Greedy Approximation with Greedy Approximation with Non-submodular Potential Non-submodular Potential

FunctionFunction

Ding-Zhu DuDing-Zhu Du

Chapter 2. Greedy Chapter 2. Greedy StrategyStrategy

III. Nonsubmodular Potential III. Nonsubmodular Potential FunctionFunction

Connected Dominating SetConnected Dominating Set

Given a graph, find a minimum node-Given a graph, find a minimum node-subset such that subset such that

each node is either in the subset or each node is either in the subset or adjacent to a node in the subset andadjacent to a node in the subset and

subgraph induced by the subset is subgraph induced by the subset is connected. connected.

HistoryHistory

TwoTwo stage Greedy ( stage Greedy (ln Δ +3ln Δ +3)-approximation )-approximation (Guha & Khullar, 1998) where (Guha & Khullar, 1998) where Δ is the Δ is the maximum degree in input graph. maximum degree in input graph.

There is no polynomial time approximation There is no polynomial time approximation with performance ratiowith performance ratio (1-a) lnΔ (1-a) lnΔ unlessunless NP NP in DTIME(n^{log log n}) in DTIME(n^{log log n}) for any a > 0. for any a > 0.

Now, we show that there is a one-stage Now, we show that there is a one-stage Greedy (Greedy (1+a)(1+ln (Δ-1) 1+a)(1+ln (Δ-1) )-approximation for )-approximation for any a > 0 with a nonsupmodular potential any a > 0 with a nonsupmodular potential function.function.

What’s the Potential What’s the Potential Function?Function?

f(A)=p(A)+q(A)f(A)=p(A)+q(A) p(A) = # of connected components p(A) = # of connected components

after adding edges incident to A after adding edges incident to A ((supmodularsupmodular))

q(A) = # of connected components q(A) = # of connected components of the subgraph induced by A of the subgraph induced by A ((nonsupmodularnonsupmodular))

ExampleExample

p(circles)=1p(circles)=1

q(circles)=2 q(circles)=2

x

-p-q is not submodular-p-q is not submodular

BackgroundBackground

There exist many greedy algorithms There exist many greedy algorithms in the literature.in the literature.

Some have theoretical analysis. But, Some have theoretical analysis. But, most of them do not.most of them do not.

A greedy algorithm with theoretical A greedy algorithm with theoretical analysis usually has a submodular analysis usually has a submodular potential function. potential function.

Is it true?Is it true?

Every previously known one-stage Every previously known one-stage greedy approximation with greedy approximation with theoretical analysis has a theoretical analysis has a submodular (or supermodular) submodular (or supermodular) potential function. potential function.

Almost, only one exception which is Almost, only one exception which is about Steiner tree. about Steiner tree.

How should we do with How should we do with nonsubmodular functions?nonsubmodular functions?

Find a space to play your Find a space to play your tricktrick

Where is the space?Where is the space?

1 2

1

1

Suppose , , ..., are selected by Greedy

Algorithm. Denote { , ..., }. Then

( ) ( ) (| | ( )) /

k

i i

i i i

S S S

C S S

f C f C E f C opt

Why the inequality true?Why the inequality true?

1

1

1, ..., 1, ...,

Because

( ) ( ) for every ,

including those in the optimal solution *.

Note that | | ( *) and | * | .

Suppose * { } and * { }.

Then ( *) ( *).

i

j

i i

opt i i

i j

s f C sf C S

S C

E f C opt C

C X X C X X

f C x f C C

1 1

Moreover, by submodularity of ,

( ) ( *). j ji i j

f

x f C x f C C

optCfE

optCfCCf

optCCf

optCfCf

optj

CfCf

i

ii

optjjiX

optjiXiS

iXiS

j

ji

ji

/))(||(

/))(*)((

/*))((

/))(()( Thus,

10 allfor

)()( So,

10

10

1

11

11

ExampleExample

In a Steiner tree, In a Steiner tree, all Steiner nodes all Steiner nodes form a dominating form a dominating set.set.

In a full Steiner In a full Steiner tree, all Steiner tree, all Steiner nodes form a nodes form a connected connected dominating set. dominating set.

ObservationsObservations

The submodularity has nothing to do The submodularity has nothing to do with sequence chosen by the greedy with sequence chosen by the greedy algorithm.algorithm.

It is only about It is only about X1, …, XoptX1, …, Xopt The ordering of The ordering of X1, …, XoptX1, …, Xopt is free to is free to

choose. choose.

When When ff is nonsubmodular is nonsubmodular

1

1, ...,

1, ...,

1, ...,

(1) We may choose such that is

submodular on { }.

(1-iterated Steiner tree)

(2) We may choose certain ordering of to

make ( *)j j

opt

op

op

i j

X X f

X X

X X

x f C C x 1 ( ) smaller.

