Approximating the Conguration-LP for
Minimizing Weighted Sum of Completion Times
on Unrelated Mahines
Maxim Sviridenko
University of Warwik
⋆
and Andreas Wiese
⋆⋆
MPI für Informatik, Saarbrüken
Abstrat. Conguration-LPs have proved to be suessful in the design
and analysis of approximation algorithms for a variety of disrete optimization problems. In addition, lower bounds based on onguration-LPs
are a tool of hoie for many pratitioners espeially those solving transportation and bin paking problems. In this work we initiate a study of
linear programming relaxations with exponential number of variables for
unrelated parallel mahine sheduling problems with total weighted sum
of ompletion times objetive. We design a polynomial
time approximaP
tion sheme to solve suh a relaxation for R|rij | wj Cj and a fully
P polynomial time approximation sheme to solve a relaxation of R|| wj Cj .
As a byprodut of our tehniques we derive a polynomial time approximation sheme for the one mahine sheduling problem with rejetion
penalties, release dates and the total weighted sum of ompletion times
objetive.
1 Introdution
In unrelated parallel mahine sheduling we are given a set of
a set of
n
jobs. Eah job
i, i.e. it
wj ∈ N, and
j
m
mahines and
pi,j ∈ N for
i, by
mahine i. The goal
is haraterized by a proessing time
pi,j
eah mahine
takes
time units to proess job
a weight
by a release time
ri,j ∈ N
for eah
j
on mahine
is to assign the jobs to the mahines and to dene a non-preemptive shedule
ri,j
or
later. Eah mahine an proess at most one job at a time. Given a shedule
S,
for eah mahine, suh that on every mahine
we denote by
Cj (S)
i
eah job
the ompletion time of eah job
j.
j
starts at time
We write
Cj
for short if
the shedule S is lear from the ontext. The objetive is to ompute a shedule
P
S ∗ whih minimizes the total weighted sum of ompletion times j wj · Cj (S ∗ ).
Using the standard sheduling notations [6℄ this problem is ommonly denoted
as
R|rij |
P
wj Cj .
Unrelated parallel mahines sheduling is one of the basi sheduling models that is extensively studied by the researhers both from experimental and
⋆
⋆⋆
M.I.Sviridenkowarwik.a.uk,
Researh
supported
by
EPSRC
grant
EP/J021814/1, FP7 Marie Curie Career Integration Grant and Royal Soiety
Wolfson Researh Merit Award.
awiesempi-inf.mpg.de
theoretial viewpoints. Various appliations ditate dierent objetives suh as
makespan, total throughput et. For the problem with the total weighted sum
of ompletion times objetive Shulz and Skutella [15℄ and Skutella [17℄ design
2-approximation
algorithms for the general problem and
gorithms for the problem where all
rij = 0,
3/2-approximation
al-
i.e. when all jobs are released at
time zero (the standard notation for this sheduling model is
R||
P
wj Cj ). Their
algorithms are based on time indexed linear programming [15℄ and onvex programming [17℄ relaxations.
On the other side, reent improvements of the performane ratios for unrelated parallel mahine sheduling problems with other objetives using linear programming relaxations with exponential number of variables (so-alled
onguration-LPs) motivated us to onsider and study suh onguration linear
programming relaxations for
R|rij |
P
wj Cj .
In partiular, for the restrited assignment speial ase of the unrelated parallel mahine sheduling problem with makespan objetive Svensson [19℄ showed
that the onguration-LP has an integrality gap stritly better than
2, improving
upon a long standing result by Lenstra, Shmoys and Tardos [12℄ that was based
on a generalized assignment linear programming relaxation. Also many reent
results for the Santa Claus problem, i.e. unrelated parallel mahine sheduling
problem with maxmin objetive, are based on onguration-LPs [2,3,10℄.
In addition to that, many sophistiated transportation problems are solved in
pratie by using onguration linear programming relaxations (see e.g. [9℄). The
following meta-algorithm is the algorithm of hoie used by many pratitioners:
1. formulate your problem using an exponential number of variables where
eah variable enodes a non-trivial piee of the solution spae (truk route,
shedule on one mahine et.);
2. generate a set of variables (or "olumns") to onsider by running a set of
heuristis and solve the linear program orresponding to this subset of variables;
3. x some of the variables to zero or one and repeat the proess.
This heuristi algorithm performs amazingly well in pratie for a wide variety
of problems whih indiates the high quality of onguration-LPs as relaxations
of the original problem at hand.
