Chapter 10 Complexity of Approximation (1) L-Reduction Ding-Zhu Du.
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Transcript of Chapter 10 Complexity of Approximation (1) L-Reduction Ding-Zhu Du.
![Page 2: Chapter 10 Complexity of Approximation (1) L-Reduction Ding-Zhu Du.](https://reader035.fdocuments.us/reader035/viewer/2022062407/56649d1a5503460f949f041b/html5/thumbnails/2.jpg)
Traveling Salesman
• Given n cities with a distance table, find a minimum total-distance tour to visit each city exactly once.
![Page 3: Chapter 10 Complexity of Approximation (1) L-Reduction Ding-Zhu Du.](https://reader035.fdocuments.us/reader035/viewer/2022062407/56649d1a5503460f949f041b/html5/thumbnails/3.jpg)
. distance
total tour witha find ,and cities those
between tabledistance a with cities Given
:TSP-Approx-
optr
n
n
r
Definition
![Page 4: Chapter 10 Complexity of Approximation (1) L-Reduction Ding-Zhu Du.](https://reader035.fdocuments.us/reader035/viewer/2022062407/56649d1a5503460f949f041b/html5/thumbnails/4.jpg)
Proof:
Given a graph G=(V,E), define a distance table on V as follows:
EvurV
Evuvud
),( if ,||
),( if ,1),(
hard.- is TSP-Approx- ,1any For NPrr
Theorem
solvable. time-polynomial being HC implies solvable
time-polynomial being TSP-Approx- r
![Page 5: Chapter 10 Complexity of Approximation (1) L-Reduction Ding-Zhu Du.](https://reader035.fdocuments.us/reader035/viewer/2022062407/56649d1a5503460f949f041b/html5/thumbnails/5.jpg)
Contradiction Argument
• Suppose r-approximation exists. Then we have a polynomial-time algorithm to solve Hamiltonian Cycle as follow:
r-approximation solution < r |V|
if and only if
G has a Hamiltonian cycle
![Page 6: Chapter 10 Complexity of Approximation (1) L-Reduction Ding-Zhu Du.](https://reader035.fdocuments.us/reader035/viewer/2022062407/56649d1a5503460f949f041b/html5/thumbnails/6.jpg)
Special Case
• Traveling around a minimum spanning tree is a 2-approximation.
solvable. time-polynomial
is TSP-Approx-2 ,inequality triangular
thesatisfies tabledistance n the Whe Theorem
![Page 7: Chapter 10 Complexity of Approximation (1) L-Reduction Ding-Zhu Du.](https://reader035.fdocuments.us/reader035/viewer/2022062407/56649d1a5503460f949f041b/html5/thumbnails/7.jpg)
• Minimum spanning tree + minimum-length perfect matching on odd vertices is 1.5-approximation
solvable. time-polynomial
is TSP-Approx-1.5 ,inequality triangular
thesatisfies tabledistance n the Whe Theorem
![Page 8: Chapter 10 Complexity of Approximation (1) L-Reduction Ding-Zhu Du.](https://reader035.fdocuments.us/reader035/viewer/2022062407/56649d1a5503460f949f041b/html5/thumbnails/8.jpg)
Minimum perfect matching on odd verticeshas weight at most 0.5 opt.
![Page 9: Chapter 10 Complexity of Approximation (1) L-Reduction Ding-Zhu Du.](https://reader035.fdocuments.us/reader035/viewer/2022062407/56649d1a5503460f949f041b/html5/thumbnails/9.jpg)
Knapsack
.any for Hence
.any for Assume
}.1 ,0{
t.s.
max
2211
2211
ioptc
iSs
x
Sxsxsxs
xcxcxc
i
i
i
nn
nn
![Page 10: Chapter 10 Complexity of Approximation (1) L-Reduction Ding-Zhu Du.](https://reader035.fdocuments.us/reader035/viewer/2022062407/56649d1a5503460f949f041b/html5/thumbnails/10.jpg)
./
such that ) ,...,(solution feasible a find
, and ,..., , ..., ,Given
:Knapsack-Approx-
11
1
11
roptxcxc
xx
Ssscc
r
nn
n
nn
Definition
![Page 11: Chapter 10 Complexity of Approximation (1) L-Reduction Ding-Zhu Du.](https://reader035.fdocuments.us/reader035/viewer/2022062407/56649d1a5503460f949f041b/html5/thumbnails/11.jpg)
}. ,max{Output
. such that Choose
.Sort
1
1
1
11
2
2
1
1
k
k
i
iG
k
i
i
k
i
i
n
n
ccc
sSsk
s
c
s
c
s
c
solvable. time-polynomial
is Knapsack -Approx-2 Theorem
Proof.
