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ia
n J
o
urna
l o
f
E
lect
rica
l En
g
ineering
a
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Co
m
pu
t
er
Science
Vo
l.
25
,
No
.
3
,
Ma
r
ch
20
22
,
p
p
.
1
7
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1795
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esco
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m
Para
llelized
so
luti
o
n t
o
t
he
as
y
mm
etric
trav
elling
sa
lesm
a
n
pro
blem using
ce
ntral pro
cess
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a
ccele
ra
tion
Ak
s
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t
Ary
a
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o
o
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n P
er
um
a
l
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Sa
nthi K
rish
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3
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e
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s,
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Art
icle
I
nfo
AB
S
T
RAC
T
A
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ticle
his
to
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y:
R
ec
eiv
ed
Au
g
18
,
2
0
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1
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ev
is
ed
J
an
11
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2
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2
2
Acc
ep
ted
J
an
24
,
2
0
2
2
Trav
e
ll
in
g
sa
les
m
a
n
p
r
o
b
lem
is
a
we
ll
re
se
a
rc
h
e
d
p
r
o
b
lem
i
n
c
o
m
p
u
ter
sc
ien
c
e
a
n
d
h
a
s
m
a
n
y
p
ra
c
ti
c
a
l
a
p
p
li
c
a
ti
o
n
s.
It
is
c
las
sified
a
s
a
NP
-
h
a
rd
p
ro
b
lem
a
s
it
s
e
x
a
c
t
so
lu
ti
o
n
c
a
n
o
n
l
y
b
e
o
b
tai
n
e
d
i
n
e
x
p
o
n
e
n
ti
a
l
ti
m
e
u
n
les
s
P
=
NP
.
T
h
e
re
a
re
d
iffere
n
t
v
a
rian
ts
o
f
t
h
e
tra
v
e
ll
in
g
sa
les
m
a
n
p
ro
b
lem
(TS
P
)
a
n
d
i
n
th
is
p
a
p
e
r,
a
sy
m
m
e
tri
c
trav
e
ll
in
g
sa
les
m
a
n
p
ro
b
lem
is
a
d
d
re
ss
e
d
sin
c
e
t
h
is
v
a
rian
t
is
q
u
it
e
o
ften
o
b
se
rv
e
d
in
re
a
l
wo
rl
d
sc
e
n
a
rio
s.
Th
e
re
a
re
a
n
u
m
b
e
r
o
f
h
e
u
rist
ic
a
p
p
ro
a
c
h
e
s
t
o
t
h
is
p
ro
b
lem
wh
ic
h
p
r
o
v
id
e
s
a
p
p
ro
x
ima
te
so
l
u
ti
o
n
s
i
n
p
o
ly
n
o
m
ial
ti
m
e
,
h
o
we
v
e
r
t
h
is
p
a
p
e
r
p
r
o
p
o
se
s
a
n
e
x
a
c
t
o
p
ti
m
a
l
so
l
u
ti
o
n
w
h
ich
is
a
c
c
e
lera
ted
with
th
e
h
e
l
p
o
f
m
u
lt
i
-
th
re
a
d
in
g
-
b
a
se
d
p
a
ra
ll
e
li
z
a
ti
o
n
.
In
o
rd
e
r
t
o
fin
d
th
e
e
x
a
c
t
o
p
ti
m
a
l
so
l
u
ti
o
n
,
we
h
a
v
e
u
se
d
th
e
h
e
ld
-
k
a
rp
a
lg
o
rit
h
m
in
v
o
lv
i
n
g
d
y
n
a
m
ic
p
ro
g
ra
m
m
in
g
a
n
d
to
re
d
u
c
e
th
e
ti
m
e
tak
e
n
to
fin
d
th
e
o
p
ti
m
a
l
p
a
th
,
we
h
a
v
e
u
se
d
a
m
u
l
t
i
-
th
re
a
d
e
d
a
p
p
ro
a
c
h
to
p
a
ra
ll
e
li
z
e
th
e
p
r
o
c
e
ss
in
g
o
f
s
u
b
-
p
ro
b
lem
s
b
y
lev
e
r
a
g
in
g
th
e
c
e
n
tral
p
r
o
c
e
ss
in
g
u
n
it
c
o
re
s
(CP
Us
).
Th
is
m
e
th
o
d
is
a
n
e
x
ten
si
o
n
o
f
a
we
ll
re
se
a
rc
h
e
d
so
lu
ti
o
n
to
t
h
e
TS
P
;
h
o
we
v
e
r,
t
h
is
m
e
th
o
d
sh
o
ws
th
a
t
s
o
lu
ti
o
n
s
to
c
o
m
p
u
tati
o
n
a
ll
y
i
n
ten
siv
e
p
r
o
b
l
e
m
s
in
v
o
l
v
i
n
g
s
u
b
-
p
ro
b
lem
s
su
c
h
a
s
th
e
a
sy
m
m
e
ti
c
trav
e
ll
in
g
sa
les
m
a
n
p
ro
b
lem
(ATS
P
)
c
a
n
b
e
a
c
c
e
lera
te
d
with
t
h
e
h
e
lp
o
f
m
o
d
e
rn
CP
Us
.
K
ey
w
o
r
d
s
:
Asy
m
m
etr
ic
tr
av
ellin
g
s
alesm
an
p
r
o
b
lem
Dy
n
am
ic
p
r
o
g
r
am
m
in
g
Held
-
k
ar
p
al
g
o
r
ith
m
Mu
ltit
h
r
ea
d
in
g
Par
alleliza
tio
n
T
h
is i
s
a
n
o
p
e
n
a
c
c
e
ss
a
rticle
u
n
d
e
r th
e
CC B
Y
-
SA
li
c
e
n
se
.
