Chapter 7Linear Programming

(3) Pipage Rounding

Ding-Zhu Du

Pipage Rounding

example

Maximum Coverage

.by hit in subsets of weight total themaximize to||with

subset a find ,integer positive a and in subsetson function

weight enonnegativ with },...,1{set a of subsets of family aGiven

XCpX

IXpCw

nIC

ILP Formulation

.10

}1,0{

,

,,...,1 , s.t.

max

1

1

j

i

n

ii

Siji

m

jjj

z

x

px

mjzx

zw

j

Alternative Formulation 1

}1,0{

, s.t.

},1min{)( max

1

1

i

n

ii

m

j Siij

x

px

xwxLj

Alternative Formulation 2

}1,0{

, s.t.

))1(1()( max

1

1

i

n

ii

m

jSi ij

x

px

xwxFj

Relationship

.10for ),1min(11

|)|(

)1(

1)1(1

11 where

)()(

jjj

j

j

j

Sii

Sii

Sii

k

Sii

j

k

Sii

Sii

xxcxck

x

Skk

x

x

ec

xcLxF

Proof.

Relationship

.1for ),1min(1

1111

|)|(

)1(

1)1(1

11 where

)()(

jj

j

j

j

Sii

Sii

k

k

Sii

j

k

Sii

Sii

xxcckk

x

Skk

x

x

ec

xcLxF

Proof.

).1

1()1

11()( So,

[0,1].in concave and increasing monotone is )(

.1

11)1( ,0)0(

.11)(Set

ez

kzzg

zg

kgg

k

zzg

k

k

k

.0)1

(1)1()(''

.01)1

(1)('

2

11

kk

zkzg

k

z

kk

zkzg

k

kk

0 1

Relaxation

.10

, s.t.

},1min{)( max

1

1

i

n

ii

m

j Siij

x

px

xwxLj

.10

00

,

,,...,1 , s.t.

max

1

1

j

i

n

ii

Siji

m

jjj

z

x

px

mjzx

zw

j

).( in time computed

becan * optimalAn 5.3nO

x

Pipage Rounding

10 * ix*ix

otherwise.

)()( if 'Set

).*,*1min(* ),*,*1min(*

),*,*1min(* ),*,*1min(*

and ,,for * Define

).( 1*0 and 1*0 Choose

z

zFyFyx

xxxzxxxz

xxxyxxxy

jkixzy

jixx

kjjjkjkk

jkjjjkkk

iii

jk

Property

.*)( ,*)(

and ,,for *)( where

.t.convex w.r is ))(( because

*)()'(

jjkk

ii

xxxx

jkixx

xF

xFxF

)0(*

),1min( where)(

),1min( where)(

22

11

xx

xxxz

xxxy

kj

jk

Theorem

.)/11(*)()/11(*)()()(

Then rounding. pipageby obtainedsolution integer thebe Let

.ionapproximat)1/( has problem coverage weight Maximum

optexLexFxFxL

x

ee

Proof

Pipage Rounding

framework

Integer Programming

.for }1,0{

for s.t.

)( max

)(

Eex

UVvpx

xF

e

vve

e

.for }1,0{

for s.t.

)( max

)(

Eex

UVvpx

xL

e

vve

e

)()(

,solution feasibleinteger For

xLxF

x

).,,(graph bipartite aConsider EUVG

Relaxation

.for 10

for s.t.

)( max

)(

Eex

UVvpx

xF

e

vve

e

.for 10

for s.t.

)( max

)(

Eex

UVvpx

xL

e

vve

e

)()( xLcxF Solve easily to obtain Optimal solution

By ɛ-convexity, obtain an integer solution from , by pipage rounding, such that

*x*x x

*)()( xFxF optcxLcxFxFxL *)(*)()()(

Pipage Rounding

.subgraph bipartiteConsider xH

V U10 ijx

cycle. aor path maximal aeither , Find R

R

V U

.in cycle aor path maximal aeither , Find xHR

otherwise )(

))(())(( if )('

feasible. is )( ],,[for Then

)).1(min,min(min

)),1(min,min(minSet

. if

, if

, if

)(

settingby )( Define

. and matchings twointo Decompose

2

211

21

2

1

2

1

21

12

21

x

xFxFxx

x

xx

xx

Mex

Mex

Rex

x

x

MMR

eMeeMe

eMeeMe

e

e

e

e

ɛ-convexity

. .t.convex w.r is ))(( ,any For xFR

))).(()),((max())((

],,[-any for Thus,

21

21

xFxFxF

Pipage Rounding

Applications

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uuau

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}.,...,1{, },1,0{

},...,1{ },1,0{

},...,1{ ,1

,

},...,1{ , s.t.

max

1

1 1

:,

1

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sets. selected

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from ones most at with from sets most at Choose

. },,...,1|{

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},...,{ 1 nuuU

nodes

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nodes

}.,...,1{, },1,0{

},...,1{ },1,0{

},...,1{ ,1

, s.t.

),1min()( max

1

1 1

1 :,

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nlx

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ij

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jij

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ijl

jy1

ijy

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jiS

1S

jy1

ijy

}.,...,1{, },1,0{

},...,1{ },1,0{

},...,1{ ,1

, s.t.

))1(1()( max

1

1 1

1 :,

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edges. only two containspath maximaleach because

convexity - has )(

).()1

1()(

yF

yLe

yF

.)1

1()( with solution

ion approximatan get can werounding, pipageWith

opte

yLy

Theorem

Thanks, End