In this paper, motivated by the above onsiderations, we dene onguration linear programming relaxations for
R|rij |
P
wj Cj . The rst question is how
to solve suh relaxations. We use the well-known onnetion between separation and optimization [7,8,13℄. We dene a dual linear program and notie that
the separation problem for the dual orresponds to an interesting NP-hard one
mahine sheduling problem with rejetions. Similar sheduling problems were
onsidered before in the literature [5,11,16℄. Unfortunately, known tehniques
annot be applied for our sheduling problem with rejetion penalties sine to
design an approximate separation orale we are allowed to relax only the piee of
objetive funtion orresponding to the weighted sum of ompletion times (but
not the rejetion penalties).
2
We explain the onnetion between approximate separation and solving our
relaxation in the next setion. The main result of this paper is a polynomial time
Reall, that a PTAS is a olletion of algorithms suh that for
P
R|rij | wj Cj .
any ε > 0 there
(1 + ε)-approximation algorithm in the
olletion. Suh
approximation sheme for solving onguration-LP relaxation of
exists a polynomial time
a sheme is alled fully polynomial time approximation sheme (FPTAS) if the
dependene on
1/ε
is polynomial. In addition to our main result we design an
FPTAS for solving onguration-LP relaxation of
R||
P
wj Cj .
We onjeture that the worst ase integrality gaps of our linear programming relaxations are stritly better than the worst ase integrality gaps of the
linear programming relaxations for
R|rij |
P
wj Cj
and
R||
P
wj Cj ,
previously
onsidered in the literature [15,17℄. We also believe that suh new relaxations
will be instrumental in building new pratial algorithms for a wide variety of
sheduling problems the same way as they were instrumental in solving many
pratial transportation problems.
1.1 Our Contribution
For any
ǫ>0
we give a polynomial time algorithm whih omputes a
approximation of the onguration-LP relaxation of
R|rij |
P
wj Cj .
(1 + ǫ)-
Key to this
is a polynomial time approximation sheme for the separation problem of the
problem with rejetion penalties and release dates,
Psheduling P
w
C
+
S j j
S̄ ej in the three-eld notation. In that problem,
one is given a mahine and a set of jobs J as above, where additionally eah
′
job j has a rejetion penalty ej . The goal is to selet a subset J ⊆ J for whih
P
we onstrut a shedule S . The objetive is to minimize
j∈J ′ wj · Cj (S) +
P
e
.
In
other
words,
we
an
deide
to
rejet,
i.e.,
not
to shedule some
j∈J\J ′ j
job j but then we have to pay ej as a penalty. Of ourse, this problem is related
P
to the same problem without rejetion, i.e., 1|rj |
j wj Cj , for whih a PTAS
dual whih is the
denoted by
1|rj |
is known [1℄. The latter algorithm is ruially based on the fat that we an
assume that not too many jobs are released at the same time. Roughly speaking,
P
j wj Cj many jobs have the same release date then
we an postpone the release of some jobs with e.g., jobs with small weight,
if in an instane of
1|rj |
sine in an optimal solution they are not sheduled straight away. However, if
rejetion is allowed this argument breaks down ompletely sine in an optimal
shedule jobs with high weight might be rejeted and ones with smaller weight
might be sheduled immediately. Note that sheduling problems with rejetion
penalties were onsidered before but all known tehniques are not appliable
for our purposes (exept a pseudo-polynomial algorithm [5℄ for
P
S̄ ej ).
1||
P
S
wj Cj +
Hene, new methods are needed to solve the separation problem. First, like
in [1℄ we split the time horizon into intervals of subsequent powers of
we show that at the loss of a fator of
1 + O(ǫ)
1 + ǫ. Then
in the objetive we an split the
whole problem into disjoint subproblems whih span
O(log n)
intervals eah. In
suh a subproblem, we enumerate over the patterns given by the big jobs and
the spae for small jobs in the optimal solution (big and small refers to the size
3
of a job with respet to the interval in whih it is sheduled). The number of
possible suh patterns is bounded by a polynomial. Note that enumerating the
assignment of the big jobs diretly would yield a quasi-polynomially number of
options whih is too muh. Then we use a linear program to assign the jobs into
the slots. In partiular, we use an LP to deide whih jobs are big and whih jobs
are small in the optimal solution (that is a very important piee of information).
We believe that our new tehniques extend the understanding of sheduling
problems with a sum-of-ompletion-time objetive. Sine we do not use any
properties that rely on the rejetion ost term in our objetive funtion, our
methods might be useful for other settings as well.
