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!
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