approximation algorithms chapter 14: rounding applied to set cover
TRANSCRIPT
![Page 1: Approximation Algorithms Chapter 14: Rounding Applied to Set Cover](https://reader036.vdocument.in/reader036/viewer/2022062712/56649c7d5503460f94931d80/html5/thumbnails/1.jpg)
Approximation AlgorithmsChapter 14: Rounding Applied to Set Cover
![Page 2: Approximation Algorithms Chapter 14: Rounding Applied to Set Cover](https://reader036.vdocument.in/reader036/viewer/2022062712/56649c7d5503460f94931d80/html5/thumbnails/2.jpg)
Overview
Set Cover– Approximation by simple rounding
• f-approx. algorithm (f: the frequency of the most frequent element).
– Approximation by randomized rounding• O(log n)-approx. algorithm (n: # elements to be covered).
Weighted Vertex Cover– 2-approx. algorithm
• Method based on half-integral solutions of the linear programming
• Each variable takes only 0, 1/2, or 1.
![Page 3: Approximation Algorithms Chapter 14: Rounding Applied to Set Cover](https://reader036.vdocument.in/reader036/viewer/2022062712/56649c7d5503460f94931d80/html5/thumbnails/3.jpg)
Set Cover
Input– Elements U={a1,…,an},
– Subsets of U:S={S1,…,Sm}.
– Cost function c: S→Q+. Output
– Subsets of S that cover all elements in U s.t. the sum of costs of chosen subsets in S is minimized.
S1 S2 S3S4
S5
2 2 2
1
4
Cost of a subset
Cost:2+2+2=6.
Cost:1+4=5.
Cost:2+1+4=7.
This is not a solutionsince an element is not covered.
a1 a2a3
a4 a5 a6
![Page 4: Approximation Algorithms Chapter 14: Rounding Applied to Set Cover](https://reader036.vdocument.in/reader036/viewer/2022062712/56649c7d5503460f94931d80/html5/thumbnails/4.jpg)
Set Cover by linear inequalities (1/4)
Objective function– Minimize the sum of costs
of subsets chosen:
Constraints– For covers
• Each element must appear in at least one chosen subset.
– For choosing subsets• Each subset is either chosen
or not chosen.
SS
xSc )(S
)( 1:
UaxSSaS
)( }1,0{ S SxS
S1 S2 S3S4
S5
2 2 2
1
4
Cost of a subset
Cost: 2+2+2=6.
Cost: 1+4=5.
Cost: 2+1+4=7.
a1 a2a3
a4 a5 a6
This is not a solutionsince an element is not covered.
![Page 5: Approximation Algorithms Chapter 14: Rounding Applied to Set Cover](https://reader036.vdocument.in/reader036/viewer/2022062712/56649c7d5503460f94931d80/html5/thumbnails/5.jpg)
Set Cover by linear inequalities (2/4)
Objective function– Minimize the sum of costs
of subsets chosen.
Constraints– For covers
– For choosing subsets
5432141222 SSSSS xxxxx
.1: ,1:
,1: ,1 :
,1: ,1:
5352
513
4241
65
43
21
SSSS
SSS
SSSS
xxaxxa
xxaxa
xxaxxa
)( }1,0{ S SxS
S1 S2 S3S4
S5
2 2 2
1
4
a1 a2a3
a4 a5 a6
This is not a solutionsince an element is not covered.
Cost of a subset
Cost: 2+2+2=6.
Cost: 1+4=5.
Cost: 2+1+4=7.
![Page 6: Approximation Algorithms Chapter 14: Rounding Applied to Set Cover](https://reader036.vdocument.in/reader036/viewer/2022062712/56649c7d5503460f94931d80/html5/thumbnails/6.jpg)
Set Cover by linear inequalities (3/4)
Objective function
Constraints– For covers
– For choosing subsets
.60401121212
4122254321
SSSSS xxxxx
.01: ,01:
,01: ,1 :
,01: ,01:
5352
513
4241
65
43
21
SSSS
SSS
SSSS
xxaxxa
xxaxa
xxaxxa
S1 S2 S3S4
S5
2 2 2
1
4
a1 a2a3
a4 a5 a6
.0,0,1,1,154321 SSSSS xxxxx
Cost of a subset
Cost: 2+2+2=6.
Cost: 1+4=5.
Cost: 2+1+4=7.
This is not a solutionsince an element is not covered.