C
o
r
r
e
s
p
o
nd
ing
A
uth
o
r
:
Ak
s
ch
at
Ar
y
a
B
T
ec
h
in
C
o
m
p
u
ter
Scien
ce
a
n
d
E
n
g
in
ee
r
in
g
,
VI
T
Un
iv
er
s
ity
Vello
r
e,
I
n
d
ia
E
m
ail:
ak
s
ch
atar
y
a1
@
g
m
ail.
c
o
m
1.
I
NT
RO
D
UCT
I
O
N
I
n
th
e
s
im
p
le
tr
av
elin
g
s
alesp
er
s
o
n
p
r
o
b
lem
(
T
SP
)
,
we
ar
e
g
iv
en
an
u
n
d
ir
ec
te
d
g
r
ap
h
=
(
,
)
an
d
(
)
>
0
f
o
r
ea
c
h
ed
g
e
∈
an
d
th
e
o
b
jectiv
e
is
to
f
in
d
a
h
a
m
ilto
n
i
an
cy
cle
with
th
e
m
in
im
u
m
co
s
t.
A
h
am
ilto
n
ian
cy
cle
v
is
its
ev
er
y
v
er
tex
in
ex
ac
tly
o
n
ce
.
I
n
th
is
p
ap
er
we
ar
e
ad
d
r
ess
in
g
th
e
asy
m
m
etr
ic
tr
av
ellin
g
s
alesm
an
p
r
o
b
lem
(
AT
SP
)
wh
ich
f
r
eq
u
en
tly
h
as
to
b
e
d
ea
lt
with
in
r
ea
l
wo
r
ld
s
ce
n
ar
io
s
.
L
et
=
(
,
)
b
e
a
g
iv
en
d
ir
e
cted
g
r
ap
h
,
with
v
er
tex
s
et
=
{
1
,
.
.
.
,
}
an
d
ar
c
s
et
=
{
(
,
)
∶
,
∈
}
.
L
et
b
e
th
e
c
o
s
t
f
o
r
th
e
ar
c
(
,
)
∈
with
=
+
∞
(
∈
)
.
A
h
am
ilto
n
ian
cir
cu
it
(
t
o
u
r
)
o
f
is
a
cir
cu
it v
is
itin
g
ea
ch
v
er
te
x
o
f
ex
ac
tly
o
n
ce
.
T
h
e
o
b
jectiv
e
o
f
th
e
AT
SP
is
to
f
in
d
a
Ha
m
ilto
n
ian
cir
cu
it
∗
=
(
,
∗
)
o
f
with
m
in
im
u
m
co
s
t =
∑
(
,
)
∈
∗
T
h
er
e
ar
e
d
if
f
er
e
n
t
v
ar
ian
ts
o
f
th
e
tr
av
ellin
g
s
alesm
an
p
r
o
b
lem
wh
ich
h
av
e
b
ee
n
ad
d
r
ess
ed
b
y
r
esear
ch
er
s
ea
r
lier
an
d
b
o
t
h
ap
p
r
o
x
im
ate
(
f
aster
)
a
n
d
ex
a
ct
(
s
lo
wer
)
s
o
lu
tio
n
s
h
av
e
b
e
en
p
r
o
v
id
ed
.
So
m
e
p
o
s
s
ib
le
s
o
lu
tio
n
s
f
o
r
s
o
m
e
o
f
th
e
o
th
er
v
a
r
ian
ts
as
p
er
ea
r
lier
r
esear
ch
ar
e
as
f
o
llo
ws:
i)
s
y
m
m
etr
ic
T
SP
:
GPU
ac
ce
ler
ated
s
o
lu
tio
n
p
r
o
v
id
ed
b
y
Kim
u
r
a
et
a
l.
in
[
1
]
,
ii)
AT
SP
:
ap
p
r
o
x
im
atio
n
alg
o
r
ith
m
s
b
y
Evaluation Warning : The document was created with Spire.PDF for Python.
I
SS
N
:
2
5
0
2
-
4
7
5
2
I
n
d
o
n
esian
J
E
lec
E
n
g
&
C
o
m
p
Sci
,
Vo
l.
25
,
No
.
3
,
Ma
r
ch
20
22
:
1
7
9
5
-
1
8
0
2
1796
d
ec
o
m
p
o
s
in
g
d
ir
ec
ted
r
eg
u
lar
m
u
ltig
r
ap
h
s
p
r
o
v
i
d
ed
b
y
Ka
p
lan
et
a
l.
in
[
2
]
,
iii)
AT
SP
with
win
d
o
ws:
ex
ac
t
s
o
lu
tio
n
th
r
o
u
g
h
a
g
r
a
p
h
tr
an
s
f
o
r
m
atio
n
p
r
o
v
i
d
ed
by
Alb
iac
h
et
a
l.
in
[
3
]
,
iv
)
AT
SP
with
r
ep
len
is
h
m
en
t
ar
cs:
p
o
ly
h
e
d
r
al
r
esu
lts
p
r
o
v
id
ed
b
y
Ma
k
an
d
B
o
lan
d
in
[
4
]
.
