UDC 519.854.2
PAVLOV A.A.
THE OPTIMALITY SIGNS OF FEASIBLE SOLUTIONS OF INTRACTABLE
COMBINATORIAL OPTIMIZATION PROBLEMS
In the article for several types of intractable single-stage scheduling problems the optimality signs of a feasible
solution are formulated which are the theoretical basis for the construction the polynomial component of the PDCalgorithms for these problems.
Introduction
Most of the discrete mathematical models constructed for the analysis and synthesis of complex
systems are either NP-complete or are not easier
than NP-complete problems. In particular, most of
the scheduling problems belong to this class [1].
Analysis of difficulties encountered in the calculations on the way of effective methods creation for
solving such problems has led to the following
problem: is it possible to exclude the search of all,
or almost all the variations in the problem? This
problem is explored in the theory of NP-complete
problems that has been formed on the basis of
works by Stephen Cook, Richard Karp, Leonid
Levin et al [2]. If the hypothesis P ≠ NP is true
then exact polynomial algorithms for solving this
class of problems do not exist [2]. It can only be
possible to set off in certain classes of NPcomplete problems the subclasses (defined by the
restrictions imposed on the parameters of a combinatorial problem) for which the exact polynomial
algorithms are generated [2]. In general, the effective are only approximate and heuristic algorithms.
In [1, 3, 4], the new approach is exposed to the
possibility of obtaining exact solutions for NPcomplete problems of sufficiently high dimension,
which was formed as a theory of PDC algorithms.
PDC-algorithm is an algorithm [1] consisting of
the polynomial and the exponential component,
which may contain conditions of the original problem decomposition into sub-problems of smaller
dimension (usually the polynomial component is a
part of the exponential component). The upper
bound of the polynomial component complexity is
known. The polynomial component is generated by
the logical-analytic conditions (p-conditions), the
fulfillment of which by a feasible solution obtained
as a result of the polynomial part of the PDC-algorithm defines it as an optimal. p-conditions are
found in a result of theoretical studies of the corresponding class of intractable combinatorial optimization problems. The average efficiency of the polynomial component of the PDC-algorithm is found
by statistical methods [1]. The polynomial compo-
nent of the PDC-algorithm is synthesized in such a
way that the sequential procedure of feasible solutions constructing was the most effective in terms
of realization of the p-conditions (the optimality
signs of feasible solutions). Sometimes the exponential component of the PDC-algorithm is replaced by an algorithm of polynomial complexity
which leads to an approximate (suboptimal) solution [1].
Despite common methodology of PDC-algorithms construction, their particular implementation for the various classes of intractable combinatorial optimization problems (in particular their
polynomial component) leads to completely different algorithms [1, 3, 4]. Thus, the construction of
the PDC-algorithm for an intractable combinatorial
optimization problem is uniquely determined by
the ability to produce for this class of problems the
constructive (simply checked) optimality signs of a
feasible solution.
In this paper a number of new single-stage
scheduling problems are formulated, for each of
them the p-conditions (optimality signs of a feasible solution) are found. The results implement a
possibility of constructing for these classes of
combinatorial optimization problems efficient
PDC-algorithms. The vast number of new singlestage scheduling problems studied below are generated by a multi-stage network scheduling problem presented in [5] which is the formal representation of the third level of the four-level model of
planning, decision making and operational control
in the network systems with limited resources [5].
Note. These presented below new single-stage
scheduling problems have not been studied on the
subject of what class of problems (P or not easier
than NP-complete) they belong. However, it does
not matter: the PDC-algorithm may appear necessary computational procedure even in cases when:
a) An exact polynomial algorithm for the problem solution is unknown;
b) An exact polynomial algorithm will be constructed (proof that the problem belongs to class
P), or if it will be proved that P = NP, but the