![Page 7: Approximation Algorithms Chapter 14: Rounding Applied to Set Cover](https://reader036.vdocument.in/reader036/viewer/2022062712/56649c7d5503460f94931d80/html5/thumbnails/7.jpg)
Set Cover by linear inequalities (4/4)
Objective function
Constraints– For covers
– For choosing subsets
.51411020202
4122254321
SSSSS xxxxx
.10: ,10:
,10: ,10 :
,10: ,10:
5352
513
4241
65
43
21
SSSS
SSS
SSSS
xxaxxa
xxaxa
xxaxxa
S1 S2 S3S4
S5
2 2 2
1
4
a1 a2a3
a4 a5 a6
.1,1,0,0,054321 SSSSS xxxxx
Cost of a subset
Cost: 2+2+2=6.
Cost: 1+4=5.
Cost: 2+1+4=7.
This is not a solutionsince an element is not covered.
![Page 8: Approximation Algorithms Chapter 14: Rounding Applied to Set Cover](https://reader036.vdocument.in/reader036/viewer/2022062712/56649c7d5503460f94931d80/html5/thumbnails/8.jpg)
LP-relaxation
Constraints– Each subset is either
chosen or not chosen.
– It takes a value bet. 0 and 1.
• From the nature of Set Cover, the upper bound can be eliminated.
– It takes a positive value.
)( }1,0{ S SxS
S1 S2 S3S4
S5
2 2 2
1
4
a1 a2a3
a4 a5 a6
)( 10 S SxS
)( 0 S SxS
Cost of a subset
Cost: 2+2+2=6.
Cost: 1+4=5.
Cost: 2+1+4=7.
This is not a solutionsince an element is not covered.
![Page 9: Approximation Algorithms Chapter 14: Rounding Applied to Set Cover](https://reader036.vdocument.in/reader036/viewer/2022062712/56649c7d5503460f94931d80/html5/thumbnails/9.jpg)
Rounding
To change a natural number into an integer.
x1 x2 x3 x40
1
Solution found by LP
x1 x2 x3 x4
0
1
x1 x2 x3 x4
0
1
Rounded solutionwith a threshold
Probabilisticallyrounded solution
Threshold
![Page 10: Approximation Algorithms Chapter 14: Rounding Applied to Set Cover](https://reader036.vdocument.in/reader036/viewer/2022062712/56649c7d5503460f94931d80/html5/thumbnails/10.jpg)
Overview
Set Cover– Approximation by simple rounding
• f-approx. algorithm (f: the frequency of the most frequent element).
– Approximation by randomized rounding• O(log n)-approx. algorithm (n: # elements to be covered).
Weighted Vertex Cover– Method based on half-integral solutions of the linear
programming• Each variable takes only 0, 1/2, or 1.• 2-approx. algorithm
![Page 11: Approximation Algorithms Chapter 14: Rounding Applied to Set Cover](https://reader036.vdocument.in/reader036/viewer/2022062712/56649c7d5503460f94931d80/html5/thumbnails/11.jpg)
Algorithm 14.1
A simple rounding algorithm A1
– f: the frequency of the most frequent element.– 1. Find an optimal solution to the LP-relaxation.
– 2. Pick all sets S for which xS 1/≧ f.
• xS becomes 1 if xS 1/≧ f .
S1 S2
S3
S4
S5
2 2
2
1
3
a1 a2
a3a4 a5
f =2.
Solutionby LP-relax.
Solution byLP-relax.
54321 SSSSS xxxxx
0.5
54321 SSSSS xxxxx
1.0
54321 SSSSS xxxxx
0.5
54321 SSSSS xxxxx
1.0
Rounded solution Rounded solution
![Page 12: Approximation Algorithms Chapter 14: Rounding Applied to Set Cover](https://reader036.vdocument.in/reader036/viewer/2022062712/56649c7d5503460f94931d80/html5/thumbnails/12.jpg)
Theorem 14.2
A1 (Algorithm 14.1) is a f -approximation algorithm for Set Cover.– We need to consider the following two properties:
• A1 outputs a sound solution, which covers all elements.
• How much is the cost of the solution with A1?
![Page 13: Approximation Algorithms Chapter 14: Rounding Applied to Set Cover](https://reader036.vdocument.in/reader036/viewer/2022062712/56649c7d5503460f94931d80/html5/thumbnails/13.jpg)
Proof of Theorem 14.2 (1/2)
A1 outputs a sound solution, which covers all elements.– For any element a, there exists a set S s.t. a is in S and
xS 1/≧ f .• From the constraints for covers.
– Therefore, every element is chosen.