![Page 12: Chapter 10 Complexity of Approximation (1) L-Reduction Ding-Zhu Du.](https://reader035.fdocuments.us/reader035/viewer/2022062407/56649d1a5503460f949f041b/html5/thumbnails/12.jpg)
solvable. time-polynomial
is Knapsack -Approx- ,1any For rr
Theorem
(PTAS). schemeion approximat
timepolynomial hasKnapsack s,other wordIn
![Page 13: Chapter 10 Complexity of Approximation (1) L-Reduction Ding-Zhu Du.](https://reader035.fdocuments.us/reader035/viewer/2022062407/56649d1a5503460f949f041b/html5/thumbnails/13.jpg)
• Classify: for i < m, ci < a= cG,
for i > m+1, ci > a.• Sort
• For
.2
2
1
1
m
m
s
c
s
c
s
c
;0)(set then , if
},,,1{
IcSs
nmI
Ii
i
1
Algorithm
![Page 14: Chapter 10 Complexity of Approximation (1) L-Reduction Ding-Zhu Du.](https://reader035.fdocuments.us/reader035/viewer/2022062407/56649d1a5503460f949f041b/html5/thumbnails/14.jpg)
).(max)(Output
.)(
set and
such that maximum choose then , If
1
1
IcIc
ccIc
sSs
mkSs
Ioutput
k
i
i
Ii
i
Ii
i
k
i
i
Ii
i
Proof. }.in 1 | },...,1{{*Let optxnmiI i
![Page 15: Chapter 10 Complexity of Approximation (1) L-Reduction Ding-Zhu Du.](https://reader035.fdocuments.us/reader035/viewer/2022062407/56649d1a5503460f949f041b/html5/thumbnails/15.jpg)
1)(
.1
)(
)()( Hence,
.*)(
*)( then , If
.*)( then , If
1*1
*1
output
output
outputoutput
kIi
im
ii
Iii
m
ii
Ic
opt
optIc
aIcoptIc
aIc
cIcoptsSs
optIcsSs
![Page 16: Chapter 10 Complexity of Approximation (1) L-Reduction Ding-Zhu Du.](https://reader035.fdocuments.us/reader035/viewer/2022062407/56649d1a5503460f949f041b/html5/thumbnails/16.jpg)
Time
.)( timegives This
./)1(2||
with hoseconsider tonly need weTherefore,
.2)/)1(2(
then,)/2(1 |I| If .2 Note
)/1(/)1(2
O
GIi
i
G
nnO
I
I
optcac
copt
![Page 17: Chapter 10 Complexity of Approximation (1) L-Reduction Ding-Zhu Du.](https://reader035.fdocuments.us/reader035/viewer/2022062407/56649d1a5503460f949f041b/html5/thumbnails/17.jpg)
MAX3SAT
clauses. satisfied of # the
maximiz toassignmentan find , 3CNF aGiven F
clauses. satisfied
/least at have toassignmentan find
, 3CNF aGiven :MAX3SAT-Aprrox-
ropt
Fr
![Page 18: Chapter 10 Complexity of Approximation (1) L-Reduction Ding-Zhu Du.](https://reader035.fdocuments.us/reader035/viewer/2022062407/56649d1a5503460f949f041b/html5/thumbnails/18.jpg)
Theorem
solvable.
time-polynomial is MAX3SAT-Approx-2
;|
0set else
|
1set hen t
) clauses(#) (clauses# if
do to1for
0
1
i
i
x
i
x
i
ii
FF
x
FF
x
xx
ni
![Page 19: Chapter 10 Complexity of Approximation (1) L-Reduction Ding-Zhu Du.](https://reader035.fdocuments.us/reader035/viewer/2022062407/56649d1a5503460f949f041b/html5/thumbnails/19.jpg)
Theorem
hard.-
is MAX3SAT-Approx- ,1constant someFor
NP
rr
This an important result proved using PCP system.
complete).-(APX
complete-SNP MAX is MAX3SAT
Theorem
![Page 20: Chapter 10 Complexity of Approximation (1) L-Reduction Ding-Zhu Du.](https://reader035.fdocuments.us/reader035/viewer/2022062407/56649d1a5503460f949f041b/html5/thumbnails/20.jpg)
Class MAX SNP (APX?)