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Признаки оптимальности допустимых решений труднорешаемых задач комбинаторной оптимизации
PDC-algorithm will appear statistically more effective. This might be in the case when the complexity
of the exact polynomial algorithm is significantly
higher than the complexity of the polynomial component of the PDC-algorithm.
Problem 1.1
Given a set of jobs J, the number of independent
machines m, for each job j  J the duration lj is
known, j = 1, n . All the jobs have the common due
date d. The processing of each job can start at any
moment of time, it will proceed without interruption until job completion. All machines operate
without interruption with the common moment of
launch (release time).
The problem is to construct a feasible schedule
of jobs j  J that has the maximal release time of
jobs r or the minimal total earliness of jobs in relation to the common due date.
As shown in [6], the optimal schedule for one of
the criteria is automatically an optimal schedule for
another criterion.
In [6] the first sign of optimality of a feasible
schedule is given: on a uniform schedule (processing times of all machines are the same) the absolute optimum is achieved for both criteria. If this
condition is not met, the optimal is a feasible
schedule that satisfies the second criterion for optimality: a feasible schedule is optimal for both criteria if Ci (where Ci, i = 1, m , is the total processing time of the machine i) it is true that for any
i ≠ j, i, j = 1, m , | Ci – Cj | = 0  b, where b is an
arbitrary rational number such that i = 1, n the
numbers li / b are integers; li > 0, i = 1, n , is the
processing time of the i-th job and is an arbitrary
rational number (an example of this schedule is
shown in Fig. 1).
imizing the release time of jobs r. Indeed, rmax does
not satisfy the inequality
rmax ≥ d – Cmin, Cmin = min Ci.
(1)
i 1, m
Indeed, Fig. 1 shows that for r ≥ d – Cmin the
constructed feasible schedule is impossible, since
the inequality is satisfied:
n
Cmin ∙ m <  li
i 1
due to the fact that li and Ci are integers, rmax ≤ d –
– Cmin – 1. But for r = d – Cmin – 1 a feasible
schedule has been built, so rmax = d – Cmin – 1.
Note. Due to the fact that Ci are integers, i, j,
i ≠ j, | Ci – Cj | are natural numbers.
The general case. b > 0 is rational number,
li > 0, i = 1, n , are rational numbers, i li / b are
natural numbers. By changing the scale this problem is reduced to the previous one: lˆi , i  1, n , are
the new durations expressed in natural numbers,
where the unit of measurement equals to b. In the
new units the feasible schedule (Fig. 1) reduces to
this special case above.
Proposition 1. The number b is the common divisor of the numbers li, i = 1, n , that is evenly divided by other common divisors aj, j = 1, k , of these
numbers, and i li / b, i li / j aj are integers.
Proof. If i  j is satisfied Ci  Cj (|Ci – Cj| =
= b > 0, see Fig. 1), then necessarily |Ci – Cj| = klal,
l = 1, k , where l kl are integers. Indeed, since
i li / al, l = 1, k , are integers, then al the number
|Ci – Cj| can be represented as klal where kl is an
integer. Hence, b is evenly divided by al, l = 1, k .
Corollary. b is the greatest common divisor of
the numbers li, i = 1, n .
Let k is the number of machines, each of which
has the total working time d – b (see Fig. 1), where
d is the common due date. Let’s denote
n
lˆ   li b .
i 1
Fig. 1.
Proof. Consider the special case of such a
schedule when li, i = 1, n , are integers and b equals
to 1. This schedule is optimal by criterion of max-
Proposition 2. a) if lˆ / m is integer then the
second criterion of optimality can not be implemented (k does not exist);
b) if lˆ / m is fractional number then k is the only
number from {1, ..., m – 1} for which lˆ / m – k / m
is integer.
Proof. Fig. 1 shows that
n
m ∙ Cmax =  li – k ∙ b
i 1
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or d1 = lˆ / m – k / m, where d1 = Cmax / b is integer,
n
 li
b  lˆ is integer. If lˆ / m is integer then for
i 1
any k = 1, m  1 d1 is fractional number, which is
impossible. If lˆ / m is fractional number then only
for one k  {1, …, m – 1}, d1 is an integer.
Problem 1.2
The problem 1.2 differs from the problem 1.1 so
that the machines start at different times
t1,н ≤ t2,н ≤ … ≤ tm,н where ti+1,н – ti,н = di, i =
= 1, m  1 , are given numbers. The problem is to
find a feasible schedule which is optimal by one of
the two criteria:
1) t1,н is maximum;
2) the total earliness of jobs against the due date
is minimum:
n
min  d  tik 
i 1
where tiк is the end of processing at machine i.
Proposition 3. If
j 1
n
m
m 1 m 1