2 The Conguration-LP
(1 + ǫ)-approximation
Our goal is to onstrut a
algorithm for solving the
onguration-LP for minimizing the weighted sum of ompletion times on unrelated mahines. Given an instane of the problem, we denote by
set of given jobs and mahines, respetively. For eah mahine
S(i)
the set of all feasible shedules for mahine
S
P
:= j∈J ′ wj · Cj (S).
jobs). For eah shedule
Wi,S
for some set of jobs
J′
i
i
J
and
M
a
we denote by
(for any subset of the given
on some mahine
i
we dene
The onguration-LP, or C-LP for short, is then dened by
min
X X
yi,S · Wi,S
i∈M S∈S(i)
X
yi,S ≤ 1
∀i ∈ M
yi,S ≥ 1
∀j ∈ J
yi,S ≥ 0
∀i ∈ M, S ∈ S(i)
S∈S(i)
X
X
i∈M S∈S(i):j∈S
where we write
j∈S
if job
j
arises in
S.
The onguration-LP has only a linear number of onstraints but an exponential number of variables. Hene, we annot solve it diretly. Therefore, instead
we solve the dual via the ellipsoid method and a polynomial time separation
routine. The dual of the onguration-LP is given by
max
X
βj −
j∈J
−αi +
X
αi
X
βj ≤ Wi,S
∀i ∈ M, ∀S ∈ S(i)
αi ≥ 0
βj ≥ 0
∀i ∈ M
∀j ∈ J.
i∈M
j∈S
4
(1)
i and given values
α
and βj , we either want to nd a shedule S ∈ S(i) suh that
i
P
−αi + P j∈S βj > Wi,S or assert that for eah shedule S ∈ S(i) it holds that
−αi + j∈S βj ≤ Wi,S . This problem is equivalent to the problem of sheduling
In the separation problem for the dual, for eah mahine
for variables
with rejetion on one mahine to minimize the weighted sum of ompletion
time plus the sum of the rejetion penalties, where for eah job
penalty
ej
equals
βj . It is NP-hard (as already 1|rj |
P
wj Cj
j
the rejetion
is NP-hard). Similar
sheduling problems were studied before under the name of sheduling with
rejetion [5℄. However, for the purpose of approximating the onguration-LP
up to an error of
1 + O(ǫ)
we use the following strategy: rst, we introdue
some modiations of the instanes and the shedules under onsideration whih
1+O(ǫ) in the objetive. Then
simplify the struture and ost at most a fator of
we formulate a relaxed linear program C'-LP with the properties that
the optimal solution of C'-LP is by at most a fator of
1 + O(ǫ)
larger than
the optimal solution of C-LP ,
C'-LP an be solved optimally in polynomial time using a separation orale,
and
any feasible solution of C'-LP an be transformed into a feasible solution of
C-LP at the ost of at most a fator of
1+ǫ
in the objetive.
We start with simpliations of the input and the onsidered shedules.
3 Restritions for Input and Considered Congurations
Let
ǫ>0
and assume for simpliity that
1/ǫ ∈ N.
We prove that we an assume
some properties for the input and the shedules under onsideration while losing
1 + O(ǫ)
only a fator of
writing
Oǫ (f (n))
in the objetive. We extend the big-O notation by
for funtions whih are in
O(f (n))
if
ǫ
is a onstant. E.g.,
Oǫ (1)
denotes onstants whih might depend on ǫ.
x
We dene Rx := (1 + ǫ) and an interval Ix = [Rx , Rx+1 ) for eah integer
|Ix | = ǫ · Rx . In
1 + ǫ. This means
x.
Observe that
the sequel, several times we will streth time by
some fator
that we take a given (e.g., optimal) shedule and
shift all work done in every interval
Sine then every job
j
is proessed for
[a, b) to the interval [(1 + ǫ)a, (1 + ǫ)b).
(1 + ǫ)pj time units, we gain slak in the
shedule whih we an use in order to obtain ertain properties. We will write at
1+ǫ
loss or at
1 + O(ǫ)
loss if we an assume a ertain property for the input
or the onsidered shedules by strething time by a fator of
1+ǫ
or
1 + O(ǫ),
respetively.
Proposition
1. At 1+ǫ loss we an work with
P
Pthe alternative objetive funtion
j∈S
wj · min {Rx : Cj (S) ≤ Rx }
instead of
j∈J
wj Cj (S).
In the next lemma we round the input data and establish thatintuitively speaking
large jobs are released late (sine they have a relatively large ompletion time
anyway).