At most f
xS の値
From the constraints of covers,the sum of the areas of is at least 1.
1/f
S s.t. a is in S
= f (1/f )=1.
f
Area of
![Page 14: Approximation Algorithms Chapter 14: Rounding Applied to Set Cover](https://reader036.vdocument.in/reader036/viewer/2022062712/56649c7d5503460f94931d80/html5/thumbnails/14.jpg)
Proof of Theorem 14.2 (2/2)
How much is the cost of A1 (COST) ?
– Let OPTLP (OPTf in the text) be the cost of a solution by the LP-relaxation.
– Let xS be a solution by the LP-relax., and yS rounded one.
• yS ≦ f xS holds since
– xS 1/≧ f, f xS 1=≧ yS if yS=1.
– xS 0, ≧ f xS 0=≧ yS if yS=0.
• Therefore, COST ≦ f OPTLP ≦f OPT..)(OPT,)(COST LP s
Ss
S
xScySc
SS
1/f
f xS xS
1
yS
1/f
f xS xS
1
yS
![Page 15: Approximation Algorithms Chapter 14: Rounding Applied to Set Cover](https://reader036.vdocument.in/reader036/viewer/2022062712/56649c7d5503460f94931d80/html5/thumbnails/15.jpg)
Example 14.3
A set consists of three connected elements in Vi.
A cost of each set is 1. f = 4. The optimal cost: 2. In the bottom figure, the
cost is 8.
V1 V2 V3
xS=1/4.
xS=1.
![Page 16: Approximation Algorithms Chapter 14: Rounding Applied to Set Cover](https://reader036.vdocument.in/reader036/viewer/2022062712/56649c7d5503460f94931d80/html5/thumbnails/16.jpg)
Overview
Set Cover– Approximation by simple rounding
• f-approx. algorithm (f: the frequency of the most frequent element).
– Approximation by randomized rounding• O(log n)-approx. algorithm (n: # elements to be covered).
Weighted Vertex Cover– Method based on half-integral solutions of the linear
programming• Each variable takes only 0, 1/2, or 1.• 2-approx. algorithm
![Page 17: Approximation Algorithms Chapter 14: Rounding Applied to Set Cover](https://reader036.vdocument.in/reader036/viewer/2022062712/56649c7d5503460f94931d80/html5/thumbnails/17.jpg)
Randomized rounding C=φ % C is a collection of picked sets. while (C doesn’t satisfy condition A)
– Find C by a manner explained later.• This C satisfies condition A with prob. more than 1/2.
end-while– [Condition A]
• C is a solution of set cover.• The cost of C is at most OPTLP・ 4clog n.
– c is some constant.
– The expectation T of executing loops in while-statement is at most 2.
.28
13
4
12
2
11 T
![Page 18: Approximation Algorithms Chapter 14: Rounding Applied to Set Cover](https://reader036.vdocument.in/reader036/viewer/2022062712/56649c7d5503460f94931d80/html5/thumbnails/18.jpg)
How to find C (1/2)
Compute a solution xS of the LP-relaxation. for i=1 to clog n
– Construct a family Ci of picked sets by choosing S with prob xS.
end-for C=∪Ci .
Solution byLP-relaxation
54321 SSSSS xxxxx
1.0
C1
S1
S2
C2
S1
S2
S4
C3
S1
S2
S5
C4
S1
S5
C
S1
S2
S4
S5
![Page 19: Approximation Algorithms Chapter 14: Rounding Applied to Set Cover](https://reader036.vdocument.in/reader036/viewer/2022062712/56649c7d5503460f94931d80/html5/thumbnails/19.jpg)
How to find C (2/2)
Compute a solution xS of the LP-relaxation. for i=1 to clog n
– Construct a family Ci of picked sets by choosing S with prob xS.
end-for C=∪Ci .
– C is not a set cover with prob. less than 1/4.
– The cost of C is more than OPTLP 4c log n with prob. less than 1/4.
Less than 1/4 Less than 1/4
More than 1/2
![Page 20: Approximation Algorithms Chapter 14: Rounding Applied to Set Cover](https://reader036.vdocument.in/reader036/viewer/2022062712/56649c7d5503460f94931d80/html5/thumbnails/20.jpg)
Prob. that element a is in Ci
Consider the example below. The prob. P that any set Si containing element a is
P=(1-xS1)(1-xS2)(1-xS3).
– xS1+xS2+xS3 1 from the constraints of covers.≧
P is maximized where xS1=xS2=xS3=1/3.