d

d

(2)
 li >  i    i  dl 
i 1
j  2  i 1
l 1
i 1

then: a) a feasible schedule which is optimal by
one of two criteria is optimal by another one;
b) the first and second signs of optimality of a feasible schedule for the problem 1.1 are also signs of
optimality of a feasible schedule for the problem 1.2.
c) Proposition 1 for the problem 1.1 is also true
for the problem 1.2. Proposition 2 for the problem 1.1
is true for the problem 1.2 if all the numbers di,
i = 1, m  1 , are evenly divided by b.
Proposition 3 clearly follows from [6], the proofs
of Propositions 1 and 2 given for the problem 1.1,
and is illustrated by Fig. 2.
Fig. 2.
Note. Condition (2) is necessary to ensure that
in any feasible schedule maxi Ci > tm,н.
3
Problem 1.3
Statement of the problem is different from the
problem 1.1 statement in that the machines can
start at arbitrary times. The problem is to find a
feasible schedule in which this would be fulfilled:
max ti,н is the minimum possible.
Proposition 4. The sign of optimality of a feasible schedule for the problem 1.3 is a fulfillment
of the following conditions:
1) The total earliness is zero;
2) The schedule is uniform (ti,н = const) or
| ti,н – tj,н | = 0  b
(3)
where b > 0 is the greatest common divisor of чисел li; li / b are integer numbers, i = 1, n . The number k of machines that have ti,н < max tj,н is the onj 1, m
ly one and is defined in the Proposition 2 (problem
1.1).
Proof of Proposition 4 obviously follows from
[6] and the research of the problem 1.1 and is illustrated by Fig. 3.
Fig. 3.
Problem 2
Given a set of jobs J, the number of independent
units m, for each job j  J the processing time lj is
known, j = 1, n . The processing of each job can
start at any moment of time, it will proceed without
interruption until completion of jobs. All machines
works without interruption. Restrictions are set for
the completion times of the machines, i.e. conditions must be met:
tiк ≤ di, d1 ≥ d2 ≥ … ≥ dm, i = 1, m
where tiк is the moment of finishing work by the
machine i.
Due dates for the jobs’ completion times are not
set.
It is needed to find a feasible schedule for which
min ti,н → max.
i 1, m
Proposition 5. If
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Признаки оптимальности допустимых решений труднорешаемых задач комбинаторной оптимизации
n
m 1
i 1
i 1
n 1 
 li   di  di 1  
m 1  m 1
j 1

    d i  d i 1    d l  d l 1  ,
i  2  i 1
l 1

then a sign of optimality of a feasible schedule is:
pt. 1) tiк = diк, i = 1, m ;
pt. 2) is identical to pt. 2) of Proposition 4, and
the number k of machines that have ti,н < max tj,н is
j 1, m
the only one if the numbers di – di+1, i = 1, m  1 ,
are evenly divided by b.
Proof of Proposition 5 follows from the proof of
Propositions 3 and 4 and is illustrated by Fig. 4.
Fig. 4.
Problem 3
Given a set of independent jobs J = { 1, n } each
of which consists of a single operation with processing time lj, j = 1, n . Given due dates dj,
j = 1, n . Interruptions during the processing are not
allowed. There is one machine for the jobs processing. Jobs enter the system simultaneously.
Note. The jobs’ numbering implements fulfillment of inequalities: d1 ≤ d2 ≤ … ≤ dn.
The problem is to construct a feasible schedule
that simultaneously satisfies:
1) The moment of starting processing of the
jobs r is the maximum rmax (criterion 1);
2) The total earliness of jobs with respect to due
dates is minimal (criterion 2).
In [7] it is shown that the optimal by criterion 1
sequence is the one ordered by non-decreasing values of the due dates. In [7] an algorithm A (having
linear of n complexity) to find rmax is proposed. It’s
also shown in [7] that the upper bound on the deviation of the total earliness with respect to the due
dates of this feasible schedule from that of feasible
schedule which is optimal by criterion 2 is
i