Me
et
in
th
e
m
id
d
le
alg
o
r
ith
m
was
u
s
ed
b
y
Kaz
u
r
o
Kim
u
r
a
et
a
l.
to
ac
ce
ler
ate
th
e
ex
ec
u
tio
n
tim
e
b
u
t
th
is
m
eth
o
d
ca
n
o
n
ly
b
e
u
s
ed
o
n
th
e
s
y
m
m
etr
ic
T
SP
b
y
lev
er
ag
in
g
th
e
s
y
m
m
etr
ic
asp
ec
t
o
f
th
e
p
r
o
b
lem
an
d
th
u
s
Kim
u
r
a
et
a
l.
in
[
1
]
ac
h
i
ev
ed
an
ac
c
eler
atio
n
b
y
a
f
ac
to
r
o
f
1
.
5
an
d
th
at
o
f
1
.
7
u
s
in
g
ma
n
-
in
-
t
h
e
-
m
id
d
le
(
MI
T
M
)
wh
e
n
n
(
n
u
mb
er
o
f
ve
r
tices)
was
o
d
d
a
n
d
ev
en
,
r
esp
ec
tiv
ely
.
Sin
ce
t
h
is
p
ap
e
r
aim
s
to
ad
d
r
ess
th
e
asy
m
m
etr
ic
tr
av
ellin
g
s
alesm
an
p
r
o
b
lem
,
we
h
av
e
n
o
t
u
s
e
d
MI
T
M,
in
s
tead
we
m
ak
e
u
s
e
o
f
th
e
f
o
llo
win
g
tech
n
iq
u
es
to
ac
ce
ler
ate
th
e
p
r
o
ce
s
s
in
g
tim
e:
i
)
m
u
lti
-
th
r
e
ad
ed
p
r
o
g
r
am
to
u
tili
ze
ce
n
tr
al
p
r
o
ce
s
s
in
g
u
n
it
(
C
PU
)
co
r
es
,
ii
)
th
r
ea
d
-
s
af
e
h
a
s
h
m
ap
to
s
to
r
e
r
esu
lts
o
f
t
h
e
d
y
n
am
ic
co
s
t f
u
n
ctio
n
.
C
PU
p
ar
alleliza
tio
n
h
as
also
b
ee
n
ac
h
iev
ed
f
o
r
o
th
er
alg
o
r
ith
m
s
lik
e
th
e
an
t
co
lo
n
y
o
p
tim
izatio
n
f
o
r
th
e
T
SP
.
L
in
g
et
a
l.
in
[
5
]
h
av
e
p
r
esen
ted
an
a
d
ap
tiv
e
p
a
r
allel
an
t
co
lo
n
y
o
p
tim
izatio
n
(
PAC
O)
alg
o
r
ith
m
u
s
in
g
m
ass
iv
ely
p
ar
allel
p
r
o
ce
s
s
o
r
s
(
MPP
s
)
.
A
m
eth
o
d
o
f
ad
ju
s
tin
g
th
e
tim
e
in
ter
v
al
ad
ap
tiv
ely
f
o
r
in
f
o
r
m
atio
n
ex
ch
a
n
g
e
ac
co
r
d
i
n
g
to
th
e
d
iv
e
r
s
ity
o
f
th
e
s
o
lu
t
io
n
s
is
also
p
r
o
p
o
s
ed
b
y
C
h
en
lin
g
et
a
l.
to
av
o
id
ea
r
ly
co
n
v
er
g
e
n
ce
an
d
im
p
r
o
v
e
th
e
q
u
ality
o
f
r
e
s
u
lts
[
5
]
.
Fejzag
i
et
a
l.
h
a
v
e
s
h
o
wn
th
at
it
is
p
o
s
s
ib
le
to
ef
f
icien
tly
p
ar
allelize
m
etah
e
u
r
is
tic
alg
o
r
ith
m
s
lik
e
AC
O
u
s
i
n
g
task
p
ar
allel
lib
r
a
r
y
[
6
]
.
Gize
m
s
E
r
m
is
et
a
l.
h
av
e
in
v
esti
g
ated
th
e
ac
ce
ler
atio
n
f
r
o
m
C
UDA
b
y
u
s
in
g
2
-
o
p
t
a
n
d
3
-
o
p
t
lo
ca
l
s
ea
r
ch
h
eu
r
is
tics
an
d
s
h
ar
ed
e
x
p
lain
ed
s
o
m
e
p
ar
alleliza
tio
n
s
tr
ateg
ies to
u
tili
ze
GPU
r
eso
u
r
ce
s
ef
f
ec
tiv
ely
[
7
]
.
Haim
Kap
lan
et
a
l.
h
as
p
r
o
v
id
ed
ap
p
r
o
x
im
atio
n
alg
o
r
ith
m
s
f
o
r
asy
m
m
etr
ic
T
SP
b
y
th
e
d
ec
o
m
p
o
s
itio
n
o
f
d
ir
ec
ted
r
eg
u
lar
m
u
ltig
r
a
p
h
s
[
2
]
.
E
x
p
er
im
en
ts
b
y
Sax
en
a
et
a
l.
in
[
8
]
s
h
o
w
th
at
p
ar
all
eliza
tio
n
to
o
ls
lik
e
Op
en
MP
an
d
C
UDA
ca
n
s
ig
n
if
ican
tly
r
ed
u
ce
th
e
ex
ec
u
tio
n
tim
e
f
o
r
g
en
etic
alg
o
r
ith
m
s
u
s
ed
in
s
o
lv
in
g
th
e
T
SP
.
R
a
s
h
id
in
[
9
]
p
r
esen
ted
a
p
ar
allel
h
eu
r
is
tic
in
teg
r
atin
g
a
g
r
ee
d
y
ap
p
r
o
ac
h
i
n
to
a
g
en
etic
alg
o
r
ith
m
with
lo
ca
l
-
s
ea
r
ch
u
s
in
g
GPU
ac
ce
le
r
atio
n
.