S1
S2S3
S5
a1
![Page 21: Approximation Algorithms Chapter 14: Rounding Applied to Set Cover](https://reader036.vdocument.in/reader036/viewer/2022062712/56649c7d5503460f94931d80/html5/thumbnails/21.jpg)
Maximum prob. a is not chosen
.21 dPPP k Suppose an element is in each of k sets.
Let
Fix d, and replace Pk as )( 121 kk PPPdP Then, the partial derivative of log g becomes
To simplify the problem,
)1(1
i
k
i
Pg
)1log(log1
i
k
i
Pg
instead ofmaximize
ki
kii
PP
PPdPP
g
1
1
1
1
1
1
1
1log
11 Pi 0
log g
iP
g
log
+ 0 -
Max
This shows Pi=Pk makes log g maximized.
This property holds for any i, then Pi=d/k.
Under the constraint that d 1, ≧ g takes the max. ((1-1/k)k) where d=1.
d/k
![Page 22: Approximation Algorithms Chapter 14: Rounding Applied to Set Cover](https://reader036.vdocument.in/reader036/viewer/2022062712/56649c7d5503460f94931d80/html5/thumbnails/22.jpg)
Prob. C is not a set cover Prob. a is not covered by using Ci is at most (1-
1/k)k.
Prob. a is not covered by C is at most (1/e)clogn.– Choose constant c s.t. (1/e)clogn 1/(4≦ n).– c 5 (4/log ≧ ≧ n)+1 (n ≧ 3).
.11
1
111
1
ek
ekk
k
![Page 23: Approximation Algorithms Chapter 14: Rounding Applied to Set Cover](https://reader036.vdocument.in/reader036/viewer/2022062712/56649c7d5503460f94931d80/html5/thumbnails/23.jpg)
Prob. C is not a set cover
Prob. at least one element is not in C is at most 1/4.
Less than 1/4n
At least one of a1, a2, a3 isnot chosen with prob. less
than 1/4.
Less than 1/4n
Less than 1/4n
a1 is not chosen. a2 is not chosen.
a3 is not chosen.
n=3
![Page 24: Approximation Algorithms Chapter 14: Rounding Applied to Set Cover](https://reader036.vdocument.in/reader036/viewer/2022062712/56649c7d5503460f94931d80/html5/thumbnails/24.jpg)
The cost of C
)in is Pr(log
)in isPr()in isPr(log
1
i
i
nc
i
CSnc
CSCS
LPOPTlog
)(log
)()in is Pr(log
)()in is Pr()](cost[
nc
Scxnc
ScCSnc
ScCSCE
SS
iS
S
![Page 25: Approximation Algorithms Chapter 14: Rounding Applied to Set Cover](https://reader036.vdocument.in/reader036/viewer/2022062712/56649c7d5503460f94931d80/html5/thumbnails/25.jpg)
Markov’s inequality (1/2)
Random variable X takes a non-negative value, and the average of X is μ.
x
xx
x
dxxXxP
dxxXxPdxxXxP
dxxXxP
)(
)()(
)(
0
0
≧0
≧ε
X
P(X=x)
x
x=ε
![Page 26: Approximation Algorithms Chapter 14: Rounding Applied to Set Cover](https://reader036.vdocument.in/reader036/viewer/2022062712/56649c7d5503460f94931d80/html5/thumbnails/26.jpg)
Markov’s inequality (2/2)
Random variable X takes a non-negative value, and the average of X is μ.
).()(
)(
)()(
)(
0
0
XPdxxXP
dxxXxP
dxxXxPdxxXxP
dxxXxP
x
x
xx
x
.)( XP
≧0
≧ε
![Page 27: Approximation Algorithms Chapter 14: Rounding Applied to Set Cover](https://reader036.vdocument.in/reader036/viewer/2022062712/56649c7d5503460f94931d80/html5/thumbnails/27.jpg)
The value of cost (C)
.OPTlog4
,OPTlog)](cost[),(cost
LP
LP
nc
ncCECX
Apply Markov’s inequality to cost (C).
.)( XP
.4
1
log4OPT
logOPT
log4OPT)log4OPT)(cost(
LP
LP
LPLP
nc
nc
ncncCP
Prob. the cost of C becomes morethan OPTLP4clog n is at most 1/4.
![Page 28: Approximation Algorithms Chapter 14: Rounding Applied to Set Cover](https://reader036.vdocument.in/reader036/viewer/2022062712/56649c7d5503460f94931d80/html5/thumbnails/28.jpg)
Each of the following two events happens with prob. less than 1/4.– C is not a set cover,
– The cost of C is more than c=OPTLP4clog n. Therefore, the event that C is a set cover and its cost is at
most c is at least 1/2.