(4)
d

r

l j i ,
  i max  l j  jmin

i

1
,
n


i 1 
j 1 



i  max  0, max l j  li  ,
 j i 1, n

where a  is the nearest lower integer from a.
Proposition 6. The sign of optimality by both
criteria for a feasible schedule is:
Pt. 1. A sequence ordered by non-decreasing
values of the due dates with rmax defined in [7] (algorithm A) is also optimal by criterion of minimization the total earliness of jobs with respect to the
due dates, if in it (4) is equal to zero.
Pt. 2. Assume (4) is not zero.
Step 1. In a feasible schedule (1, 2, ..., n) with
rmax of [7] find the minimum natural index p for
which: in the schedule (1, 2, ..., n) there are jobs
with numbers from the set {i = 1, p  1 } for which
inequality tpк ≤ di is true where tpк is the completion
time of the job p. Reorder these jobs (together with
the job p) in decreasing order of their durations. It
is shown in [7] that such rearrangement reduces the
total earliness if there are disordered jobs of different durations.
Step 2. Then we find the next (in ascending order) smallest natural index p for which the procedure described above can be implemented and a set
of jobs that will be arranged in non-decreasing order is not the same as the previous one (for further
values of the index p to previous).
Such steps may be no more than n – 1. Assume
the resulting total earliness is equal to . Then, if
n 1 
i


l j i    0 , (5)
  di  rmax   l j  jmin
i 1, n 

i 1 
j 1 

then the obtained feasible schedule is optimal for
both criteria.
Proof of Proposition 6 follows from the proof of
Theorems 4 and 5 [7].
Corollary. If a feasible schedule obtained in
Pt. 2 of Proposition 6, has the expression (5) greater than zero then the obtained feasible schedule is
strictly optimal for the first criterion, sub-optimal
for the second criteria with the upper bound of deviation from the optimal value of the second criterion given by formula (5).
Problem 4.1
Given m independent parallel machines of equal
performance working without interruptions that
process n jobs (li is the processing time of job i,
i = 1, n ). Jobs should be completed till due dates di.
The start times of machines are arbitrary. The
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problem is to construct a feasible schedule that
minimizes the following criterion:
ri1  max{ min ri } ;
i 1, n
ril  max{ min ri , i  1, n, i  jk , k  1, l  1}, l  2, n , (6)
i
where i1 is the number of the machine that has the
earliest start time in an optimal schedule (it is the
latest for all feasible schedules); il, l  2, n , is the
number of machine that has the next in value earliest start time after the machines ik, k  1, l  1 (it is
the latest for all feasible schedules with fixed
rik , k  1, l  1 ).
Obviously, the inequalities di – li ≥ 0, i = 1, n ,
are true. Renumber jobs in the non-decreasing
order of numbers di – li, and let these inequalities
satisfied: d1 – l1 < d2 – l2 < … < dn – ln.
Constructing the #1 optimality sign of a feasible
schedule.
Pt. 1. To the first machine assign as first the job
with the index 1. r1 = d1 – l1. Number r1 corresponds to the maximum value of ri1 in (6). Let the
following inequalities satisfied: di + lj > dj,
i = 1, m , j = i  1, m . Then assign to the machine j,
j = 2, m , first the job with the index j and the moment of machine starting rj = dj – lj.
Proposition 7. An arbitrary feasible schedule for
which the Pt. 1 satisfied is optimal by criterion (6).
Proof. Indeed, if in any feasible schedule an algorithmic procedure of Pt. 1 is not met, it leads
immediately to fulfillment of this:
 rl1 
 r1 
 
 
   ≽   
r 
r 
 m
 lm 
in accordance with the preorder – lexigraphical order, where r1  rm  corresponds to an arbitrary feasible schedule for which the Pt. 1 is done,