(Connected dominating set)

if C

IdeaIdea

Organize optimal solution in such an Organize optimal solution in such an ordering such that every prefix ordering such that every prefix subsequence induces a connected subsequence induces a connected subgraph, i.e.,subgraph, i.e.,

For prefix subsequences A and B,For prefix subsequences A and B,

ΔΔxxq(B)-Δq(B)-Δxxq(A) ≤ 1 for A supset of Bq(A) ≤ 1 for A supset of B

Algorithm CDS1Algorithm CDS1

.return

};{

and )( maximize to vertex a choose

do 2)( while

;

C

xCC

Cfx

Cf

C

x

Connected Dominating SetConnected Dominating Set

TheoremTheorem. Algorithm CDS1 gives an . Algorithm CDS1 gives an

(2+ ln Δ)-approximation, where Δ is (2+ ln Δ)-approximation, where Δ is the maximum node degree.the maximum node degree.

ProofProof

.1)()*(

Thus,

.1)()*(

Then }.,...,{* and },...,{ Denote

subgraph. connected a induces },...,{ ,any for and

set dominating connected minimum a be }{Let

algorithm.Greedy by in turn selected are ,..., Suppose

1

1

11

1

1

1

iyjiy

iyjiy

jjii

i

opt

g

CfCCf

CqCCq

yyCxxC

yyi

, ...,yy

xx

jj

jj

)1

1(

Then ).(1 Denote

.)(1

)()(

/))(1(

/))(*)(1(

/))*(1(

/))(()( Thus,

1 allfor

)()( So,

1

1

11

1

1

1

optaa

Cfopta

opt

CfoptCfCf

optCfopt

optCfCCfopt

optCCfyopt

optCfyCfx

optj

CfCfx

ii

ii

iii

i

ii

optjji

optjii

iyi

j

ji

ji

.

12 Then

.) hence and 2

then, i.e., exist,t doesn' such If(

.

satisfying onelargest thebe to Choose

)/11(

/0

0

/00

opti

i

optiii

eaopt

optig

optgoptn

optai

aopt

i

eaoptaa

.)2)1( || :(

)./1(2/|)|1(/

))/(ln )/1(2(

12

Thus,

)/(ln

0

0

0

/0

optVnote

optoptVoptopta

optaoptopt

ioptg

optaopti

eaopt opti

Connected Dominating SetConnected Dominating Set

TheoremTheorem. For any a >1, there is a . For any a >1, there is a

a(1+ ln Δ)-approximation for the a(1+ ln Δ)-approximation for the minimum connected dominating set, minimum connected dominating set, where Δ is the maximum node where Δ is the maximum node degree.degree.

AlgorithmAlgorithm

.return

;

and ||/)( maximize to

vertices12most at of subset a choose

do 2)( while

;

C

XCC

XCf

kX

Cf

C

X

AnalysisAnalysis

.1||||

and ,1 allfor connected is

connected, is every

,12|| 1 ,11for 12||1

such that ..., , into *

set dominating connected minimum a Decompose

. Denote Algorithm.

Greedy by thechosen are ..., , Suppose

1

1

1

1

1

hoptYY

hjYY

Y

kYhikYk

YYC

XXC

XX

h

j

i

hi

h

ii

g

<k <k <k

> k

1)(

1)( )(

*)(*)(

*)(

* Denote

11

1

.1

iY

iYiY

jiYjiY

jiY

jj

Cf

CqCp

CCqCCp

CCf

YYC

j

jj

jj

j

hj

j

hj

jiY

hj

j

hj

iY

i

iX

j

iY

i

iX

Y

CCfh

Y

Cf

X

Cf

hjY

Cf

X

Cf

j

j

i

ji

1

1

1

1

1

1

1

||

*)()1(

||

)(

||

)(

1for ||

)(

||

)(

1

1

)1/(||11

1

1

1

1

1

)1

||1(

1||

Then ).1()( Denote

1

)1()(

1

))(*)(()1(

||

)(

hoptXi

iii

i

i

ii

ii

i

ii

i

iX

i

i

eahopt

Xaa

hopt

a

X

aa

hCfa

hopt

hCf

hopt

CfCCfh

X

Cf

1||'

'

such that '|| Write

.' and Denote

such that Choose

)1(ln )1( ||||

).1()1()( Note

1

1

1

1

1

1

0

)1/(|)||||(|01

|11

hopt

a

X

aa

d

b

d

b

ddX

aoptboptab

aopta

ia

hnhoptXX

hnhfa

eaa

i

i

ii

i

ii

ii

ii

hoptXXXi

ii

))1(

ln1)(1

1(|||| Thus,

)1(*)()(

)()(' ||||'

)1(ln )1( ||||

))1/(1(

1

1

11

12

1

)1/(|)|||(0

)1/(

1

opt

hn-

koptXX

hoptCfCfaopt

CfCfbXXd

opt

hn-hoptdXX

ea

eahoptdaopt

hopt

a

d

b

d

opta

g

ii

gigi

i

hoptXXd

hoptdii

ii

i

))1ln(1)(1

1(||||

Therefore,

1)1(

have we,1 Since

.2)1(

that induction by proved becan It

1

kXX

opt

hn

h

optn

g

EndEnd

Thanks!Thanks!