5
Lemma 1 ([1℄). At 1 + O(ǫ) loss we an assume that all proessing times and
release dates are powers of 1 + ǫ and rj ≥ ǫ · pj for eah job j .
Let S be a shedule. We dene a job j to be large in S if j starts in an interval
Ix suh that pj > ǫ ·Ix and small in S otherwise. In the following lemmas we will
streth time several times in order to gain free spae that we will use in order
to enfore ertain properties of the shedules under onsideration. A tehnial
problem is that when we streth time by a fator
beome small. To avoid this, we streth time
that is needed by
all
one
1+ǫ
then a large job an
O(1)
by the total fator
(1 + ǫ)
subsequent lemmas. In the resulting shedule we lassify
jobs to be small or large as dened above. We will write in the statements of the
1 + ǫ loss when we mean that we use an extra spae of
Ix in order to ensure some property. In fat, when strething
1 + O(ǫ) we gain an idle period of total length ǫ · Ix only in
subsequent lemmas at
ǫ·Ix
in eah interval
time by a fator
intervals
Ix
where a job nishes. However, it will turn out that only in those
intervals we need this extra spae.
Lemma 2. At 1 + ǫ loss we an restrit to shedules where for eah interval Ix
eah large job j , starting during Ix , is started at a time t = Rx (1 + k · ǫ3 )
for some k ∈ {0, ..., ǫ12 }, and
there is a time interval [a, b) ⊆ Ix in whih no large jobs are sheduled and
no small jobs are sheduled in Ix \ [a, b) (note that the interval [a,b) ould be
empty).
Proof.
We rst ensure the seond property by moving the large and small jobs
of the shedule for
interval
[a, b) ⊆ Ix .
Ix
suh that all small jobs are sheduled in a (onseutive)
Note that sine we hanged the objetive funtion (Proposi-
tion 1) this does not inrease the objetive value. In eah interval Ix there an
Ix
ǫ·Ix = 1/ǫ jobs that start in Ix as large jobs. Then, using a free spae
1
3
of at most ǫ Rx ǫ = ǫ · Ix , we move the start time of eah large job to the next
value t of the form of the lemma statement.
⊓
⊔
be at most
In the next lemma we establish that eah job is pending for at most
Oǫ (log n)
intervals.
Lemma 3. At 1 + ǫ loss we an assume that there is an integer integer K ∈
Oǫ (log n) suh that eah
Ix+K the latest.
job released at some time Rx nishes during the interval
Proof.
1
Reall that due to Lemma 1 we have pj ≤
ǫ · rj for eah job j . Hene,
ǫ
K
pj ≤ n · rj · (1 + ǫ) for some integer K ∈ Oǫ (log n). Sine at most n jobs are
released at eah time Rx , all these jobs t into an empty spae of ǫ · Ix+K in the
interval
Ix+K .
⊓
⊔
1
The ruial part is now to deouple the instane into bloks of O( ǫ log n) onseutive intervals eah suh that eah blok represents a subinstane whih is
independent of all the other bloks.
We will use the next lemma to identify groups of
C
onseutive intervals
eah, suh that eah two groups are separated by at least
6
c
intervals and all
intervals
not
in some group ontribute only negligibly towards the objetive.
For any desired separation
suitable value
c
and any bound
δ
on the ontribution we an nd a
C.
While the next lemma works for any values
1
the reader may think of c = O(log n) and C = O( log n).
ǫ
c
and
δ,
for later
Lemma 4. Consider any (frational) solution to C-LP. For every δ > 0 and
every integer c there exist a value C ∈ O( 1δ c) and an oset a ∈ {0, ..., c + C}
suh that all jobs released or sheduled during an interval Ix with a + k · C ≤
x ≤ a + k · C + c for some integer k ontribute only a δ -fration to the overall
objetive.
Proof.
Can be shown using the pigeon hole priniple.
⊓
⊔
c := K + L where L is the smallest integer suh that ǫ12 ≤ (1 + ǫ)L .
For a value δ ∈ Oǫ (1) to be dened later let C and a denote the values given
1
by Lemma 4 for c and δ . Note that sine c + C ∈ O( c) we an try all possible
δ
values for a in polynomial time. For the remainder of our reasoning we assume
that we guessed a orretly. We all an interval Ix a gap-interval if a + k · C ≤
x ≤ a + k · C + c for some integer k .
We dene
Lemma 5. For any ǫ > 0 there is a value δ0 > 0 suh that if δ ≤ δ0 then
at 1 + O(ǫ) loss we an assume that in eah gap-interval only small jobs are
exeuted.