solvable. time-polynomial
is PR-Approx-such that 1
constant a exists thereif SNP MAX tobelongs
PR problemon minimizatior on maximizatiA
rr
![Page 21: Chapter 10 Complexity of Approximation (1) L-Reduction Ding-Zhu Du.](https://reader035.fdocuments.us/reader035/viewer/2022062407/56649d1a5503460f949f041b/html5/thumbnails/21.jpg)
L-reduction
|))(()(|
|)())((|
such that on of solutions feasible to
)(on of solutions feasible from maps L2)(
);())((
such that of
instances to of instances from maps (L1)
such that 0, constants twoand , and
functions computable time-polynomial twoare thereif
. and problemson optimizati woConsider t
xhoptyobjb
xoptygobj
x
xhg
xoptaxhopt
h
bagh
PL
![Page 22: Chapter 10 Complexity of Approximation (1) L-Reduction Ding-Zhu Du.](https://reader035.fdocuments.us/reader035/viewer/2022062407/56649d1a5503460f949f041b/html5/thumbnails/22.jpg)
x )(xh
)(yg yxon of
solutions
feasible
)(on of
solutions
feasible
xh
![Page 23: Chapter 10 Complexity of Approximation (1) L-Reduction Ding-Zhu Du.](https://reader035.fdocuments.us/reader035/viewer/2022062407/56649d1a5503460f949f041b/html5/thumbnails/23.jpg)
VC-b
y.cardinalit its minimize cover to
vertexa find ,most at degree with graph aGiven bG
Theorem .3-VC-VC ,1any For PLbb
.3-VC-VC that trivialisit ,3any For PLbb
![Page 24: Chapter 10 Complexity of Approximation (1) L-Reduction Ding-Zhu Du.](https://reader035.fdocuments.us/reader035/viewer/2022062407/56649d1a5503460f949f041b/html5/thumbnails/24.jpg)
. degree with graph aconsider ,4For bGb
1 2
3
45
1 2
3
45G G’
v
vc
vc
![Page 25: Chapter 10 Complexity of Approximation (1) L-Reduction Ding-Zhu Du.](https://reader035.fdocuments.us/reader035/viewer/2022062407/56649d1a5503460f949f041b/html5/thumbnails/25.jpg)
'. ofcover - vertexa is )()(
ofcover - vertexa is
Gcc
GS
vSvvSv
'. ofcover - vertexminimum a is )()(
ofcover - vertexminimum a is
Gcc
GS
vSvvSv
)()12(2)()'( GoptbnmGoptGopt
![Page 26: Chapter 10 Complexity of Approximation (1) L-Reduction Ding-Zhu Du.](https://reader035.fdocuments.us/reader035/viewer/2022062407/56649d1a5503460f949f041b/html5/thumbnails/26.jpg)
).'(|'|)(|)'(|
Thus,
.|||)('|' that Note
'. ofcover - vertexa is )'(Then
}.'|{)'(
define ,' of 'cover x each verteFor
GoptSGoptSg
cccSSc
GSg
ScvSg
GS
vvvv
v
![Page 27: Chapter 10 Complexity of Approximation (1) L-Reduction Ding-Zhu Du.](https://reader035.fdocuments.us/reader035/viewer/2022062407/56649d1a5503460f949f041b/html5/thumbnails/27.jpg)
Properties
. , pL
pL
pL
.-Approx--Approx-
:1,1 then , If
sr
srpT
pL
(P1)
(P2)
![Page 28: Chapter 10 Complexity of Approximation (1) L-Reduction Ding-Zhu Du.](https://reader035.fdocuments.us/reader035/viewer/2022062407/56649d1a5503460f949f041b/html5/thumbnails/28.jpg)
x )(xh
)(' yg y
xon of
solutions
feasible
)(on of
solutions
feasible
xh
))((' xhh
))('( ygg
))(('on of
solutions
feasible
xhh
h
pL p
L
'h
g 'g
![Page 29: Chapter 10 Complexity of Approximation (1) L-Reduction Ding-Zhu Du.](https://reader035.fdocuments.us/reader035/viewer/2022062407/56649d1a5503460f949f041b/html5/thumbnails/29.jpg)
. , pL
pL
pL
|)))(('()(|'
|))(())('(|
