Chapter 8 PD-Method and Local Ratio
-
Upload
ifeoma-young -
Category
Documents
-
view
42 -
download
2
description
Transcript of Chapter 8 PD-Method and Local Ratio
.** Moreover,ly.respective ,* and *say solutions, optimal haveboth they
then solution, feasible have dual theand primal both the Ifsolution.
feasible no has dual theiff valueminimum has primal The. value
maximum has dual theiffsolution feasible no has primal The
bycxyx
Theorem
.),(Then
.0 and i.e., , with associatedsolution optimalan is Suppose
feasible. dual is S,
.),(),()(
Then .Set
.0
satisfying basis feasible optimalan obtaincan wemethod, hicallexicagrapby theat recall statement, 3rd theshow To
.from follows statements First two
11
1
1,
1
1
1
cxxAAcxcxAAccyAxyb
xbAxIx
y
cccAAccAAAcyA
Acy
AAcc
I
ybcx
IIIIIIIIII
III
IIIIIIIIII
II
IIII
Proof.
Complementary Slackness Condition
.)(
optimal. are and both case, asuch In
.0)(such that solution feasible dual a exists there
iff optimal is solution feasible primalA
xyAcybcx
yx
xyAcy
x
Remark
Complementary Slackness Condition
.optimal. are and both case, asuch In
.0)( ,0)(
such that solution feasible dual a exists there iff optimal is solution feasible primalA
ybyAxcxyx
bAxyxyAc
yx
Remark
Primal c-s ConditionDual c-s Condition
Idea0)()1( xyAcAxy
.01 if increaseonly ,much too)1( increase not toorder in
0)( with choose
0)( keep
;1 to0 some change ; feasible primalnot is
xayAxy
yacj
xyac
xxx
ii
jj
jj
j
jj
iteration?each in allat increased benot would)1(you think Do AxyQuestion:
.Output
while.-end;1
)(for and )(for set
;1 and if set
min
such that )( choose
}0|{)(
and 0}|{)(set
begin do feasible primenot is while
.0 0, 0,set Initially,
11
11
)(
1)(
)(
1
1
00
kA
ki
ki
ki
ki
kr
kj
kj
m
kIiij
m
i
kiijj
kJjm
kIiir
m
i
kiirr
n
j
kjij
kj
k
xx
kkkIiyykIiyy
xrjxx
a
yac
a
yac
kJr
xaikI
xjkJ
x
kyx
))(( nmnO
.feasible-dual
still is Hence, . Thus, .0 ),(
),(for that Note ).(for , of choiceby and
0 that note we,1for (a) see To .1consider weNow,
.0for hold they Suppose . trivially trueare (b) and (a) all Initially,on induction by it prove We
and 0, (b)
feasible,-dual is (a)
:following thehave wek,any For
1
11
1
1
1
1
1
kj
m
i
kiij
m
i
kiijij
j
m
i
kiij
ki
ki
kjj
m
i
kiij
k
ycyayaakJj
kIikJjcya
yyk+k +
kk.
xcya
y
Lemma
Proof
0. = Therefore, . know we, Since
.0 ,hypothesisinduction By . then , If
. allfor that note we,1for (b) see To
and 0, (b)
1
1
1
11
1
1
1
1
1
kj
kjr
m
i
kiir
kjj
m
i
kiijj
m
i
kiij
m
i
kiij
m
i
kiij
kjj
m
i
kiij
xxrjcya
xcyacya
jyayak+
xcya
optf
yfyf
Ax = ycx
x
xzAyc
xxh
afoptfcx
hm
i
hi
AhA
Aj
Ahh
hA
n
jijmi
A
)1(
Thus, .10 that Note
0)(
and Then .iterations at stops Algorithm Suppose
.max = where
1
11
Lemma
Proof
Primal and Dual
.0 ,0
where01
1 s.t. min
Ac
xAxcx
.0 ,0
where0,0
s.t.11 max
Ac
zyczyA
zy
1. Is redundant?z Yes!!!
Complementary-Slackness
LP. dualfor optimal is and LP, primalfor optimal is
ifonly and if0)()1()1(
0)()1()1(1
yx
xzyAcxzAxyxzyAcxzAxyycx
.Output
while.-end;1
,0);(max set
)(for and )(for set
;1 and if set
min
such that )( choose
}0|{)(
and 0}|{)(set
begin do feasible primenot is while
.0 0,),( 0,set Initially,
1
11
11
11
)(
1)(
)(
1
1
000
k
m
ij
kiij
kj
ki
ki
ki
ki
kr
kj
kj
m
kIiij
m
iiijj
kJjm
kIiir
m
iiirr
n
j
kjij
kj
k
x
kk
cyaz
kIiyykIiyy
xrjxx
a
yac
a
yac
kJr
xaikI
xjkJ
x
kzyx
))(( nmnO
.Output
while.-end;1
)(for and )(for set
;1 and if set
min
such that )( choose
}1|{)(
and 0}|{)(set
begin do feasible primenot is while
.0 0, 0,set Initially,
11
11
)(
1)(
)(
1
1
00
kA
ki
ki
ki
ki
kr
kj
kj
m
kIiij
m
i
kiijj
kJjm
kIiir
m
i
kiirr
n
ji
kjij
kj
k
xx
kkkIiyykIiyy
xrjxx
a
yac
a
yac
kJr
bxaikI
xjkJ
x
kyx
do covered,not edgean exists thereiftices}chosen vernot {)( kJ
edge} coverednot {)( kI
of degree edge uncovered of weight remaining
jj
ight vertex weupdate
.Output
while.-end;1
)(for
and )(for set
;1 and if set
min
such that )( choose
}1|{)(
and 0}|{)(set
begin do feasible primenot is while
.0 0, 0,set Initially,
1
1
11
)(
1)(
)(
1
1
00
kA
ki
ki
ki
ki
kr
kj
kj
m
kIiij
m
i
kiijj
kJjm
kIiir
m
i
kiirr
n
ji
kjij
kj
k
xx
kkkIiyy
kIiyy
xrjxx
a
yac
a
yac
kJr
bxaikI
xjkJ
x
kyx
.while.-end
)at edges uncovered of (# xeach verteat weight update
};{
)(at edge uncovered of # of weight remaining
minimize to vertex choose edges}; uncovered{
};in not vertices{ exists edge uncovered while
. Initially,
jcc
rCCr
rCr
ICJ
C
jj
A Special Case
.2||2and || weight totalhascover ex every vert because is This
ion.approximat-2 iscover ex every vert ),deg( that caseIn
1 optEccE
jc
n
j
Weight Decomposition
.)(deg))(deg(
Hence,
).(deg))(deg(:parts in two decomposed is weight iteration,each In
1cov
1cov
1
covcov
n
jjeredun
n
jjeredunj
n
jjj
ereduneredunjj
xjxjcxc
jjcc