Mo
s
t
o
f
th
e
p
r
ev
io
u
s
wo
r
k
h
a
v
e
p
r
esen
ted
an
ap
p
r
o
x
im
ate
alg
o
r
ith
m
f
o
r
th
e
g
en
e
r
al
T
SP
o
r
an
ex
ac
t
alg
o
r
ith
m
with
o
u
t
C
PU
p
ar
alleliza
tio
n
f
o
r
th
e
ASTP.
I
n
th
is
p
ap
er
we
p
r
esen
t
an
ex
ac
t
alg
o
r
ith
m
f
o
r
th
e
asy
m
m
etr
ic
T
SP
u
tili
zin
g
C
PU
p
ar
alleliza
tio
n
an
d
th
r
ea
d
-
s
af
e
h
ash
m
ap
to
ac
ce
ler
ate
th
e
ex
ec
u
tio
n
p
r
o
ce
s
s
.
Alr
ash
d
an
et
a
l.
h
av
e
u
s
ed
e
n
h
an
ce
d
c
r
o
s
s
o
v
er
o
p
er
atio
n
u
s
in
g
g
en
etic
alg
o
r
ith
m
with
th
eir
p
r
o
b
ab
ilit
ies
in
o
r
d
er
to
cr
ea
te
an
ef
f
icien
t
m
e
th
o
d
to
p
r
o
v
id
e
a
n
ea
r
o
p
tim
al
s
o
lu
tio
n
f
o
r
th
e
AT
SP
[
1
0
]
.
A
T
wo
-
way
p
ar
allel
s
lim
e
m
o
ld
alg
o
r
ith
m
b
y
f
lo
w
an
d
d
is
tan
ce
(
T
PS
MA
)
is
p
r
o
p
o
s
ed
b
y
L
iu
et
a
l.
in
[
1
1
]
in
o
r
d
er
to
s
o
lv
e
s
lim
e
m
o
ld
alg
o
r
ith
m
’
s
p
r
o
b
lem
o
f
p
o
o
r
l
o
ca
l
o
p
tim
izatio
n
.
Asc
h
eu
er
et
a
l.
h
as
p
r
o
v
i
d
ed
a
c
o
m
p
u
tatio
n
al
s
tu
d
y
wh
ich
h
as
in
d
icate
d
th
at
m
o
s
t
AT
SP
with
tim
e
w
in
d
o
ws
in
s
tan
ce
s
r
an
g
in
g
till
5
0
–
7
0
n
o
d
e
s
ca
n
b
e
o
p
ti
m
ally
s
o
lv
ed
u
s
in
g
b
r
an
ch
an
d
c
u
t
[
1
2
]
.
Kan
g
et
a
l.
p
r
o
p
o
s
e
an
e
f
f
ec
tiv
e
m
eth
o
d
o
f
c
o
n
s
tr
u
ctiv
e
cr
o
s
s
o
v
er
s
u
ch
t
h
at
lar
g
e
n
u
m
b
er
o
f
g
en
es
ca
n
b
e
ef
f
ec
tiv
ely
ev
o
l
v
ed
b
y
ex
p
l
o
itin
g
th
e
GPUs
p
ar
allel
co
m
p
u
tin
g
p
o
wer
a
n
d
an
ef
f
ec
tiv
e
p
a
r
allel
ap
p
r
o
ac
h
to
g
en
etic
T
SP
wh
er
e
cr
o
s
s
o
v
er
m
eth
o
d
s
ca
n
n
o
t
b
e
ea
s
ily
im
p
l
em
en
ted
in
p
ar
allel
f
ash
io
n
[
1
3
]
.
Vasilch
ik
o
v
h
as
s
h
o
wn
th
at
th
e
litt
le
a
l
g
o
r
i
t
h
m
a
l
s
o
h
a
s
g
o
o
d
p
o
t
e
n
t
i
a
l
f
o
r
r
e
c
u
r
s
i
v
e
-
p
a
r
a
l
l
el
c
o
m
p
u
t
a
t
i
o
n
s
a
n
d
c
a
n
b
e
u
s
ed
w
i
t
h
a
c
o
m
b
i
n
e
d
a
p
p
r
o
a
c
h
[
1
4
]
.
S
a
m
p
l
e
i
n
s
t
a
n
c
es
f
o
r
t
h
e
T
SP
(
a
n
d
r
e
l
at
e
d
p
r
o
b
l
e
m
s
)
f
r
o
m
v
a
r
i
o
u
s
s
o
u
r
c
es
a
n
d
o
f
v
a
r
i
o
u
s
t
y
p
e
s
a
r
e
p
r
o
v
i
d
e
d
b
y
T
SP
l
i
b
i
n
[
1
5
]
.
W
e
h
av
e
a
l
s
o
m
a
d
e
u
s
e
o
f
t
h
e
d
a
t
a
s
e
t
s
p
r
o
v
i
d
e
d
b
y
T
SP
l
ib
.
S
v
e
n
s
s
o
n
et
a
l
.
h
av
e
p
r
o
v
id
ed
a
co
n
s
tan
t
f
ac
t
o
r
ap
p
r
o
x
im
atio
n
alg
o
r
ith
m
b
y
th
e
r
ed
u
ctio
n
t
o
s
u
b
to
u
r
p
a
r
titi
o
n
co
v
e
r
(
a
n
ea
s
ier
p
r
o
b
lem
o
b
tain
ed
wh
e
n
th
e
g
e
n
e
r
a
l
c
o
n
n
e
v
t
i
v
i
t
y
r
e
q
u
i
r
e
m
e
n
t
s
a
r
e
r
e
l
a
x
e
d
s
i
g
n
if
i
c
a
n
t
l
y
i
n
t
o
l
o
c
a
l
c
o
n
n
e
c
t
i
v
ity
c
o
n
d
i
t
i
o
n
s
)
[
1
6
]
.