|)()))('((|
)('))((')))(('(
xhhoptyobjbb
xhoptygobjb
xoptyggobj
xoptaaxhoptaxhhopt
![Page 30: Chapter 10 Complexity of Approximation (1) L-Reduction Ding-Zhu Du.](https://reader035.fdocuments.us/reader035/viewer/2022062407/56649d1a5503460f949f041b/html5/thumbnails/30.jpg)
PTASPTAS
MAX SNPMAX SNP
PTAS
PTAS ,
p
L
![Page 31: Chapter 10 Complexity of Approximation (1) L-Reduction Ding-Zhu Du.](https://reader035.fdocuments.us/reader035/viewer/2022062407/56649d1a5503460f949f041b/html5/thumbnails/31.jpg)
.for ion approximat-)1( is )(then
,for ion approximat-)(1 is If
))((
)))(()((1
)(
)())((1
)(
))((
problems.on minimizati are and Both :1
abyg
y
xhopt
xhoptyobjab
xopt
xoptygobj
xopt
ygobj
case
![Page 32: Chapter 10 Complexity of Approximation (1) L-Reduction Ding-Zhu Du.](https://reader035.fdocuments.us/reader035/viewer/2022062407/56649d1a5503460f949f041b/html5/thumbnails/32.jpg)
.for ion approximat-)1( is )(then
,for ion approximat-)(1 is If
)(
))())(((1
))((
))())(((1
)(
)())((1
)(
))((
problem.on maximizati
a is and problemon minimizati a is :2
abyg
y
yobj
yobjxhoptab
xhopt
yobjxhoptab
xopt
xoptygobj
xopt
ygobj
case
![Page 33: Chapter 10 Complexity of Approximation (1) L-Reduction Ding-Zhu Du.](https://reader035.fdocuments.us/reader035/viewer/2022062407/56649d1a5503460f949f041b/html5/thumbnails/33.jpg)
.for ion approximat-)1( is )(then
,for ion approximat-)(1 is If
))(())(()(
1
1
)())(()(
1
1
)())((
)(
))((
)(
on.minimizati a is andon maximizati a is :3
)(
abyg
y
xhoptxhoptyobj
ab
xoptygobjxopt
xoptygobjopt
xopt
ygobj
xopt
case
x
![Page 34: Chapter 10 Complexity of Approximation (1) L-Reduction Ding-Zhu Du.](https://reader035.fdocuments.us/reader035/viewer/2022062407/56649d1a5503460f949f041b/html5/thumbnails/34.jpg)
.for ion approximat-)1( is )(then
,for ion approximat-)(1 is If
)()())((
1
1
))(()())((
1
1
)())(()(
1
1
)())((
)(
))((
)(
problems.on maximizati are and Both :4
)(
abyg
y
yobjyobjxhopt
abxhopt
yobjxhoptab
xoptygobjxopt
xoptygobjopt
xopt
ygobj
xopt
case
x
![Page 35: Chapter 10 Complexity of Approximation (1) L-Reduction Ding-Zhu Du.](https://reader035.fdocuments.us/reader035/viewer/2022062407/56649d1a5503460f949f041b/html5/thumbnails/35.jpg)
MAX SNP-complete (APX-complete)
.
SNP, MAXany for and SNP MAX if
complete-SNP MAX is problemon optimizatiAn
pL
Theorem
PTAS. no has then , if i.e.,
hard,- is -Approx- ,1
, problem complete-SNP MAX
NPP
NPrr
![Page 36: Chapter 10 Complexity of Approximation (1) L-Reduction Ding-Zhu Du.](https://reader035.fdocuments.us/reader035/viewer/2022062407/56649d1a5503460f949f041b/html5/thumbnails/36.jpg)
MAX3SAT-3
clauses. satisfied of # themaximize
toassignmentan find times,most threeat
appears bleeach varia that 3CNF aGiven F
complete.-SNP MAX is 3-MAX3SAT
Theorem
3-MAX3SATMAX3SAT pL
![Page 37: Chapter 10 Complexity of Approximation (1) L-Reduction Ding-Zhu Du.](https://reader035.fdocuments.us/reader035/viewer/2022062407/56649d1a5503460f949f041b/html5/thumbnails/37.jpg)
VC-4 is MAX SNP-complete
graph. a is where
,)(construct , 3CNFeach For
4-VC of inputs3-SAT3MAX of inputs:
4-VC3-SAT3MAX
G
GFfF
f
pL
Proof.
