Partial Vertex Cover Joshua Wetzel Department of Computer Science Rutgers University–Camden [email protected]

April 2, 2009

Joshua Wetzel

Partial Vertex Cover

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Vertex Cover

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

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

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Vertex Cover: Example

24

50

24

50 10

15

10

15

6

30

6

30

18

18

12

12

COST = 97

COST = 108

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Partial Vertex Cover

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Vertex Cover: IP Formulation

xv ← 1 if v is in our cover, 0 otherwise X min wv xv v ∈V

s.t. xa + xb ≥ 1, ∀ e = (a, b) xv ∈ {0, 1}, ∀ v ∈ V

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LP Relaxation

Integer programs have been shown to be NP-hard Relax the integrality constraints xv ∈ {0, 1}, ∀ v ∈ V min

X

wv xv

v ∈V

s.t. xa + xb ≥ 1, ∀ e = (a, b) xv ≥ 0, ∀ v ∈ V

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Partial Vertex Cover

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Constructing the Dual LP

Primal LP: min

X

wv xv

v ∈V

s.t. xa + xb ≥ 1, ∀ e = (a, b)

(ye )

xv ≥ 0, ∀ v ∈ V

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Partial Vertex Cover

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Primal LP and Dual LP

Dual LP:

Primal LP: min

X

max

wv xv

X

ye

e∈E

v ∈V

s.t.

s.t.

xa + xb ≥ 1, ∀ e = (a, b) xv ≥ 0, ∀ v ∈ V

Joshua Wetzel

X

ye ≤ wv , ∀ v ∈ V

e : e hits v

ye ≥ 0, ∀ e ∈ E

Partial Vertex Cover

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Primal-Dual Method

DualFeasible ≤ DualOPT = PrimalOPT PrimalOPT ≤ OPTIP Construct the dual LP Construct an algorithm that manually tightens dual constraints to obtain a ’maximal’ dual solution

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Clarkson’s Algorithm

Inititally all edges are uncovered. While ∃ an uncovered edge in G: raise ye for all uncovered edges P simultaneouslty until a vertex, v , becomes full (i.e y e = wv ) e:e hits v

C ← C ∪ {v } any e that touches v is covered.

Return C as our vertex cover

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Partial Vertex Cover

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Clarkson’s Algorithm: Example X

ye ≤ wv

e:e hits v 0 24

50

10 0

0

0

0 15

6 0 0

30

18

0

0

0 12

Raise each ye uniformly until a vertex is full. Joshua Wetzel

Partial Vertex Cover

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Clarkson’s Algorithm: Example X

ye ≤ wv

e:e hits v 2 24

50

10 2

2

2

2 15

6 2 2

30

18

2

2

2 12

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Partial Vertex Cover

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Clarkson’s Algorithm: Example X

ye ≤ wv

e:e hits v 3 24

50

10 3

2

2

3 15

6 2 3

30

18

3

3

3 12

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Partial Vertex Cover

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Clarkson’s Algorithm: Example X

ye ≤ wv

e:e hits v 12 24

50

10 12

2

2

12 15

6 2 3

30

18

3

3

3 12

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Partial Vertex Cover

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Clarkson’s Algorithm: Example X

ye ≤ wv

e:e hits v 19 24

50

10 12

2

2

19 15

6 2 3

30

18

3

3

3 12

COST = 83 Joshua Wetzel

Partial Vertex Cover

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Clarkson’s Algorithm: Analysis

19 24

50

10 12

2

Dual Obj. Fn: X max ye

2

e

19 15

6 2

3

30

18

3

3

3

Our Cost = wt(red vertices) X ≤ 2 ye

12

X

e hits red

≤ 2 ye ≤ wv

e:e hits v

X

ye

e

= 2DFS ≤ 2OPT

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Partial Vertex Cover

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Clarkson’s Algorithm: Tight Example

1

1

1

1 1

1 1

1

1

6

1

1

6 1

1

1

1

COSTOPT = 6

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1 1

1

COSTClarkson = 12

Partial Vertex Cover

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Partial Vertex Cover

Input: Graph, G = (V , E) Non-negative integer weights for vertices Integer, k

Objective: Find the least cost set of vertices in G that will cover at least k edges.

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Key Issue

In full vertex cover OPT covers all edges and hence we know which edges to cover. In partial vertex cover, we do not know which k edges OPT covers. When k is part of the input, the techniques for full-coverage do not directly apply.

