using homogeneous weights for approximating the partial cover problem reuven bar-yehuda
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Using Homogeneous Weights for Using Homogeneous Weights for Approximating Approximating
the Partial Cover Problemthe Partial Cover Problem
Reuven Bar-Yehuda
The partial set cover problem 2
The Partial Set Cover problemThe Partial Set Cover problemGiven:
Find: min
s.t.
Rt
REl
RV
EVH
:
:
),(
)(C
tCEl
VC
))((
The partial set cover problem 3
Minimal t-coverMinimal t-cover
C is a t-cover (a feasible solution)
if and
C is a minimal t-cover
if , is not a t-coverCv
VC tCEl ))((
}{\ vC
The partial set cover problem 4
Example (a simple graph)Example (a simple graph)
3
4
2
1
1 1
26
1
1
7
4 3
7
5
6
13t
The partial set cover problem 5
Example: 13-coverExample: 13-cover
3
4
2
1
1 1
26
1
1
7
4 3
7
5
6
The partial set cover problem 6
Example: minimal 13-coverExample: minimal 13-cover
3
4
2
1
1 1
26
1
1
7
4 3
7
5
6
The partial set cover problem 7
Example: optimal 13-coverExample: optimal 13-cover
3
4
2
1
1 1
26
1
1
7
4 3
7
5
6
The partial set cover problem 8
Example (a hyper-graph)Example (a hyper-graph)
7
5
10
2
1
6
2
2
1
4
VE
a
b
c
de={a,b,c,d}
11t
The partial set cover problem 9
Example: minimal 11-coverExample: minimal 11-cover
7
5
10
2
1
6
2
2
1
4
VE
a
b
c
d
The partial set cover problem 10
Example: optimal 11-coverExample: optimal 11-cover
7
5
10
2
1
6
2
2
1
4
VE
a
b
c
d
The partial set cover problem 11
Homogeneous Weight functionHomogeneous Weight function
Definition:
Property: any minimal cover is
a “good” approximation
))}((,min{)( vEltvVv
The partial set cover problem 12
Homogeneous Weight functionHomogeneous Weight function
Given:
Claim: C is a minimal t-cover
(define: )
Rt
REl
RV
EVH
:
:
),(
tCt E )(
eEeE max
The partial set cover problem 13
Proof:Proof:
Lemma 1: If C is a t-cover then
Lemma 2: If C is a minimal t-cover then
tC )(
tC E )(
The partial set cover problem 14
Proof:Proof:
Lemma 1: If C is a t-cover then
If √
Else
tC )(
Cv Cv
tCElvElvC ))(())(()()(
tvCv )(
The partial set cover problem 15
Proof:Proof:
Lemma 2 (a simple graph): If C is a minimal t-cover then
Case 1:
Case 2:
(case 1)
tC 2)(
2C
ttCvCvC CvCv
2)(max)()(
tvCv )(
}{vC
The partial set cover problem 16
Proof:Proof:
Lemma 2 (a simple graph): If C is a minimal t-cover then
Case 3:
tC 2)(
))(()2( tvC Cv
The partial set cover problem 17
L2: If C is a minimal t-cover then
Case 3:
Define:
tC 2)(
))(()2( tvC Cv
))('2(2
)()'(2
)())((2
)()(2))((
)(
}2:{},1:{
,)(
1
1
1
12
21
2121
Eltt
Eltt
ElCEl
ElElvEl
C
CeeECeeE
EEEECE
Cv
tCElt ))(('
The partial set cover problem 18
To complete the proof:
Let , and
C is a minimal t-cover
But
'2)( 1 tEl
Cv )()( 11 vEEvE
tvCEl })){\((
tvElvCEl ))((})){\(( 1
'))(( 1 tvEl
'2')( 1 ttCEl
The partial set cover problem 19
Example: hyper-graph, Example: hyper-graph, C is a minimal t-cover, C is a minimal t-cover, δ(C) > 2tδ(C) > 2t
1,9
1,9
2,5
3,5
4
4
1
1
1
1
VE
a
b
c
d
5,4e
12t
The partial set cover problem 20
Proof:Proof:
Lemma 2 (hyper-graph): If C is a minimal t-cover then
Case 1:
Case 2:
(case 1)
tC E )(
EC
ttCvCvC ECvCv
)(max)()(
tvCv )(
}{vC
The partial set cover problem 21
L2: If C is a minimal t-cover then
Case 3:
Define:
tC E )(
))(()( tvC CvE
))('(
)()'(
)()1()'(
)()1())((
)()())((
)(
}1:{},1:{
,)(
1
1
1
1
12
21
2121
Eltt
Eltt
Eltt
ElCEl
ElElvEl
C
CeeECeeE
EEEECE
EE
E
EE
EE
CvE
tCElt ))(('
The partial set cover problem 22
To complete the proof:
Let , and
C is a minimal t-cover
But
')( 1 tEl E
Cv )()( 11 vEEvE
tvCEl })){\((
tvElvCEl ))((})){\(( 1
'))(( 1 tvEl
'')( 1 ttCEl E
The partial set cover problem 23
Homogeneous Weight functionHomogeneous Weight function
Claim: C is a minimal t-cover, C* optimal
Proof:By Lemma 2:
By Lemma 1:
)()( *CC E
tC E )(
)( *CE
The partial set cover problem 24
Weight DecompositionWeight Decomposition
Local-Ratio Theorem:
C is an approximation w.r.t.