![Page 38: Chapter 10 Complexity of Approximation (1) L-Reduction Ding-Zhu Du.](https://reader035.fdocuments.us/reader035/viewer/2022062407/56649d1a5503460f949f041b/html5/thumbnails/38.jpg)
1x
))(( 143231 xxxxxx
2x 3x 4x1x 2x 3x 4x
1c
13c
11c 12c2c
23c
22c21c
![Page 39: Chapter 10 Complexity of Approximation (1) L-Reduction Ding-Zhu Du.](https://reader035.fdocuments.us/reader035/viewer/2022062407/56649d1a5503460f949f041b/html5/thumbnails/39.jpg)
hard.-SNP MAX isCover -Set
Theorem
Proof. Cover-Set3-VC pL
. ofcover set a is }|{
ofcover - vertexa is
Then }.|{
set aconstruct ,each For
3.-VC of instancean be ),(Let
ECvsS
GVC
evEes
Vv
EVG
vC
v
![Page 40: Chapter 10 Complexity of Approximation (1) L-Reduction Ding-Zhu Du.](https://reader035.fdocuments.us/reader035/viewer/2022062407/56649d1a5503460f949f041b/html5/thumbnails/40.jpg)
Theorem
. unlessCover -Setfor
ionapproximat-) (ln time-polynomial no is There
PNP
no
).(
unlessCover -Setfor ion approximat- ln
time-polynomial no is there,1For
) (log nOnDTIMENP
n
Theorem
Proved using PCP system
![Page 41: Chapter 10 Complexity of Approximation (1) L-Reduction Ding-Zhu Du.](https://reader035.fdocuments.us/reader035/viewer/2022062407/56649d1a5503460f949f041b/html5/thumbnails/41.jpg)
).(
unless MCDSfor ion approximat- ln
time-polynomial no is there,1For
) (log nOnDTIMENP
n
Theorem
MCDS
y.cardinalit its minimize set to
dominating connected a find ,graph aGiven G
![Page 42: Chapter 10 Complexity of Approximation (1) L-Reduction Ding-Zhu Du.](https://reader035.fdocuments.us/reader035/viewer/2022062407/56649d1a5503460f949f041b/html5/thumbnails/42.jpg)
1x 2x 3x 4x 5x 6x
1S 2S 3S
}.,,{
},,,,{},,,{
6543
543123211
xxxS
xxxxSxxxS
![Page 43: Chapter 10 Complexity of Approximation (1) L-Reduction Ding-Zhu Du.](https://reader035.fdocuments.us/reader035/viewer/2022062407/56649d1a5503460f949f041b/html5/thumbnails/43.jpg)
. unless CLIQUEfor
ionapproximat- time-polynomial no is there,1For
PNP
ns s
subgraph.) complete a is (A y.cardinalit its
maximize toclique a find,graph aGiven
clique
G
CLIQUE
Theorem
Proved with PCP system.
![Page 44: Chapter 10 Complexity of Approximation (1) L-Reduction Ding-Zhu Du.](https://reader035.fdocuments.us/reader035/viewer/2022062407/56649d1a5503460f949f041b/html5/thumbnails/44.jpg)
disjunct?- is
,1integer an and matrix binary aGiven
:-coin is problem following theProve
dM
dM
NP
matrix?binary -by-
disjunct- a thereis ,0,, integersGiven
:in is problem following theProve 2
nt
dtdn
p
1
2
Exercises
![Page 45: Chapter 10 Complexity of Approximation (1) L-Reduction Ding-Zhu Du.](https://reader035.fdocuments.us/reader035/viewer/2022062407/56649d1a5503460f949f041b/html5/thumbnails/45.jpg)
rows.) of # maximum thedeletingby submatrix
disjunct-2 a find ,matrix aGiven :DS-2-(Min
2. size has pool
every that case specialin hard- is DS-2-Min
:Prove
M
NP
3
![Page 46: Chapter 10 Complexity of Approximation (1) L-Reduction Ding-Zhu Du.](https://reader035.fdocuments.us/reader035/viewer/2022062407/56649d1a5503460f949f041b/html5/thumbnails/46.jpg)
hint
DS-2-MinVC pm
![Page 47: Chapter 10 Complexity of Approximation (1) L-Reduction Ding-Zhu Du.](https://reader035.fdocuments.us/reader035/viewer/2022062407/56649d1a5503460f949f041b/html5/thumbnails/47.jpg)
• Min-2-DS is MAX SNP-complete in the case that all given pools have size at most 2.
4 Prove that
![Page 48: Chapter 10 Complexity of Approximation (1) L-Reduction Ding-Zhu Du.](https://reader035.fdocuments.us/reader035/viewer/2022062407/56649d1a5503460f949f041b/html5/thumbnails/48.jpg)
5. Is TSP with triangular inequality MAX SNP-complete?
![Page 49: Chapter 10 Complexity of Approximation (1) L-Reduction Ding-Zhu Du.](https://reader035.fdocuments.us/reader035/viewer/2022062407/56649d1a5503460f949f041b/html5/thumbnails/49.jpg)