Proof.
By strething time one by a fator of
Ix+L
Ix by L intervals to
L we have that Ix = ǫRx ≤ ǫ2 ·Ix+L .
We shift the large jobs sheduled in eah gap-interval
the future. Observe that due to the hoie of
to t all large jobs nished in
Ix
1 + 2ǫ we gain enough spae in the interval
(we need at most Rx + Ix ≤ 2ǫIx+L time).
Sine we move only large jobs, it still holds that eah job released at a time
Rx nishes during the interval Ix+K the latest. By hoosing δ to be at most
δ0 := ǫ/(1 + ǫ)L then δ · (1 + ǫ)L ≤ ǫ and the inrease of the total ost is bounded
by ǫ · OP T .
⊓
⊔
For any integer
lie in the same
k
we say that all intervals
gap-blok.
Ix
with
a+k·C ≤ x≤ a+k·C +c
Lemma 6. For any ǫ > 0 there is a value δ1 > 0 suh that if δ ≤ δ1 then at 1+ǫ
loss we an enfore that at the end of eah gap-interval Ix there is an auxiliary
interval of length ǫ·Ix . All jobs released during the gap-blok and proessed within
the same gap-blok are only allowed to be proessed in the auxiliary intervals.
These auxiliary intervals are not allowed to proess any other job. Also, eah job
nishes at most K + L intervals after its release.
Proof.
B = {Ia+k·C , ..., Ia+k·C+c }. Denote by JB all (small)
B . Similarly as in Lemma 5 we
shift them by L = Oǫ (1) intervals to the future suh that all jobs from JB ,
sheduled during an interval Ix ∈ B , have a total volume of ǫ · Ix+L , assuming
⊓
⊔
an appropriate upper bound for δ .
Consider a gap-blok
jobs whih are released and sheduled during
7
We hoose
bloks,
δ := min{δ0 , δ1 }. The above lemmas split the overall problem into
j with Ra+k·C ≤ rj < Ra+(k+1)·C for some integer k . In
one for all jobs
the next denition we summarize the problem we are faing in eah blok.
Denition 1 (Blok-Problem). We are given m unrelated mahines, a set of
jobs J and an integer k suh that Ra+k·C ≤ rj < Ra+(k+1)·C for all j ∈ J . We
want to nd a feasible shedule whih on eah mahine
during eah interval Ix ∈ {Ia+k·C , ..., Ia+k·C+c − 1} may use only an interval
of length ǫ · Ix at the end of Ix (and may shedule only small jobs there),
during eah interval Ix ∈ {Ia+k·C+c , ..., Ia+(k+1)·C − 1} may use the entire
interval Ix , and
during eah interval Ix ∈ {Ia+(k+1)·C , ..., Ia+(k+1)·C+c−1 } may use the entire
interval Ix apart from an interval of length ǫ · Ix at the end of Ix and may
shedule only small jobs during Ix .
The objetive is to minimize the weighted sum of ompletion times.
Now eah integer
k
indues a blok of the above form.
Lemma 7. If there is a polynomial time (1 + ǫ)-approximation algorithm for
solving the onguration-LP for the blok-problem then there is a polynomial
time (1 + ǫ)-approximation algorithm for solving the overall onguration-LP.
4 A Relaxation of the Conguration-LP
For eah mahine
i
denote by
S ′ (i) ⊆ S(i)
the set of shedules obeying the
restritions dened in Setion 3. Reall that this restrition osts us only a
fator of
1 + O(ǫ) in the objetive. Sine we split the overall problem into disjoint
bloks, it sues to be able to solve the onguration-LP for one single blok
(see Lemma 7).
For solving the separation problem we relax the notion of a onguration
S ′′ (i) ⊇ S ′ (i).
and in partiular enlarge the set of allowed ongurations to a set
Then, we will show that we an solve the resulting separation problem exatly.
′′
We will denote by C'-LP the onguration-LP using ongurations in S (i).
Finally, we show that when given a solution of C'-LP, while losing only a fator
of
in
1+ǫ
S(i).
we an ompute a solution to C-LP, i.e., whih uses only ongurations
The rst important observation is that for eah interval
onstantly many possible
set of
O(ǫ)
for the big jobs. A pattern
Ix there are only
P for big jobs is a
integers whih denes the start and end times of the big jobs whih
are exeuted during
after
patterns
Ix .
Note that suh a job might start before
Ix
and/or end
Ix .