A
zi
m
i
e
t
a
l
.
h
av
e
p
r
esen
ted
a
n
ew
m
o
d
el
u
s
in
g
s
im
u
lated
an
n
ea
lin
g
with
m
u
ltip
le
tr
an
s
p
o
r
ter
s
f
o
r
th
e
T
SP
[
1
7
]
.
A
n
e
w
h
y
b
r
id
alg
o
r
ith
m
f
o
r
th
e
p
r
o
b
ab
ilis
tic
tr
av
elin
g
s
alesm
an
p
r
o
b
lem
(
PTSP
)
is
p
r
o
p
o
s
ed
b
y
Ma
r
in
a
k
is
b
ased
o
n
g
r
ee
d
y
r
an
d
o
m
ize
d
ad
a
p
tiv
e
s
ea
r
ch
p
r
o
ce
d
u
r
e
(
GR
ASP),
p
ar
ticle
s
war
m
o
p
tim
izatio
n
(
PS
O)
an
d
ex
p
a
n
d
in
g
n
eig
h
b
o
r
h
o
o
d
s
ea
r
ch
(
E
NS)
s
tr
ateg
y
[
1
8
]
.
Han
et
a
l.
Hav
e
s
o
lv
ed
th
e
la
r
g
e
-
s
ca
le
co
lo
r
e
d
tr
av
ellin
g
s
alesm
an
p
r
o
b
lem
u
s
in
g
a
n
im
p
r
o
v
ed
an
t
co
lo
n
y
o
p
tim
izatio
n
(
I
AC
O)
alg
o
r
ith
m
in
[
1
9
]
.
E
r
e
m
ee
v
et
a
l.
h
av
e
v
er
if
ied
in
[
2
0
]
t
h
e
u
s
ef
u
ln
ess
o
f
a
p
ar
allel
ad
ap
tiv
e
an
t
c
o
lo
n
y
c
o
m
m
u
n
ities
f
o
r
th
e
d
y
n
am
ic
t
r
av
ellin
g
s
alesm
an
p
r
o
b
lem
(
DT
SP
)
.
E
r
em
ee
v
et
a
l.
h
av
e
p
r
o
p
o
s
ed
a
n
ew
m
e
m
etic
alg
o
r
ith
m
f
o
r
th
e
asy
m
m
etr
ic
tr
av
ellin
g
s
alesm
an
p
r
o
b
lem
(
AT
SP
)
with
o
p
tim
al
r
ec
o
m
b
i
n
atio
n
in
[
2
0
]
.
R
ash
id
an
d
Mo
s
teir
o
h
av
e
p
r
o
v
id
e
d
a
n
o
v
el
s
o
lu
tio
n
in
[
2
1
]
th
at
in
teg
r
ates
lo
ca
l
-
s
ea
r
ch
h
eu
r
is
tics
,
a
g
r
ee
d
y
alg
o
r
ith
m
a
n
d
a
g
en
etic
al
g
o
r
ith
m
.
Od
ili
et
a
l
.
in
[
2
2
]
p
r
esen
t
a
co
m
p
a
r
ativ
e
p
er
f
o
r
m
an
ce
an
aly
s
is
o
f
s
o
m
e
o
f
th
e
m
etah
eu
r
is
tic
alg
o
r
ith
m
s
lik
e
th
e
im
p
r
o
v
ed
ex
tr
em
al
o
p
tim
izatio
n
(
I
E
O)
,
af
r
ican
b
u
f
f
al
o
o
p
tim
izatio
n
alg
o
r
ith
m
(
AB
O)
,
m
ax
-
m
in
an
t
s
y
s
tem
(
MM
AS)
,
th
e
h
e
u
r
is
tic
r
an
d
o
m
ize
d
in
s
er
tio
n
alg
o
r
ith
m
(
R
AI
)
an
d
co
o
p
er
ativ
e
g
en
etic
an
t
s
y
s
tem
(
C
GAS)
to
s
o
lv
e
th
e
AT
SP
.
Fo
s
in
et
a
l.
h
av
e
p
r
esen
ted
a
n
ew
p
ar
allel
iter
ated
lo
ca
l
s
ea
r
ch
ap
p
r
o
ac
h
in
[
2
3
]
with
2
-
o
p
t
an
d
3
-
opt
o
p
er
a
to
r
s
f
o
r
s
y
m
m
etr
ic
T
SP
,
u
s
in
g
GPU
ac
ce
ler
atio
n
.
Li
et
a
l.
h
av
e
p
r
o
v
id
ed
an
im
p
r
o
v
ed
m
u
ltico
r
e
b
ased
p
ar
allel
b
r
an
ch
an
d
b
o
u
n
d
alg
o
r
ith
m
t
o
s
o
lv
e
class
ic
T
SP
with
its
s
h
o
r
tco
m
in
g
s
in
[
2
4
]
.
R
ico
-
Gar
cia
et
a
l.
h
av
e
p
r
o
v
id
e
d
a
p
ar
allel
im
p
lem
en
tatio
n
o
f
th
e
d
is
cr
e
te
teac
h
in
g
lear
n
in
g
-
b
ased
o
p
tim
izatio
n
alg
o
r
ith
m
(
DT
L
B
O)
b
y
u
tili
zin
g
a
m
u
ltico
r
e
GPU
en
v
ir
o
n
m
en
t
i
n
o
r
d
e
r
to
im
p
r
o
v
e
th
e
p
er
f
o
r
m
an
ce
o
f
th
e
al
g
o
r
ith
m
a
n
d
t
o
o
b
tain
s
u
b
o
p
tim
al
o
r
o
p
tim
al
s
o
lu
tio
n
s
to
th
e
tr
a
v
elin
g
s
alesm
an
p
r
o
b
lem
[
2
5
]
.