Less than 1/4 Less than 1/4
More than 1/2
![Page 29: Approximation Algorithms Chapter 14: Rounding Applied to Set Cover](https://reader036.vdocument.in/reader036/viewer/2022062712/56649c7d5503460f94931d80/html5/thumbnails/29.jpg)
Overview
Set Cover– Approximation by simple rounding
• f-approx. algorithm (f: the frequency of the most frequent element).
– Approximation by randomized rounding• O(log n)-approx. algorithm (n: # elements to be covered).
Weighted Vertex Cover– Method based on half-integral solutions of the linear
programming• Each variable takes only 0, 1/2, or 1.• 2-approx. algorithm
![Page 30: Approximation Algorithms Chapter 14: Rounding Applied to Set Cover](https://reader036.vdocument.in/reader036/viewer/2022062712/56649c7d5503460f94931d80/html5/thumbnails/30.jpg)
Weighted vertex cover
Weighted vertex cover– Input: graph with weights on vertices G=(V,E).– Output: A ⊆V.
• For any (u,v)∈E, u∈A or v∈A .
• The sum of weights of v∈A is minimized.
14 1
2 55
14 1
2 55
v1 v2 v3
v4 v5 v6
v1 v2 v3
v4 v5 v6
![Page 31: Approximation Algorithms Chapter 14: Rounding Applied to Set Cover](https://reader036.vdocument.in/reader036/viewer/2022062712/56649c7d5503460f94931d80/html5/thumbnails/31.jpg)
Weighted vertex cover
Definition– Input: Graph G=(V,E).– Output: A ⊆V.
• For any edge (u,v)∈E, u∈A or v∈A .
• The sum of weights of v∈A is minimized.
14 1
2 55
a1 a2
a3
a6
a4
a5
14 1
2 55
a1 a2
a3
a6
a4
a5
v1 v2 v3
v4 v5 v6
v1 v2 v3
v4 v5 v6
![Page 32: Approximation Algorithms Chapter 14: Rounding Applied to Set Cover](https://reader036.vdocument.in/reader036/viewer/2022062712/56649c7d5503460f94931d80/html5/thumbnails/32.jpg)
Formulation by linear inequalities
Objective function– Minimize:
Constraints– For covers– For choosing edges
Vv
vxvc )(
Evuxx vu ),( ,1Vvxv },1,0{
14 1
2 55
a1 a2
a3
a6
a4
a5
14 1
2 55
a1 a2
a3
a6
a4
a5
v1 v2 v3
v4 v5 v6
v1 v2 v3
v4 v5 v6
![Page 33: Approximation Algorithms Chapter 14: Rounding Applied to Set Cover](https://reader036.vdocument.in/reader036/viewer/2022062712/56649c7d5503460f94931d80/html5/thumbnails/33.jpg)
Formulation by linear inequalities
Objective function– Minimize:
Constraints– For covers
– For choosing edges
654321525114 vvvvvv xxxxxx
}.1,0{,,,,,654321vvvvvv xxxxxx
14 1
2 55
a1 a2
a3
a6
a4
a5
14 1
2 55
a1 a2
a3
a6
a4
a5
v1 v2 v3
v4 v5 v6
v1 v2 v3
v4 v5 v6
.1,1,1
,1,1,1
655463
413221
vvvvvv
vvvvvv
xxxxxx
xxxxxx
![Page 34: Approximation Algorithms Chapter 14: Rounding Applied to Set Cover](https://reader036.vdocument.in/reader036/viewer/2022062712/56649c7d5503460f94931d80/html5/thumbnails/34.jpg)
LP-relaxation
Objective function– Minimize:
Constraints– For covers– For choosing edges
Vv
vxvc )(
Evuxx vu ),( ,1
Vvxv ,0
14 1
2 55
a1 a2
a3
a6
a4
a5
14 1
2 55
a1 a2
a3
a6
a4
a5
v1 v2 v3
v4 v5 v6
v1 v2 v3
v4 v5 v6
![Page 35: Approximation Algorithms Chapter 14: Rounding Applied to Set Cover](https://reader036.vdocument.in/reader036/viewer/2022062712/56649c7d5503460f94931d80/html5/thumbnails/35.jpg)
LP-relaxation
Objective function– Minimize:
Constraints– For covers
– For choosing edges
654321525114 vvvvvv xxxxxx
.0,,,,,654321vvvvvv xxxxxx
14 1
2 55
a1 a2
a3
a6
a4
a5
14 1
2 55
a1 a2
a3
a6
a4
a5
v1 v2 v3
v4 v5 v6
v1 v2 v3
v4 v5 v6
.1,1,1
,1,1,1
655463
413221
vvvvvv
vvvvvv
xxxxxx
xxxxxx
![Page 36: Approximation Algorithms Chapter 14: Rounding Applied to Set Cover](https://reader036.vdocument.in/reader036/viewer/2022062712/56649c7d5503460f94931d80/html5/thumbnails/36.jpg)
Extreme point solution
The optimal solution of Linear Programming. The solution which cannot be expressed as
convex combination of two other feasible solution.– Convex combination: A linear equation s.t. the sum of
its coefficients is 1.• z is a convex combination of x and y, where z =0.8x+0.2y.