Proposition 2. For eah interval Ix there are only N ∈ Oǫ (1) many possible
patterns. There are only N Oǫ (log n) ∈ O(poly(n)) possible ombinations for all
patterns in one blok together.
8
Note that a pattern for an interval alone does not dene what exat job is
exeuted during
Ix ,
it desribes only the start and end times of the big jobs.
S ′′ (i). It ontains eah fra-
Now we dene the relaxed set of ongurations
tional job assignment whih an be obtained with the following proedure. Fix a
pattern
Px
for eah interval
pattern. Denote by
eah interval
interval
It .
It
Q(P)
denote by
Ix
in the blok and denote by
P
the overall (global)
the set of slots for big jobs whih are given by
rem(t)
P.
For
the remaining idle time for small jobs in eah
We allow any frational assignment of jobs to slots and the idle time
in the intervals whih
assigns at most one frational unit of eah job,
assigns at most one frational unit of big jobs to eah slot,
assigns small jobs frationally to the idle time of eah interval
exeeding
It ,
while not
rem(t).
Formally, we allow any feasible solution to the following linear program (the term
size(s)
denotes the length of the slot
X
xt,j +
t
X
s
and
begin(s)
denotes its start time).
xs,j ≤ 1
∀j ∈ J
(2)
xs,j ≤ 1
∀s ∈ Q(P)
(3)
∀t
(4)
s∈Q(P)
X
j∈J
X
pj · xt,j ≤ rem(t)
j∈J
xt,j ≥ 0
∀t ∀j ∈ J : rj ≤ Rt ∧ pj ≤ ǫ · It
xs,j ≥ 0
∀s ∈ Q(P), ∀j ∈ J :
pj ≤ size(s) ∧ rj ≤ begin(s).
Note that we introdue a variable
is small during
It .
xt,j
only if
j
is available at time
Similarly, we introdue a variable
is available in the interval where
So for eah mahine
i,
s
xs,j
only if
j
Rt
ts into
(5)
and
s
j
and
starts.
eah global pattern
P
to the above LP we introdue a onguration in
and eah frational solution
S ′′ (i) (formally, there is an
innite number of feasible solutions to the above LP but we are only about
basi solutions or verties it the orresponding polyhedron). We denote by C'LP the onguration-LP we obtain by taking C-LP as dened in Setion 2 but
′′
allowing the ongurations in S (i), rather than the ongurations in S(i). For
P
′′
a onguration in S ∈ S (i) we dene its weight Wi,S :=
j∈J xt,j · Rt+1 +
P
P
j∈J
s∈Q(P) xs,j ·end(s) where for eah slot
time of the interval in whih slot s ends.
s we denote by end(s) the nishing
Lemma 8. The optimal solution of C'-LP is by at most a fator of 1 + O(ǫ)
larger than the optimal solution of C-LP.
Proof.
S ′ (i) (rather than allowing all ongura′
′′
tions in S(i)) loses at most a fator of 1 + O(ǫ) in the objetive. As S (i) ⊆ S (i)
Restriting to ongurations in
9
allowing all onguations in
S ′′ (i)
rather than only the ones in
S ′ (i)
does not
⊓
⊔
lose anything in the objetive.
′′
The benet of allowing all ongurations in S (i) (rather than only ongura′
tions in S (i)) is that we an solve the separation problem of the dual exatly.
′′
When separating the dual, we need to either nd a shedule S ∈ S (i) suh that
P
P
P
s∈Q(P) xs,j ) > Wi,S or asserts that no suh shedule
t xt,j +
j∈S βj (
exists. For eah mahine i we do the following: we enumerate all patterns for the
′′
big jobs. For eah pattern P , we nd the onguration S ∈ S (i) whih follows
−αi +
P
for eah mahine
i
P
P
P
βj (
j∈S
and eah pattern
and whih optimizes
t
P
P
s∈Q(P) xs,j ) − Wi,S . Formally,
we solve the above LP with the linear
xt,j +
objetive funtion
max
X
j∈J
X
X
X X
X
xs,j · end(s).
xs,j ) −
βj (
xt,j · Rt+1 −
xt,j +
t
s∈Q(P)
We all the overall linear program the
j∈J
j∈J s∈Q(P)
Slot-LP. We remark that a similar LP has
reently been used in [4℄.
Lemma 9. For eah ǫ > 0 there is a polynomial time algorithm whih solves
the separation problem of C'-LP exatly.