Ho
wev
er
,
m
o
s
t
o
f
th
ese
m
et
h
o
d
s
m
en
tio
n
ed
in
Evaluation Warning : The document was created with Spire.PDF for Python.
I
n
d
o
n
esian
J
E
lec
E
n
g
&
C
o
m
p
Sci
I
SS
N:
2502
-
4
7
5
2
P
a
r
a
lleliz
ed
s
o
lu
tio
n
to
th
e
a
s
ymm
etri
c
t
r
a
ve
llin
g
s
a
lesma
n
p
r
o
b
lem
u
s
in
g
ce
n
tr
a
l …
(
A
ks
ch
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1797
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e
af
o
r
em
e
n
tio
n
ed
p
ap
er
s
p
r
o
v
id
e
o
n
ly
an
ap
p
r
o
x
im
ate
s
o
lu
tio
n
o
r
d
o
n
o
t
co
n
s
id
er
th
e
asy
m
m
etr
ic
v
er
s
io
n
o
f
th
e
T
SP
with
p
ar
alleliza
tio
n
.
Ou
r
m
eth
o
d
h
as
s
h
o
wn
th
at
th
e
o
p
tim
al
s
o
lu
tio
n
o
f
AT
SP
an
d
s
im
ilar
co
m
p
u
tatio
n
ally
in
ten
s
iv
e
p
r
o
b
lem
s
with
s
u
b
-
p
r
o
b
lem
s
ca
n
b
e
ac
ce
ler
ated
with
p
r
alleliz
atio
n
u
s
in
g
m
o
d
er
n
C
PU
s
.
2.
M
E
T
H
O
D
2
.
1
.
T
heo
re
t
ica
l
a
na
ly
s
is
W
e
h
av
e
u
s
ed
th
e
Held
-
Kar
p
alg
o
r
ith
m
o
n
a
d
ataset
o
f
n
n
o
d
es
to
f
i
n
d
a
n
e
x
ac
t
s
o
lu
tio
n
t
o
th
is
n
o
d
e
s
et.
B
ef
o
r
e
Par
alleliz
in
g
th
e
alg
o
r
ith
m
,
we
n
ee
d
to
p
er
f
o
r
m
th
e
th
e
th
eo
r
etica
l
an
aly
s
is
o
f
th
e
s
tan
d
ar
d
alg
o
r
ith
m
.
T
h
e
tim
e
an
d
s
p
ac
e
co
m
p
lex
ity
ca
n
b
e
ca
lc
u
lated
as f
o
llo
ws.
T
im
e
co
m
p
lex
ity
:
L
et
th
e
g
iv
en
s
e
t
o
f
n
o
d
es
b
e
V
∈
{
1
,
2
,
…
}
with
1
as
th
e
in
itial
n
o
d
e.
Fo
r
ev
er
y
o
t
h
er
n
o
d
e
s
u
ch
th
at
≠
1
,
th
e
aim
is
to
f
in
d
t
h
e
m
in
im
u
m
co
s
t
p
ath
with
1
as
th
e
s
tar
tin
g
n
o
d
e,
as
th
e
en
d
in
g
n
o
d
e
s
u
ch
th
at
all
o
th
er
n
o
d
es
ar
e
v
is
ited
ex
ac
tly
o
n
ce
.
Fo
r
a
s
et
o
f
s
ize
k
,
we
co
n
s
id
er
k
-
2
s
u
b
s
ets ea
ch
o
f
s
ize
k
-
1
s
u
c
h
t
h
at
all
s
u
b
s
ets d
o
n
’
t h
a
v
e
ℎ
in
th
em
.
T
h
u
s
,
b
y
e
v
alu
atin
g
th
e
s
u
m
o
f
m
i
n
im
u
m
co
s
t
p
ath
f
o
r
ea
ch
s
u
b
s
et
o
f
k
-
1
n
o
d
es
s
tar
tin
g
with
t
h
e
in
itial n
o
d
e
we
g
et
th
e
tim
e
co
m
p
lex
ity
.
T
h
is
is
g
iv
en
b
y
(
1
)
:
−
1
+
∑
(
−
1
)
−
1
=
2
×
(
−
1
)
(
1
)
Als
o
th
e
o
cc
u
r
r
e
n
ce
s
o
f
co
m
p
u
tatio
n
s
f
o
r
th
e
n
ex
t
p
h
ase
is
g
iv
en
b
y
(
2
)
:
∑
=
(
−
1
)
2
−
1
=
2
−
1
(
2
)
th
u
s
f
r
o
m
(
1
)
a
n
d
(
2
)
,
(
1
)
r
ed
u
ce
s
to
(
3
)
:
(
−
1
)
(
−
2
)
2
−
3
+
(
−
1
)
(
3
)
on
f
u
r
th
er
r
ed
u
ctio
n
we
g
et
th
e
tim
e
co
m
p
lex
ity
as O
(
2
2
)
Sp
ac
e
co
m
p
lex
ity
:
T
h
e
Hel
d
-
Kar
p
alg
o
r
ith
m
is
ex
ec
u
te
d
in
ex
p
o
n
en
tial
tim
e
b
u
t
s
till
o
f
f
er
s
r
elativ
ely
f
aster
ex
ec
u
tio
n
co
m
p
ar
ed
to
ex
h
a
u
s
tiv
e
en
u
m
er
atio
n
.
