ p
i
+ p
k
)
+
w
j
w
i
f (p
k
+ p
j
) (5)
f (p
j
+ p
k
) <ν
ki
(p
j
) :=

1 −
w
i
w
k

f (p
j
+ p
i
+ p
k
)
+
w
i
w
k
f (p
j
+ p
i
) (6)
We use two of three above inequalities. Without loss
of generality, we consider Expressions (5) and (6).
Thus, we have
ICORES 2022 - 11th International Conference on Operations Research and Enterprise Systems
210

f (p
k
+ p
j
) + f (p
i
+ p
k
)
<ν
i j
(p
k
) + ν
ki
(p
j
)
=

1 −
w
j
w
i

f (p
i
+ p
j
+ p
k
)
+

1 −
w
i
w
k

f (p
i
+ p
j
+ p
k
)
+
w
j
w
i
f (p
k
+ p
j
) +
w
i
w
k
f (p
j
+ p
i
)
=

2 −
w
j
w
i
−
w
i
w
k

f (p
i
+ p
j
+ p
k
) +
w
j
w
i
f (p
k
+ p
j
)
+
w
i
w
k
f (p
j
+ p
i
)
=

−w
j
w
k
−w
2
i
+ 2w
i
w
k
w
i
w
k

f (p
i
+ p
j
+ p
k
)
+
w
j
w
i
f (p
k
+ p
j
) +
w
i
w
k
f (p
j
+ p
i
) (7)
We reorder the Expression (7) and have
w
k
w
i

w
j
w
k
+ w
2
i
−2w
i
w
k
w
i
w
k

f (p
k
+ p
i
+ p
j
)

−
w
k
w
i

w
j
−w
i
w
i
f (t + p
k
+ p
j
) − f (t + p
k
+ p
i
)

< f (p
j
+ p
i
),
where the left-term is greater than ν
jk
(p
i
) by Lemma
1. This implies that k ≺
(p
i
)
j, which contradicts the
case assumption given by Expression (4).
For the case w
k
> w
i
, which covers the order prior-
ity weights o
2
, o
3
and o
6
, the proof considers Expres-
sions (4) and (6) in a similar way, which contradicts
Expression (5). To end the analysis for the set A, we
have w
j
> w
k
covering order priority weights o
1
, o
4
and o
6
. Here, the proof is also by contradiction. Ex-
pressions (4) and (5), contradicting Expression (6).
Symmetrically, we prove the case when the set B
is considered. Suppose S
2
,S
4
and S
6
with respect the
local precedence among the jobs, we have:
1. S
2
= ikj ∧S
6
= kji imply k ≺
`(t)
j for t = p
i
,0
2. S
2
= ik j ∧S
4
= jik imply i ≺
`(t)
k for t = p
j
,0
3. S
4
= jik ∧S
6
= kji imply j ≺
`(t)
i for t = p
k
,0
In particular,
f (p
i
+ p
k
) <ν
k j
(p
i
) :=

1 −
w
j
w
k

f (p
j
+ p
i
+ p
k
)
+
w
j
w
k
f (p
i
+ p
j
) (8)
f (p
j
+ p
i
) <ν
ik
(p
j
) :=

1 −
w
k
w
i

f (p
j
+ p
i
+ p
k
)
+
w
k
w
i
f (p
j
+ p
k
) (9)
f (p
k
+ p
j
) <ν
ji
(p
k
) :=

1 −
w
i
w
j

f (p
j
+ p
i
+ p
k
)
+
w
i
w
j
f (p
k
+ p
i
) (10)
Here, we analyse the different cases. For the case
where w
i
> w
k
, we have Expressions (9) and (10) that
contradicts Expression (8), covering the order prior-
ity weights o
1
, o
4
and o
5
. For the case w
j
> w
i
, we
consider Expressions (10) and (8), which contradicts
Expression (9). This covers the order priority weights
o
3
, o
4
and o
6
. Finally, the order priority weights o
2
, o
3
and o
5
are covered by the case w
k
> w
j
. This uses Ex-
pressions (8) and (9), contradicting Expression (10).
Therefore, only two of three sequences of set re-
spect the local precedence among the jobs. Specifi-
cally, the pairs sequences are:
{S
1
,S
3
} {S
1
,S
5
} {S
3
,S
5
} {S
2
,S
4
} {S
2
,S
6
}
{S
4
,S
6
} {S
1
,S
6
} {S
2
,S
3
} {S
4
,S
5
},
which concludes the proof.
From Theorem 1, we obtain the following corol-
lary.
Corollary 1. Consider three jobs consecutively exe-
cuted. The local precedence property is satisfied at
most for two pairs of jobs at any time in [t
0
,t
n
].
Finally, Corollary 2 shows a new upper bound,
which represents a dramatic improvement over the n!
different schedules of the search space.
Corollary 2. Given an algorithm, which uses some
search tree procedure to solve the scheduling prob-
lem. The total number of leaves of the tree is upper
bounded by n!/3
n/3
Proof. The proof follows the observation from Theo-
rem 1, which implies that the number of nodes in the
third level of the tree is upper bounded by 2