and rl 1  rl m  are start times of machines for

any feasible schedule.
Note. If in accordance with Pt. 1 all jobs are assigned on a smaller number of devices then the
corresponding rj are formally set equal to +∞.
Constructing the #2 optimality sign of a feasible
schedule.
Pt. 2. To the first machine assign as first the job
with index 1. r1 = d1 – l1. (d1 – l1 is maximum possible value of ri1 in (6)). Assume k2 – 1 is the maximum natural number for which the following inequality is true:
5
l
d1   l j  dl , l  1, k2  1 .
(7)
j 2
Then to the first machine are sequentially assigned jobs with indexes 1, 2, ..., k2 – 1. The job
with index k2 is assigned as first to the second machine. If the inequality
min{ d1 
k 2 1
 l j  lk 1, dk
2
j 2
2
 lk 2 1}  d k 2 1
is not satisfied then to the third machine as first
assigned the job with index k2 + 1 at the moment of
time r3  d k 2 1  lk 2 1 . In this case k3 = k2 + 1. Otherwise, the inequalities should be satisfied:
min{ d1 
max{ d1 
k 2 1
 l j , dk }  lk 1  dk 1 ,
j 2
2
2
2
(8)
k 2 1
 l j , dk }  lk 1  dk 1 .
j 2
2
2
2
(9)
Then the job with index k2 + 1 is assigned to the
machine corresponding to the minimum in (8). If
(8) is satisfied and (9) is violated then the optimality sign #2 of a feasible schedule for this individual
tasks is violated. Similarly, sequentially assigned to
the first or the second machine are the jobs with
indexes k 2  2, k3  1 (k3 is the maximum natural
number possible). In this case, if the current job
can be assigned to a machine with a smaller processing start time, then assigning it to a machine
with bigger processing start time should lead to a
violation of the due date (analog of the inequalities
(8), (9)). The job with the index k3 is assigned as
first to the third machine at the time r3  d k3  l k3 .
Similarly, there is a further consistent assignment of jobs with indexes k3 + j to the machines. A
necessary condition for the optimality sign #2 is
the requirement that the assigning job can be assigned to only one machine (with a minimum release time) from the current set of machines onto
which the assignment of jobs is done. If the job can
be assigned to none of the current set of machines,
it is assigned as first to the next machine in a time
equal to the due date of the job minus its duration.
The distribution of work ends either when all the
jobs are assigned to l machines (l < m) or with the
assignment to the m-th machine as first the job km,
rm  dkm  lkm . The distribution of jobs in accordance with Pt. 2 is complete.
Let J1 is a set of jobs which includes all of the
jobs with indexes from the set { 1, km } or { 1, kl } if
the jobs were distributed to l machines (l < m). Impose the following condition to be true. Let’s enu-
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Признаки оптимальности допустимых решений труднорешаемых задач комбинаторной оптимизации
merate all the jobs from the set J1 in the order of
their assignment to the machines. Then for each
job with the index j ( j  2, km  kl ) should be satisfied:
dj – tjк < lp, p  l  1, km  kl ,
(10)
where tjк is the completion time of the job j. Then
the following proposition is true:
Proposition 8. An arbitrary feasible schedule
that satisfies Pt. 2 and the condition (10) is optimal
by criterion (6), i.e.
 rl1 
 r1 
 
 
(11)
   ≽   
r 
r 
 m
 lm 
in accordance with the preorder – lexigraphical or
der, where r1  rm  corresponds to an arbitrary feasible schedule for which the Pt. 2 and the