T
h
is
is
co
m
p
en
s
ated
b
y
u
s
in
g
a
lo
t
m
o
r
e
s
p
ac
e
th
an
ex
h
a
u
s
tiv
e
en
u
m
e
r
atio
n
.
T
h
e
s
p
ac
e
c
o
m
p
lex
ity
is
g
iv
en
b
y
(
4
)
:
−
1
+
∑
−
1
=
2
×
(
−
1
)
(
4
)
=
(
−
1
)
2
−
2
on
r
ed
u
ctio
n
we
g
et
th
e
s
p
ac
e
co
m
p
lex
ity
as O(
2
)
.
2
.
2
.
P
r
o
ce
s
s
ing
a
rc
hite
ct
ure
W
e
h
av
e
u
tili
ze
d
C
PU
p
ar
all
eliza
tio
n
to
ac
h
iev
e
f
aster
ex
ec
u
tio
n
tim
e.
T
h
e
ar
c
h
itectu
r
e
o
f
a
C
PU
with
m
u
ltip
le
co
r
es
is
r
ep
r
ese
n
ted
b
y
Fig
u
r
e
1
.
A
m
u
lti
-
c
o
r
e
p
r
o
ce
s
s
o
r
is
a
ty
p
e
o
f
p
r
o
c
ess
o
r
th
at
co
n
tain
s
m
u
ltip
le
co
r
es
o
r
p
r
o
ce
s
s
in
g
u
n
ites
o
n
th
e
s
am
e
ch
i
p
.
T
h
is
k
in
d
o
f
p
r
o
ce
s
s
o
r
is
d
if
f
er
en
t
f
r
o
m
a
s
u
p
er
s
ca
lar
p
r
o
ce
s
s
o
r
,
wh
ic
h
ca
n
is
s
u
e
m
u
ltip
le
in
s
tr
u
ctio
n
s
p
er
cl
o
ck
c
y
cle
f
r
o
m
o
n
e
in
s
tr
u
ctio
n
s
tr
ea
m
(
th
r
ea
d
)
a
n
d
co
n
tain
s
m
u
ltip
le
ex
ec
u
tio
n
u
n
its
.
Ho
wev
er
,
m
u
ltip
le
in
s
tr
u
ctio
n
s
p
er
clo
ck
cy
cle
f
r
o
m
m
u
ltip
le
in
s
t
r
u
ctio
n
s
tr
ea
m
s
is
is
s
u
ed
b
y
a
m
u
lti
-
co
r
e
p
r
o
ce
s
s
o
r
.
E
v
er
y
co
r
e
in
a
m
u
lti
-
co
r
e
p
r
o
ce
s
s
o
r
p
o
ten
tially
ca
n
b
e
s
u
p
er
s
ca
lar
t
o
o
,
im
p
ly
in
g
t
h
at
o
n
ea
ch
clo
ck
cy
cle,
m
u
ltip
le
in
s
tr
u
ctio
n
s
ca
n
b
e
is
s
u
ed
f
r
o
m
a
s
in
g
le
th
r
ea
d
b
y
ea
c
h
co
r
e.
Simu
ltan
e
o
u
s
m
u
ltit
h
r
ea
d
in
g
(
I
n
tel’
s
h
y
p
er
t
h
r
ea
d
in
g
tech
n
o
lo
g
y
is
a
n
ex
am
p
le)
was
an
ea
r
ly
f
o
r
m
o
f
p
s
eu
d
o
-
m
u
lti
-
co
r
e
ar
ch
itectu
r
e.
A
p
r
o
ce
s
s
o
r
ca
p
ab
le
o
f
co
n
cu
r
r
e
n
t
m
u
ltit
h
r
ea
d
in
g
in
clu
d
es
m
u
ltip
le
ex
ec
u
tio
n
u
n
its
in
th
e
s
am
e
p
r
o
ce
s
s
in
g
u
n
it,
th
u
s
it
ca
n
b
e
s
aid
th
at
it
h
as
a
s
u
p
er
s
ca
lar
ar
ch
itectu
r
e
an
d
ca
n
is
s
u
e
m
o
r
e
th
an
o
n
e
in
s
tr
u
ctio
n
p
er
cl
o
ck
cy
cle
f
r
o
m
m
u
ltip
l
e
th
r
ea
d
s
.
Ho
wev
er
,
tem
p
o
r
al
m
u
ltit
h
r
ea
d
in
g
c
an
is
s
u
e
o
n
e
in
s
tr
u
ctio
n
at
a
tim
e
f
o
r
m
m
u
ltip
le
th
r
ea
d
wh
e
r
e
a
s
in
g
le
ex
ec
u
tio
n
u
n
it
in
th
e
s
am
e
p
r
o
ce
s
s
in
g
u
n
it
is
in
cl
u
d
ed
.
2
.
3
.
P
a
ra
lleliza
t
io
n
Par
alleliza
tio
n
is
ac
h
iev
ed
b
y
m
ap
p
i
n
g
s
u
b
-
p
r
o
b
lem
s
c
r
ea
ted
b
y
th
e
f
ir
s
t
r
ec
u
r
s
iv
e
ca
ll
to
th
r
ea
d
s
wh
ich
will
b
e
r
u
n
n
in
g
in
p
ar
al
lel.
