Vertex Cover - Linear Progamming and Approximation Algorithms

Linear Programming TheoryVertex CoverLP-Rounding

Primal-Dual

Vertex CoverLinear Progamming and Approximation Algorithms

Joshua Wetzel

Department of Computer ScienceRutgers University–Camden

[email protected]

March 24, 2009

Joshua Wetzel Vertex Cover 1 / 52


Primal-Dual

What is Linear Programming?

Optimize a linear objective fn subject to linear constraints.

Examples:

min 7x1 + x2 + 3x3

s.t.

x1 − x2 + 3x3 ≥ 10

5x1 + 2x2 − x3 ≥ 6

x1, x2, x3 ≥ 0

max 6x1 + 2x2 − 4x3

s.t.

5x1 + x2 − 2x3 ≤ 14

2x1 − 2x2 + 4x3 ≤ 20

x1, x2, x3 ≥ 0



Primal-Dual

Feasible Solutions: Upper Bounding OPT

Linear Program:

min 7x1 + x2 + 3x3

s.t.

x1 − x2 + 3x3 ≥ 10

5x1 + 2x2 − x3 ≥ 6

x1, x2, x3 ≥ 0

Feasible Solutions:x1 = 2, x2 = 1, x3 = 4

7(2) + 1(1) + 5(4) = 35

x1 = 85 , x2 = 1

2 , x3 = 3

7(85) + 1(1

2) + 5(3) = 26.7

All constraints are satisfied.

OPT ≤ 35, OPT ≤ 26.7



Primal-Dual

Finding a Lower Bound on OPT

Linear Program:

min 7x1 + x2 + 3x3

s.t.

x1 − x2 + 3x3 ≥ 10

5x1 + 2x2 − x3 ≥ 6

x1, x2, x3 ≥ 0

OPT ≥ ?



Primal-Dual

Lower Bounding OPT: Adding the Constraints

min 7x1 + x2 + 3x3

s.t.

x1 − x2 + 3x3 ≥ 10

+ 5x1 + 2x2 − x3 ≥ 6

6x1 + x2 + 2x3 ≥ 16

OPT ≥ 16



Primal-Dual

Lower Bounding OPT

min 7x1 + x2 + 3x3

s.t. x1 − x2 + 3x3 ≥ 10 (y1)

+ 5x1 + 2x2 − x3 ≥ 6 (y2)

x1( y1 + 5y2 ) + x2( −y1 + 2y2 ) + x3( 3y1 − y2 ) ≥ 10y1 + 6y2

10y1 + 6y2 is a lower bound on OPT if:

y1 + 5y2 ≤ 7

−y1 + 2y2 ≤ 1

3y1 − y2 ≤ 3



Primal-Dual

LP to Lower Bound OPT

max 10y1 + 6y2

s.t.

y1 + 5y2 ≤ 7

−y1 + 2y2 ≤ 1

3y1 − y2 ≤ 3

y1, y2 ≥ 0



Primal-Dual

Primal LP and Dual LP

Primal LP:

min 7x1 + x2 + 3x3

s.t. x1 − x2 + 3x3 ≥ 10

5x1 + 2x2 − x3 ≥ 6

x1, x2, x3 ≥ 0

Dual LP:

max 10y1 + 6y2

s.t. y1 + 5y2 ≤ 7

−y1 + 2y2 ≤ 1

3y1 − y2 ≤ 3

y1, y2 ≥ 0

Every primal LP has a corresponding dual LP.

If the primal is a min problem, the dual is a max problem.

There is a dual constraint corresponding to each primalvariable.



Primal-Dual

LP Duality Theorems

DualFeasible ≤ DualOPT = PrimalOPT ≤ PrimalFeasible

The Primal LP:

min 7x1 + x2 + 3x3

s.t. x1 − x2 + 3x3 ≥ 10

5x1 + 2x2 − x3 ≥ 6

x1, x2, x3 ≥ 0

The Dual LP:

max 10y1 + 6y2

s.t. y1 + 5y2 ≤ 7

−y1 + 2y2 ≤ 1

3y1 − y2 ≤ 3

y1, y2 ≥ 0



Primal-Dual

Vertex Cover

Input:Given G = (V , E)Non-negative weights on vertices

Objective:Find a least-weight collection of vertices such that eachedge in G in incident on at least one vertex in the collection.