5
Feasible Solution to the Original Conguration-LP
Sine we an solve the dual of C'-LP exatly in polynomial time (using the
ellipsoid method together with our separation orale), we an also ompute in
polynomial time an optimal solution of C'-LP itself. It remains to show that any
solution to C'-LP an be transformed to a feasible solution to C-LP (i.e., using
S(i) for eah mahine i) while losing at most a fator of
1+ǫ. To ahieve this we show that by taking eah onguration S ∈ S ′′ (i) arising
only ongurations in
in the omputed solution for C'-LP and replaing it by a set of ongurations
in
S(i),
eah with a suitable oeient
yi,S .
We hoose these ongurations and
oeients suh that eah job is still assigned to the same extent as in
the total ost inreases at most by a fator of
S
and
1 + ǫ.
The main step is to prove the following lemma.
Lemma 10. Let S ∈ S ′′ (i) be a onguration, dened by a solution x to the
SP
Slot-LP. In polynomial time we an ompute a set of ongurations
1 , ..., SB
P
λ
=
and
oeients
λ
,
...,
λ
suh
that
for
eah
job
j
we
have
ℓ
1
B
t xt,j +
ℓ:j∈S
ℓ
P
P
P
s xs,j and
ℓ λℓ · Wi,Sℓ ≤ (1 + ǫ) · Wi,S and
ℓ λℓ = 1.
S ∈ S ′′ (i), desribed by a global
pattern P and a vetor x for the Slot-LP. We interpret S as a frational mathing
in a bipartite graph. Here we borrow some ideas from [12℄. For eah job j arising
in S we introdue a vertex vj . For eah slot s for a big job we introdue a vertex
ws . If in S some job j is (frationally) assigned to s then we add the edge (vj , ws )
Consider a mahine
i
and a onguration
10
jl to ws . For
m eah interval It with
P
idle time for small jobs, we introdue kt :=
verties wt,1 , ..., wt,kt ,
j xt,j
representing this idle time. For dening the edges of the verties wt,ℓ we do the
following proedure: assume that the jobs (frationally) assigned to It as small
jobs are labeled {1, ..., nt } and they are ordered by non-inreasingly by proessing
with weight
wj · end(s) and
assign
xs,j
units of
times. We iterate over the jobs in this order. While doing this we maintain the
ℓ suh that all verties wt,1 , ..., wt,ℓ−1 have one
wt,ℓ has α units of jobs assigned to
it for some value α ∈ [0, 1), and the verties wt,ℓ+1 , ..., wt,kt have no job assigned
to them. Consider a job j . If xt,j ≤ 1 − α then we assign j ompletely to wt,ℓ
and introdue an edge (vj , wt,ℓ ) with weight wj · Rt+1 . If xt,j > 1 − α then we
assign α units of j to wt,ℓ and the remaining xt,j − α units to the next vertex
wt,ℓ+1 . In that ase we introdue edges (vj , wt,ℓ ) and (vj , wt,ℓ+1 ) with weights
wj · Rt+1 and wj · Rt+2 , respetively. Denote by G the resulting graph.
invariant that there is some value
(frational) unit of jobs assigned to it, vertex
Lemma 11. For any integral mathing M in G there is a shedule S ′ ∈ S(i)
whose ost is at most by a fator 1 + ǫ larger than the weight of M . Given M ,
the orresponding shedule S ′ an be omputed in polynomial time.
Using the above lemma, we an ompute a onvex ombination of shedules
in
S(i)
whose total ost is not muh bigger than the ost of
S
and whih assigns
(frationally) the same jobs to the same extent.
Proof (of Lemma 10). From mathing theory we know that the bipartite mathing polytope is integral. This implies that the frational assignment indued by
S
an be written as a onvex ombination of integral mathings
eah having a suitable oeient
λℓ ,
M1 , ..., MB ,
see e.g., [14℄. This representation an be
omputed in polynomial time. For eah mathing
Mℓ
we dene
Sℓ ∈ S(i) to be
Sℓ and the
⊓
⊔
the resulting shedule as given by Lemma 11. Then the shedules
oeients
λℓ
have the properties laimed in the lemma.
Using Lemma 10 we ompute a solution
y
to C'-LP. Initially, we set
ȳ := 0.
ȳ
for C-LP, given we have a solution
i. For eah S ∈ S ′′ (i)
Consider a mahine
yi,S > 0, we ompute the onvex ombination S1 , ..., SB with oeients
λ1 , ..., λB aording to Lemma 10. Then for eah ℓ ∈ {1, ..., B} we inrease the
variable ȳi,Sℓ by λℓ · yi,S . Hene, we obtain our main theorem.
with
Theorem 1. For any ǫ > 0 there is a polynomial time algorithm whih omputes
a (1 + ǫ)-approximation of the onguration-LP for sheduling jobs on unrelated
mahines to minimize the weighed sum of ompletion time.