Feasible solution
Convex combinationof feasible solution
![Page 37: Approximation Algorithms Chapter 14: Rounding Applied to Set Cover](https://reader036.vdocument.in/reader036/viewer/2022062712/56649c7d5503460f94931d80/html5/thumbnails/37.jpg)
Half-integral solution
Solution of Linear Programming s.t. each value takes 0, 1/2 or 1.
![Page 38: Approximation Algorithms Chapter 14: Rounding Applied to Set Cover](https://reader036.vdocument.in/reader036/viewer/2022062712/56649c7d5503460f94931d80/html5/thumbnails/38.jpg)
2-approximation algorithm
Compute an extreme point solution x. Choose any vertex s.t its corresponding value
takes 1/2 or 1.– If x is an extreme point solution, each variable takes 0,
1/2, or 1. (Lemma 14.4)
![Page 39: Approximation Algorithms Chapter 14: Rounding Applied to Set Cover](https://reader036.vdocument.in/reader036/viewer/2022062712/56649c7d5503460f94931d80/html5/thumbnails/39.jpg)
Lemma 14.4
x: a solution of weighted vertex cover obtained by Linear Programming.
If x is not half-integral, x can be expressed as convex combination of two other feasible solution.– x is not an extreme point solution.
– Outline of its proof• Construct y and z s.t. x is not half-integral and x=1/2(y+z).
– x can be expressed by convex combination of y and z.
• Show y and z are feasible solutions.
![Page 40: Approximation Algorithms Chapter 14: Rounding Applied to Set Cover](https://reader036.vdocument.in/reader036/viewer/2022062712/56649c7d5503460f94931d80/html5/thumbnails/40.jpg)
Proof of Lemma 14.4 (1/3)
Construct other solutions y and z from x, each of them takes 0, 1/2, or 1.
54321 vvvvv xxxxx
1.0x
0.5
1.0y
0.5
1.0z
0.5
+ε
- ε
- ε
+ε
V+={v3}.54321 vvvvv xxxxx
54321 vvvvv xxxxx
vi s.t. xvi > 1/2.
V - ={v4}. vi s.t. xvi < 1/2.
+εin y, - εin z.
- εin y, +εin z.
zyx 2
1holds.
![Page 41: Approximation Algorithms Chapter 14: Rounding Applied to Set Cover](https://reader036.vdocument.in/reader036/viewer/2022062712/56649c7d5503460f94931d80/html5/thumbnails/41.jpg)
Proof of Lemma 14.4 (2/3)
Are y and z feasible solutions?– In any feasible solution, xu+xv 1 holds.≧
xu
xv
1/2 1
1/2
1
Feasible solution
yu
yv
1/2 1
1/2
1
Change from x to y
zu
zv
1/2 1
1/2
1
Change from x to z
Where εis set to a small value,y and z are feasible solutions.
![Page 42: Approximation Algorithms Chapter 14: Rounding Applied to Set Cover](https://reader036.vdocument.in/reader036/viewer/2022062712/56649c7d5503460f94931d80/html5/thumbnails/42.jpg)
Proof of Lemma 14.4 (3/3)
When xu+ xv=1,– xu= xv=1/2.
• yu= yv=zu= zv=1/2 (no change).
– xu=0, xv=1.• yu=0, yv=1, zu=0, zv=1 (no change).
– xu<1/2, xv>1/2.• yu+ yv= xu+ε + xv -ε =1,• zu+ zv= xu-ε + xv +ε =1.
Then, y and z are feasible solutions, and any solution can be expressed by a half-integral solution.