Primal-Dual

Vertex Cover: Example

24

15

50

30

12

10

18

6

COST = 97

24

15

50

30

10

18

6

12

COST = 108



Primal-Dual

Approximation Algorithms

NP-hard problems.No optimal poly-time algorithms are known

β-approximation alg., A, for a minimization problem Ppoly-time algorithm.for every instance I of P, A produces solution of cost atmost β ·OPT (I)OPT (I)?



Primal-Dual

Approximation Algorithms

compute a lower bound on OPT .

compare cost of our solution with the lower bound.

lowerbound

OPT upperbound

β



Primal-Dual

Vertex Cover

Input:Given G = (V , E)Non-negative weights on vertices

Objective:Find a least-weight collection of vertices such that eachedge in G in incident on at least one vertex in the collection



Primal-Dual

Unweighted Vertex Cover: Algorithm

Find a maximal matching in G

Include in our cover both vertices incident on each edge ofthe matching



Primal-Dual

Unweighted Vertex Cover: Example

11

1

1

1

1

1

1



Primal-Dual


1

1

1

1

1

1

1

1



Primal-Dual


1

1

1

1

1

1

1

1

Cost = 6



Primal-Dual

Analysis: Feasibility

1

1

1

1

1

1

1

1

Every black edge shares a vertex with a green edge



Primal-Dual

Analysis: Approximation Guarantee

1

1

1

1

1

1

1

1

OPT has to choose atleast one endpoint fromeach green edge.

We choose both endpointsfor each green edge.

Hence:

Our Cost ≤ 2OPT



Primal-Dual

Unweighted Vertex Cover: Tight Example

1

1

1

1

1

1

1

1

OPT = 3

1

1

1

1

1

1

1

1

COSTAlg = 6



Primal-Dual

Bad Example

200 1 200

OPT = 1

200 1 200

CostAlg = 201



Primal-Dual

Vertex Cover: IP Formulation

xv ← 1 if v is in our cover, 0 otherwise

min∑

v∈V

wvxv

s.t.

xa + xb ≥ 1, ∀ e = (a, b)

xv ∈ {0, 1}, ∀ v ∈ V



Primal-Dual

LP Relaxation

Integer programs have been shown to be NP-hard

Relax the integrality constraints xv ∈ {0, 1}, ∀ v ∈ V

min∑

v∈V

wvxv

s.t.

xa + xb ≥ 1, ∀ e = (a, b)

xv ≥ 0, ∀ v ∈ V



Primal-Dual

Why Do This?

LP can be solved in polynomial time

Every solution to the IP is also a solution to the LP

Hence:

OPTLP ≤ OPTIP

We can use OPTLP as a lower bound on OPTIP



Primal-Dual

LP Solution: Example

xa + xb ≥ 1, ∀ e = (a, b)

xv ≥ 0, ∀ v ∈ V

1/2 1/2

1/2

OPTLP = 1.5

xa + xb ≥ 1, ∀ e = (a, b)

xv ∈ {0, 1}, ∀ v ∈ V

1

1 0

OPTIP = 2



Primal-Dual

LP-Rounding Algorithm

x∗ ← optimal LP soln.

x̂v ← 1 if x∗

v ≥12 , otherwise x̂v ← 0

Include v in our cover iff x̂v = 1



Primal-Dual

Analysis: Feasibility

min∑

v∈V

wvxv

s.t.

xa + xb ≥ 1, ∀ e = (a, b)

xv ≥ 0, ∀ v ∈ V

x∗

a ≥12

or x∗

b ≥12

x̂a = 1 or x̂b = 1



Primal-Dual

Analysis: Approximation Guarantee

Our Cost =∑

v∈C

wv =∑

v∈V

wv x̂v

≤∑

v∈V

wv (2x∗

v )

= 2∑

v∈V

wvx∗

v

= 2OPTLP

≤ 2OPTIP



Primal-Dual

LP-Rounding: Tight Example

1/2 1/2

1/2

OPTLP = 1.5

1

1 1

Our Cost = 3



Primal-Dual

Primal-Dual Method

PrimalOPT ≤ OPTIP

Solving the LP is expensive.