n
3

and for
a multiplying argument, the total number of leaves of
the tree is upper bounded by n!/3
n/3
3 PENALTY FUNCTIONS WHERE
THE PROBLEM BECOMES
EASY
In this section, we study the penalty functions where
the problem becomes easy. We show that for any in-
creasing convex penalty functions, an instance of n
On the Local Dominance Properties in Single Machine Scheduling Problems
211

jobs with equal Smith ratios, i.e. w
i
/p
i
equal to a
constant for all job i = 1,. . . , n, admits an optimal
schedule where the jobs are ordered in non-increasing
weight.
Theorem 2. Consider a strict convex penalty function
f (t) and two jobs i, j ∈ J . If w
i
/p
i
= w
j
/p
j
and p
i
>
p
j
, then i ≺
`
j
Proof. Let A,B be two arbitrary job sequences. Sup-
pose that i, j ∈ J are adjacent in an optimal schedule
and let t be the largest completion time of the jobs in
A. The claim states that the order i, j generates a cost
strictly smaller than the order j, i, i.e.
F(A jiB) > F(Ai jB)
≡p
j
f (t + p
j
) + p
i
f (t + p
i
+ p
j
) > p
i
f (t + p
i
)
+ p
j
f (t + p
i
+ p
j
)
≡(p
i
−p
j
) f (t + p
i
+ p
j
) + p
j
f (t + p
j
) > p
i
f (t + p
i
)
≡(1 −
p
j
p
i
) f (t + p
i
+ p
j
) +
p
j
p
i
f (t + p
j
) > f (t + p
i
)
≡ν
i j
(t) > f (t + p
i
),
which holds for p
i
> p
j
and f strictly convex.
From Theorem 2, we obtain the following state-
ment.
Corollary 3. Consider a strictly convex penalty func-
tion f (t) and jobs with equal Smith ratios. This prob-
lem admits an optimal schedule where the jobs are
ordered in non-increasing weight.
Note that Corollary 3 applies to the problem of
minimizing the sum of weighted mean squared devia-
tion of the completion times with respect to a common
due date, where the penalty function is strict convex.
4 FUTURE RESEARCH
In literature, previous NP-hardness proofs for the
scheduling problem with some convex and concave
penalty function involved almost equal ratio instances
(see (Jinjiang, 1992; V
´
asquez, 2014) for example).
We now provide as future research some insights to
show the computational complexity of the problem of
minimizing the sum of weighted mean squared devia-
tion of the completion times with respect to a common
due date, whose status is open for jobs with arbitrary
weights (V
´
asquez, 2014).
In practice, we define an instance I
C
as follows:
Consider the strict convex penalty function f (t) :=
(t −d)
2
and a set J with 2n +1 jobs where B ⊆J with
B = {1,...,2n} with equal Smith ratios and p
i
> p
i+1
for i = 1,. .., 2n −1 and a job k = 2n + 1 with w
k
>p
k
,
p
k
≥ max
i∈B
p
i
, and
√
w
k
min
i∈B
p
i
≥ p
k
. Based on
the global properties from Corollary 1 in (Pereira
and V
´
asquez, 2017) and Theorem 1 in (Bansal et al.,
2017), and Theorem 1, in the optimal solution of these
instances I
C
, the completion time of job k belongs
to the interval (d −p
k
−min
i∈B
p
i
,d + min
i∈B
p
i
), all
jobs preceding k are scheduled in a non-increasing
processing time, and all jobs following k are sched-
uled in a non-increasing processing time.
Based on Theorem 2, we focus on the necessary
condition so that the problem does not become easy
given by the sequence of three jobs that respects the
local precedence among them, which is defined as fol-
lows: the job k and two jobs i
0
+ 1 and i
0
, which are
immediately executed before and after job k, respec-
tively. Given the above sequence, we consider Theo-
rem 1 and have that at most one of these job sequences
defined by a) k,i
0
,i
0
+ 1, b) i
0
,k, i
0
+ 1 or c) i
0
,i
0
+ 1,k,
respects the local precedence among them. Clearly,
the sequences a) and b) and, the sequences b) and
c) are reciprocally excluded by local precedence be-
tween jobs k and i
0
and, jobs k and i
0
+1, respectively.
However, we note that we can exclude the sequence c)
by choosing sequence a), and vice-versa. This exclu-
sion is based on the definition of jobs i
0
,i
0
+1, Lemma
3, the necessary condition and Corollary 1. This al-
lows us reduce the search space, restricting the cases
to be analyzed.
ACKNOWLEDGEMENTS
The authors are grateful for partial support by ANID,
Beca de Postdoctorado en el Extranjero, Folio:
74200020 and ANID, Proyecto FONDECYT, Folio:
1211640.
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