Using these tehniques, we obtain a PTAS for the sheduling problem with
rejetion on one mahine. The details are in the full version of this paper.
Theorem
2. There
P
P is a polynomial time approximation sheme for the problem
1|rj |
S
wj Cj +
S̄
ej .
For the setting where all jobs have equal release dates there is an optimal
pseudopolynomial time algorithm and an FPTAS for the orresponding problem
11
of sheduling with rejetion [5℄. This an be turned into an FPTAS for solving
the onguration-LP in that setting.
Theorem 3. For any ǫ > 0 there is an algorithm with running time O(poly(n, 1ǫ ))
whih omputes a (1 + ǫ)-approximative solution to C-LP if all jobs are released
at time t = 0.
Referenes
1. F. Afrati, E. Bampis, C. Chekuri, D. Karger, C. Kenyon, S. Khanna, I. Milis, M.
Queyranne, M. Skutella, C. Stein and M. Sviridenko, Approximation Shemes for
Minimizing Average Weighted Completion Time with Release Dates, in Proedings
of Symposium on Foundations of Computer Siene (FOCS 1999), pp.32-43.
2. A. Asadpour, U. Feige and A. Saberi, Santa laus meets hypergraph mathings,
ACM Transations on Algorithms 8(3), pp. 1 24 (2012).
3. N. Bansal and M. Sviridenko, The Santa Claus problem, in STOC 2006, pp. 31-40.
4. V. Bonifai and A. Wiese. Sheduling unrelated mahines of few diferent types.
CoRR abs/1205.0974, 2012.
5. D. Engels, D. Karger, S. Kolliopoulos, S. Sengupta, R. Uma and J. Wein, Tehniques for sheduling with rejetion, J. of Algorithms 49(1), pp. 175-191 (2003).
6. R. Graham, E. Lawler, J.Lenstra, and A. H. G. Rinnooy Kan (1979). Optimization
and approximation in deterministi sequening and sheduling: a survey. Annals
of Disrete Mathematis 5, 287326.
7. M.D. Grigoriadis, L.G. Khahiyan, L. Porkolab and J. Villavienio. Approximate
max-min resoure sharing for strutured onave optimization. SIAM Journal on
Optimization, 11:10811091, 2001.
8. M. Grötshel, L. Lovász and A. Shrijver. Geometri algorithms and ombinatorial
optimization. Springer-Verlag, Berlin, 1988.
9. O. Gunluk, T. Kimbrel, L. Ladanyi, B. Shieber and G. Sorkin, Vehile Routing
and Stang for Sedan Servie, Transportation Siene, 40, pp. 313326 (2006).
10. B. Haeupler, B. Saha and A. Srinivasan, New Construtive Aspets of the Lovász
Loal Lemma, J. ACM 58(6): pp. 128 (2011).
11. H. Hoogeveen, M. Skutella and G. Woeginger, Preemptive sheduling with rejetion, Mathematial Programming 94(2-3), pp. 361-374 (2003).
12. J. Lenstra, D. Shmoys and E. Tardos, Approximation Algorithms for Sheduling
Unrelated Parallel Mahines, Mathematial Programming 46, pp. 259-271 (1990).
13. S.A. Plotkin, D. Shmoys and E. Tardos. Fast approximation algorithms for frational paking and overing problems. Math. of Oper. Res., 20:257301, 1995.
14. A. Shrijver. Combinatorial Optimization: Polyhedra and Eieny. Springer,
2003.
15. A. Shulz and M. Skutella, Sheduling Unrelated Mahines by Randomized Rounding, SIAM J. Disrete Math. 15(4) pp. 450-469 (2002).
16. S. Seiden, Preemptive multiproessor sheduling with rejetion, Theoretial Computer Siene 262(1), pp. 437-458 (2001).
17. M. Skutella, Convex quadrati and semidenite programming relaxations in
sheduling, J. ACM 48(2) pp. 206-242 (2001).
18. W. E. Smith. Various optimizers for single-stage prodution. Naval Researh and
Logistis Quarterly, 3:5966, 1956.
19. O. Svensson, Santa Claus shedules jobs on unrelated mahines, in Proeedings of
STOC 2011, pp. 617-626.
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