DualFeasible ≤ DualOPT = PrimalOPT ≤ PrimalFeasible

Better Alternative:Construct the dual LPConstruct an algorithm that manually tightens dualconstraints to obtain a ’maximal’ dual solution



Primal-Dual

Constructing the Dual LP

Primal LP:

min∑

v∈V

wvxv

s.t.

xa + xb ≥ 1, ∀ e = (a, b) (ye)

xv ≥ 0, ∀ v ∈ V



Primal-Dual

Primal LP and Dual LP

Primal LP:

min∑

v∈V

wvxv

s.t.

xa + xb ≥ 1, ∀ e = (a, b)

xv ≥ 0, ∀ v ∈ V

Dual LP:

max∑

e∈E

ye

s.t. ∑

e : e hits v

ye ≤ wv , ∀ v ∈ V

ye ≥ 0, ∀ e ∈ E



Primal-Dual

Bar-Yehuda and Even Algorithm

Inititally all edges are uncovered.While ∃ an uncovered edge in G:

Choose an arbitrary edge, eRaise the value of ye for that edge until one of its incidentvertices , v , becomes full (i.e

∑e:e hits v

ye = wv )

C ← C ∪ {v}Any edge that touches v is considered to be covered

Return C as our vertex cover



Primal-Dual

Bar-Yehuda and Even Algorithm: Example

24

15

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30

12

10

18

00

0

00

0

0

0

0

0

6



Primal-Dual

Bar-Yehuda and Even Algorithm: Example∑

e:e hits v

ye ≤ wv

24

15

50

30

12

10

18

00

0

00

0

0

0

0

0

6

Arbitrarily choose e and raise ye until a vertex is fullJoshua Wetzel Vertex Cover 37 / 52


Primal-Dual


e:e hits v

ye ≤ wv

24

15

50

30

12

10

18

60

0

00

0

0

0

0

0

6



Primal-Dual


e:e hits v

ye ≤ wv

24

15

50

30

12

10

18

60

0

00

0

0

15

0

0

6



Primal-Dual


e:e hits v

ye ≤ wv

24

15

50

30

12

10

18

60

0

00

0

0

15

0

18

6



Primal-Dual


e:e hits v

ye ≤ wv

24

15

50

30

10

18

60

0

012

0

0

15

0

18

6

12



Primal-Dual


e:e hits v

ye ≤ wv

24

15

50

30

10

18

60

0

012

0

0

15

17

18

6

12



Primal-Dual

Bar-Yehuda and Even Algorithm: Example

24

15

50

30

10

18

60

0

012

0

0

15

17

18

6

12

Cost = 107



Primal-Dual

Bar-Yehuda and Even Algorithm: Analysis

24

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30

10

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0

012

0

0

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6

12

∑

e:e hits v

ye ≤ wv

Dual Obj. Fn:

max∑

e

ye

Our Cost = wt(red vertices)

≤ 2∑

e hits red

ye

≤ 2∑

e

ye

= 2DFS

≤ 2OPT



Primal-Dual

Bar-Yehuda and Even Algorithm: Tight Example

1 1

1

11

1 6

COSTOPT = 6

1 1

1

11

1 61

11

1

1 1

COSTBar-Yehuda = 12



Primal-Dual

Integrality Gap

1/2 1/2

1/2

OPTLP = 1.5

1

1 0

OPTIP = 2

For a complete graph of n vertices

OPTLP = n/2

OPTIP = n − 1

limn→∞

OPTIPOPTLP

= limn→∞

n−1(n/2) = 2



Primal-Dual

Reference

R. Bar-Yehuda and S. Even. A linear time approximationalgorithm for the weighted vertex cover problem. J. ofAlgorithms 2:198-203, 1981.



Primal-Dual

Thank You.


Vertex Cover - Linear Progamming and Approximation Algorithms

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