T
h
is
is
illu
s
tr
ated
in
Fig
u
r
e
2
wh
er
e
(
,
)
is
th
e
co
s
t
to
o
p
tim
ally
v
is
it
all
v
er
tices in
a
s
et
with
N
n
o
d
es st
ar
tin
g
f
r
o
m
n
o
d
e
i
a
n
d
(
,
)
is
th
e
co
s
t to
tr
av
el
f
r
o
m
n
o
d
e
i
t
o
n
o
d
e
j
.
Evaluation Warning : The document was created with Spire.PDF for Python.
I
SS
N
:
2
5
0
2
-
4
7
5
2
I
n
d
o
n
esian
J
E
lec
E
n
g
&
C
o
m
p
Sci
,
Vo
l.
25
,
No
.
3
,
Ma
r
ch
20
22
:
1
7
9
5
-
1
8
0
2
1798
I
f
th
er
e
ar
e
k
s
u
b
-
p
r
o
b
lem
s
cr
ea
ted
b
y
th
e
f
ir
s
t r
ec
u
r
s
iv
e
ca
ll a
n
d
t th
r
ea
d
s
m
ap
p
ed
to
th
e
m
th
en
ea
ch
th
r
ea
d
i
will r
u
n
s
u
b
-
p
r
o
b
lem
s
s
u
ch
th
at
,
=
{
⌊
⌋
−
(
0
,
−
1
)
=
Fig
u
r
e
1
.
Mu
lti
c
o
r
e
C
PU
ar
ch
itectu
r
e
Fig
u
r
e
2
.
T
h
r
ea
d
m
ap
p
in
g
to
r
ec
u
r
s
iv
e
ca
ll
Evaluation Warning : The document was created with Spire.PDF for Python.
I
n
d
o
n
esian
J
E
lec
E
n
g
&
C
o
m
p
Sci
I
SS
N:
2502
-
4
7
5
2
P
a
r
a
lleliz
ed
s
o
lu
tio
n
to
th
e
a
s
ymm
etri
c
t
r
a
ve
llin
g
s
a
lesma
n
p
r
o
b
lem
u
s
in
g
ce
n
tr
a
l …
(
A
ks
ch
a
t
A
r
ya
)
1799
T
h
e
ch
allen
g
e
was
to
cr
ea
te
a
co
m
m
o
n
t
h
r
ea
d
-
s
af
e
s
tr
u
cu
r
e
to
ca
ch
e
th
e
in
ter
m
e
d
iar
y
r
esu
lts
f
r
o
m
th
e
r
ec
u
r
s
iv
e
ca
lls
.
T
h
is
d
ata
s
tr
u
ctu
r
e
s
h
o
u
l
d
b
e
s
h
ar
ed
wit
h
all
th
e
t
h
r
ea
d
s
.
Fo
r
th
is
p
u
r
p
o
s
e,
we
h
a
v
e
u
s
ed
J
av
a’
s
C
o
n
cu
r
r
en
tHash
Ma
p
with
J
av
a
th
r
ea
d
s
to
ac
h
ie
v
e
th
r
ea
d
-
s
af
e
p
ar
alleliza
tio
n
.
T
h
e
h
ash
m
ap
cr
ea
tes
a
n
em
p
ty
,
n
ew
m
ap
with
th
e
s
p
ec
if
ied
in
itial
ca
p
ac
ity
,
co
n
cu
r
r
e
n
cy
an
d
lo
a
d
f
ac
to
r
lev
el.
T
h
e
im
p
lem
en
tatio
n
o
f
in
itial
ca
p
ac
ity
p
er
f
o
r
m
s
in
ter
n
al
s
izin
g
to
ac
co
m
m
o
d
ate
th
e
s
e
m
an
y
elem
en
ts
wh
er
ea
s
th
e
im
p
lem
en
tatio
n
o
f
co
n
cu
r
r
en
cy
tr
ies
to
d
o
th
e
s
am
e.
I
n
itial
co
n
cu
r
r
en
c
y
lev
el
p
ar
am
eter
s
an
d
ca
p
ac
ity
p
ar
am
eter
o
f
C
o
n
cu
r
r
en
tHash
Ma
p
c
o
n
s
tr
u
c
to
r
(
o
r
Ob
ject)
i
n
J
av
a
a
r
e
s
et
to
1
6
b
y
d
ef
a
u
lt.
As
we
ar
e
p
ar
allelizin
g
th
e
p
r
o
ce
s
s
with
9
th
r
ea
d
s
we
d
o
n
o
t
n
ee
d
to
ch
a
n
g
e
th
e
p
ar
a
m
eter
s
.
T
h
u
s
,
in
s
tead
o
f
u
s
in
g
a
m
ap
wid
e
lo
ck
,
C
o
n
cu
r
r
en
tHash
Ma
p
m
ain
tain
s
a
lis
t
o
f
lo
ck
s
b
y
d
e
f
au
lt
s
u
c
h
th
at
th
e
in
itial
ca
p
ac
ity
is
e
q
u
al
to
t
h
e
n
u
m
b
er
o
f
lo
ck
s
.
E
ac
h
lo
c
k
is
u
s
ed
to
lo
ck
o
n
a
s
in
g
le
b
u
ck
et
o
f
th
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Evaluation Warning : The document was created with Spire.PDF for Python.
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AT
S
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g
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p
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ig
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wh
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f
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h
y
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p
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ally
in
s
ten
s
iv
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p
r
o
b
lem
s
s
u
ch
as th
e
AT
SP
.
Evaluation Warning : The document was created with Spire.PDF for Python.
I
n
d
o
n
esian
J
E
lec
E
n
g
&
C
o
m
p
Sci
I
SS
N:
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-
4
7
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(
A
ks
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t
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1801
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F
E
R
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NC
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[
1
]
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.
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1802
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