C is an approximation w.r.t.
C is an approximation w.r.t.
( )
0
E
E
E
The partial set cover problem 25
Algorithm Cover ( )Algorithm Cover ( )
If return
If return “no solution”
If return Cover( )
If return
Cover( )
0t
E
}0)(:{0 vvV 0V
tlEV ,,,,
tlVEEVV ,,),(\,\ 00
}0)(:{0 vvV 0V
))((,,),(\,\ 000 VEltlVEEVV 0V
The partial set cover problem 26
For each
C = Cover( )
For each
If is a t-cover
Return C
tlEV ,,,, 2
)()(min vvVv
Vv
)()()(
)()(
12
1
vvv
vv
Cv
}{\ vC }{\ vCC
The partial set cover problem 27
ExampleExample
1
1
2
1
1
10
1
4
2 23
1
199t
3
The partial set cover problem 28
ExampleExample
1
1
2
1
1
10
1
4,4
2,3 23,9
1,6
19,9
)(
)(min
9
v
v
t
Vv
3 ω,δ
The partial set cover problem 29
ExampleExample
1
1
2
1
1
10
1
10/3
3/2 43/2
0
35/2
9t
3 ω2
The partial set cover problem 30
ExampleExample
1
1
2
10
10/3,3
3/2,2 43/2,3
35/2,3
ω,δ
)(
)(min
369
v
v
t
Vv
The partial set cover problem 31
ExampleExample
1
1
2
10
13/12
0 77/4
61/4
ω2
3t
The partial set cover problem 32
ExampleExample
2
10
13/12,1
77/4,1
61/4,1
ω,δ
)(
)(min
123
v
v
t
Vv
The partial set cover problem 33
ExampleExample
1
1
2
1
1
10
1
4
2 23
1
199t
3
The partial set cover problem 34
ExampleExample
1
1
2
1
1
10
1
4
2 23
1
199t
3
The partial set cover problem 35
Algorithm CoverAlgorithm Cover
Claim: Algorithm Cover is a approximation
for the t-cover problem
Proof:
(L3: Algorithm Cover has at most iterations)
By induction on the number of iterations.
Base: for the empty set is optimal
Step: by the Local-Ratio Theorem
E
V2
0t
The partial set cover problem 36
Time complexityTime complexity
Lemma 4: Algorithm Cover can be implemented
in time
Proof:
vertex deletion adjacent edges deletion
For each such edge,
l(e) is subtracted from its |e| vertices
At most iterations, each iteration
)(2
HVO
)( HO
V2 )(VO
The partial set cover problem 37
Related workRelated work
t-VC: (simple graph)
Bshouty and Burroughs, 1998, 2-approximation
Hochbaum, 1998 Here: t-VC with edge lengths: (simple graph)
Hochbaum, 1998, 3-approximation
Here: 2-approximation
))log()log((2
VEVVEO
)(2
VO
)(2
VO
The partial set cover problem 38
Related workRelated work
t-SC:
Burroughs, 1998, ( )-approximation
t-SC with edge lengths:
Here: approximation time
)ln( V
)(2
HVO E