condition (10) are fulfilled, and rl 1  rl m  are
the processing start times of machines for any feasible schedule.
Note 1. If, in accordance with Pt. 2 became
loaded l machines (l < m) then in (11) rl+j,
j  1, m  l , are formally set to the values +∞.
Indeed, from the logic of the jobs distribution to
machines (Pt. 2), and also in conjunction with the
fulfillment of the conditions (10), (8)  (9) and
their analogues at the subsequent stages of distribution, it follows that any change in the order of assignment of jobs (except cases when in (8), (9) and
their analogues the minimum is not the only one)
leads to degradation in lexigraphical order of the
processing start times of the machines.
Note 2. If all of the jobs are distributed to l machines, l < m, then the Pt. 2 in the Proposition 7 is
finished with the assignment as first to the l-th machine the job with the number kl.
Note 3. Signs of optimality #1 and #2 let us construct the polynomial component of the PDC-algorithm for the Problem 4.1: the algorithm of polynomial complexity is constructed to obtain a feasible schedule in which jobs from a set J1 are preassigned in accordance with the Proposition 8
(with regard to Note 2). In this case, the algorithm
must take into account when assigning not yet distributed jobs the time reserves dj – tjк,
k  1, km  kl , that remained after the assignment
of the jobs of the set J1 according to the Proposition 8. If the polynomial algorithm has built a feasible schedule then for it the optimality sign (Proposition 8) is true, and this schedule is optimal by
lexigraphical criterion (6). If it fails to build a fea-
sible schedule then the Problem 4.1 is solved by
the exponential component of PDC-algorithm or an
approximate or heuristic polynomial algorithm that
approximates the exponential component of the
PDC-algorithm.
Note 4. Let’s present one of the most statistically effective strategies for constructing an approximate polynomial algorithm. An approximation algorithm is identical to the polynomial part of the
PDC-algorithm (Note 3) except that the requirement is removed in the optimality sign #2 that a job
can be assigned to only one machine of the current
set of machines (conditions (8), (9) and their generalization). That is, the current job is assigned to
the machine with the smallest release time, though
perhaps it could be assigned as first without violation of its due date to other machines from the current set. It’s impossible to claim that in this case a
feasible schedule is optimal, but with the fulfillment of the condition (10) it leads to the fact that if
the current job is not assigned to the machine with
a minimum release time, it can only be assigned
only as first to one of the current machines set.
And this means that the number of feasible variants
of the initial schedule is considerably reduced. It is
obvious that in the simulation of arbitrary individual problems, for which the condition (10) is satisfied, the logic of the algorithm to assign a next job
to the machine from the current set of machines
with the minimum release time almost always
leads to the fulfillment of the condition (11). Thus,
constructing a feasible schedule with modified in
such way the optimality sign #2 (the empirical sign
of optimality) almost always leads to a strict solution of the Problem 4.1 by lexigraphical criteria
(6).
Problem 4.2
It differs from the Problem 4.1 in that for jobs
with indexes from the set J1  J = { 1, n } the limitations for the completion time of the jobs tlк
l = 1, n are not the due dates dl (tlк ≤ dl), but the
limitations in the form tlк  [dl – l, dl] l  J1, see
[5], l > 0.
In particular, if l is small then this restriction
corresponds to the practical implementation of the
formal restriction to complete the job just in time
(tlк = dl).
For the Problem 4.2 the optimality sign #1 remains unchanged (Proposition 7). The optimality
sign #2, the Proposition 8, as well as the empirical
sign of optimality change in the following way.
Either indexes of all the jobs assigned in accordance with Pt. 2 to the machines, beginning from the
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second machine, do not belong to J1, or for these
jobs this is satisfied:
tlк  [dl – l, dl] l  J1.
Problem 5.1
There are m independent parallel machines of
different performance running without interruption
that execute n jobs ( li j is the processing time of the
job i at the machine j). The jobs should be completed by the due dates di. Processing start times of
the machines are arbitrary. The problem is to build
a feasible, optimal by criterion (6) schedule.
For the problem 5.1 we can obviously generalize the optimality sign #1 of a feasible schedule
proposed for the Problem 4.1.
Constructing the #1 optimality sign of a feasible
schedule.
Pt. 1. Consider the following monotonic nondecreasing sequence of numbers:
di1  li1j1 , di2  li2j2 , ..., dim  limjm , where
di1  li1j1  min {( di  min li j ), i  1, n, j  1, m} (15)
i
j
jp
di p  li p  min {(di  min li j ), i  1, n, j  1, m,
i
7
dil  li p  di p p  l  1, m, l  1, m  1 .
jl
Then the following proposition is true:
Proposition 9. An arbitrary feasible schedule,
in which the job il is assigned as first to the machine jl, l  1, m , at the time r jl  dil  liljl , is an optimal schedule by criterion (6).
Problem 5.2
This is a generalization of the Problem 5.1,
identical to the generalization of the Problem 4.1 to
the Problem 4.2.
For the Problem 5.2 the optimality sign of a feasible schedule (Proposition 9) remains valid. Indeed, all ri  [di – l, di], i  1, m , by construction.
Conclusions
The paper presents the basics of the Theory of
PDC-algorithms. New formulations of the singlestage scheduling problems were made. The optimality signs of a feasible schedule for them were
found which are the basics of PDC-algorithms constructing.
j
i  {i1 , i p 1}, j  { j1, j p 1}}
In this case, at each machine achieved only one
minimum. Let all the inequalities are satisfied:
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