Chapter 3 Restriction

(2) Greedy k-restricted Steiner tree

Ding-Zhu Du

A general result on greedy algorithmWith non-integer potential function

Consider a monotone increasing, submodular function

Rf E 2:}0)(:|{)( AfExEAf x

Consider the following problem:

)( subject to

)()( min

fA

xcAcAx

where REc : is a nonnegative cost function

.output

};{ and

)(

)( maximize to choose

do )( while

;

A

xAA

xc

AfEx

fA

A

x

Greedy Algorithm G

Theorem

Suppose in Greedy Algorithm G, selected x always satisfies

1)(/)( xcAfx

Then its p.r. opt

ff )(*ln1

where )(for )(* fAAff

Proof. Let },...,,{ 21 gg xxxA

}.,...,,{ 21 ii xxxA

},...,,{* 21 kyyyA

be obtained by

Greedy Algorithm G.

Denote

Let be an optimal solution.

Denote ).(* iAi Afa

)(

)(*)(

)(*)(

)()(

1

11

*1*1

ix

ii

ii

iAiAii

Af

AfAAf

AfAAf

AfAfaa

i

)(

)(

)(

)(max

)(

)(11

11

1 11

i

ix

j

iy

kjk

j j

k

j iyi

xc

Af

yc

Af

yc

Af

opt

aijj

)(11

i

iii

xc

aa

opt

a

optxcxc

optxci

iii

i

i

ea

eaopt

xcaa

/))()((0

/)(11

1

))(

1(

0)()*()(* gggAg AfAAfAfa

Note that

g

ii

g

iix

g

optxcAf

fAfAffa

i11

1

00

.)()(

)()()(*

There exists i such thatii aopta 1

.1)(

)(

''

''

'

'

such that ''')(

. ''')(

thatso '' ,'

1

1

1

1

1

1

i

ix

i

ixii

ii

xc

Af

c

a

c

a

ccxc

aaAfaa

aoptaoptaa

i

i

Let

Let

.)(

)(

' 1

1

opt

a

xc

Af

c

opta i

i

ixi i

Note that

.ln)()('

)'

1(

01

/))()('(0

1

opt

aoptxcxcc

eaopt

caopt

i

optxcxcci

i

So

.

)()(''

)()(''

1211

11

2

2

opt

aaaaaopt

AfAfa

xcxcc

ggiii

gxix

gi

gi

Note

Hence,

).*

ln1()(opt

foptAc g

),:()()( HPmstPmstHf

is the length of MST on P after terminals in eachconnected component of H are contracted into a point.

Consider

where ):( HPmst

Consider

the set of all full component of size at most k.

kQ

Theorem. is a monotone increasing submodular

function on f

.kQ

).()(

by from induced is 2:

. of subsets of collection a is

.submodular is 2 : Suppose

SgAf

gRf

EC

Rg

As

C

E

)()( BfAfBA xx

)()( SgSgBA BsxAsx

.submodular and increasing monotone is g

For k >2, consider each

as a set of edges in a spanning treeon terminals.

kQT

)(Te

)).(()( TefAfAT

For ,kQA

)()()(},{ AfAfAf yxyx

)(}){( AfxAf yy

submodular increasing monotone is f

iff

i.e.,

}){:():()(}){( xAPmstAPmstAfxAf

}),{:():(

)(}),{(

yxAPmstAPmst

AfyxAf

}){:():()(}){( yAPmstAPmstAfyAf

)()( BfAfBA xx iff

}){:():(

)(}){(

xAPmstAPmst

AfxAf

x

x

For k=2,

is the length of a longest edge in the path connectingtwo endpoints of , in MST(A).x

}),{:():(

)(}),{(

yxAPmstAPmst

AfyxAf

. and

cycles twocontains ):(

21 CC

yxAPMST

).(\in edgelongest thebe ''Let

. cycle a contains ')(

).(in edgelongest thebe 'Let

3

321

21

yxCe

CeyxCC

yxCCe

).''(length)'(length ee

y

x

x

x

x

}),{:():(

)(}),{(

yxAPmstAPmst

AfyxAf

y

x x

x

}),{:():(

)(}),{(

yxAPmstAPmst

AfyxAf

Theorem Greedy Algorithm G is

)()ln1()()ln1(

)()(

)()

)(

)()()(

ln1()())(

)(ln1(

21

2

PsmtPsmt

PsmtPsmt

Psmt

Psmt

PsmtPsmtPmst

PsmtPsmt

Pmst

kk

k

kk

k

))(

)(ln1(

Psmt

Pmst

k

-approximation for .)(Psmtk

Greedy Algorithm G produces approximation solutionfor SMT with length at most

Loss(T)

TTv

QTTeTvmstTloss k

ofset vertex the)(

for ))(:)(()(

Loss(T)

)(2

1)( TlengthTloss

Operation BA

B

A

BA

)).(()()(

, and eeSteiner tr aFor

KTmstTcAg

QAT

AKT

k

Function )(AgT

T 2K

1K

Lemma

)'()'(

.,.

)'()')(()(

.,.

)'()'()(

.,.

)'()()'(

)( KgKg

ei

KgKKTmstmstKTmst

ei

KgKKTmstKTmst

ei

KgKgKKg

TKTMST

T

T

TTT

Function )'( and )'( )( KgKg KTMSTT

T

'KK

Lemma

function. dpolymatroi a is )(AgT

Proof.

)()(

)'()(

})',({

)')(())((

)(})',{()(

'

))(())((

))((

',

AgAg

KgKg

KKg

KKYTmstYTmst

AgKKAgAg

TKTK

YTMSTYTMST

YTMST

AYAY

TTTKK

AYAY

AY

.output

);( and

)(

)( maximize to choose

do ]0)([ while

);(

A

KTMSTT

Kc

KgQK

KgQK

PMSTT

Tk

Tk

Greedy Algorithm

)()()(

then,0)(,any for If

ee.Steiner tr restricted- a be Let

TlossPsmtTc

KgQK

kT

k

Tk

Lemma

Proof

)()(

)()(

.0)()()(

. )( Suppose

1

11

1

TlossPsmt

KKTmstTc

KgKgKKg

KKPSMT

k

p

pTTpT

pk

.output

);( and

)(

)( maximize to choose

do ]0)([ while

);(

A

KTMSTT

Kloss

KgQK

KgQK

PMSTT

Tk

Tk

Greedy Algorithm

.output

));((

)( and

size)smaller with one choose tied,(if

)(

)( maximize to choose

do ]0)([ while

)(

);(

A

KTMSTH

KTMSTT

Kloss

KgQK

KgQK

PMSTH

PMSTT

Hk

Hk

Robin-Zelikvosky

What is ? )(K

)(Kpoint. a into contracted is

)(in edgeEach KLoss

Lemma

)()()()(

iteration.

th theof end at the thebe and

iterationth at the selected be Let

11 iiiiH

i

i

HcHcKlossKg

iHH

iK

i