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Clarkson’s Algorithm Fails

Input: k = 1 1

1

5

1

1

1

1

1

5/6

5/6

5/6 5/6

1

1

1

COSTOPT = 1

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5

5/6

1

5/6 1

COSTClarkson = 5

Partial Vertex Cover

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Related Work

Bshouty & Burroughs 1998: solve the LP, modify it and then round the modified solution. 2-approximation for partial vertex cover. Hochbaum 1998: 2-approximation for partial vertex cover. Bar-Yehuda 1999: “local ratio” method, 2-approximation for partial vertex cover. Mestre 2005: 2-approximation primal-dual algorithm for partial vertex cover with improved running time

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Vertex Cover: IP Formulation

xv ← 1 if v is in our cover, 0 otherwise X min wv xv v ∈V

s.t. xa + xb ≥ 1, ∀ e = (a, b) xv ∈ {0, 1}, ∀ v ∈ V

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Partial Vertex Cover

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Partial Vertex Cover: IP formulation xv ← 1 if vertex v is chosen in the cover, 0 otherwise. ye ← 1 if edge e is not covered, 0 otherwise. X min wv xv v ∈V

s.t. ye + xa + xb ≥ 1, ∀e = (a, b) X ye ≤ m − k e∈E

xv ∈ {0, 1}, ∀v ∈ V ye ∈ {0, 1}, ∀e ∈ E

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Partial Vertex Cover

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Patial Vertex Cover: LP formulation

Relax the integrality constraints. xv ∈ {0, 1} → xv ≥ 0, ∀ v ∈ V ye ∈ {0, 1} → ye ≥ 0, ∀ e ∈ E

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Constructing the Dual LP

min

X

wv xv

v ∈V

s.t. ye + xa + xb ≥ 1, ∀ e = (a, b) X ye ≤ (m − k)

(ue ) (z)

e∈E

xv ≥ 0, ∀ v ∈ V ye ≥ 0, ∀ e ∈ E

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Partial Vertex Cover

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Primal LP and Dual LP

Primal LP: min

X

Dual LP:

wv xv

max

v ∈V

X

ue − z(m − k)

e∈E

s.t.

s.t. ye + xa + xb ≥ 1, ∀e X ye ≤ m − k

X

ue ≤ wv , ∀v ∈ V

e : e hits v

e∈E

ue ≤ z, ∀ e ∈ E

xv ≥ 0, ∀ v ∈ V

ue ≥ 0, ∀ e ∈ E

ye ≥ 0, ∀ e ∈ E

z ≥0

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Intuition for Primal-Dual Input: k = 1 1

1

5

1

1

1

1

1

5/6

5/6

5/6 5/6

1

1

1

COSTOPT = 1

5

5/6

1

5/6 1

COSTClarkson = 5

In last iteration, Clarkson’s Alg. may choose more edges than we need at a very high cost We must somehow bound the cost of last vertex chosen Joshua Wetzel

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Primal-Dual Algorithm

For each vertex, v, in G Guess v to be the heaviest vertex in OPT , called vh Cv ← {vh } Raise weight of all heavier vertices in G to ∞ Remove all edges touching vh from G k ′ ← k − deg(vh ) Run Clarkson on this instance until k ′ edges are covered. Cchoices ← Cchoices ∪ CV

Return the lowest cost cover, C

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Primal-Dual Algorithm: Example

24

50

10

15

18

15

17

12

Input: G = (V , E ),

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k =8

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Primal-Dual Algorithm: Example X

ue ≤ wv

e:e hits v 0 INF

INF

10

0

0

0 15

INF 0

15

0 INF

Vh

0

0

12

k′ = 6 Joshua Wetzel

Partial Vertex Cover

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Primal-Dual Algorithm: Example X

ue ≤ wv

e:e hits v 4 INF

INF

10

4

4

4 15

INF 4

15

4 INF

Vh

4

4

12

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Partial Vertex Cover

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Primal-Dual Algorithm: Example X

ue ≤ wv

e:e hits v 6 INF

INF

10

6

6

6 15

INF 6

15

4 INF

Vh

4

4

12

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Partial Vertex Cover

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Primal-Dual Algorithm: Example X

ue ≤ wv

e:e hits v 11 INF

INF

10

11

11

6 15

INF 11

15

4 INF

Vh

4

4

12

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Partial Vertex Cover

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Primal-Dual Algorithm: Example X

ue ≤ wv

e:e hits v INF

INF

INF

INF

11

10 6

15

INF INF

15

4 INF

Vh

4

4

12

COST = ∞ Joshua Wetzel

Partial Vertex Cover

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Primal-Dual Algorithm: Example X

ue ≤ wv

e:e hits v INF INF

INF

10

INF

INF

10 INF

INF

INF INF

INF INF

12

Vh

COST = ∞ Joshua Wetzel

Partial Vertex Cover

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Primal-Dual Algorithm: Example X

ue ≤ wv

e:e hits v 11 INF

INF

10

11

11

6 11

15

INF 4

15

17

Vh 4

4 12

COST = 69 Joshua Wetzel

Partial Vertex Cover

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Primal-Dual Algorithm: Example X

ue ≤ wv

e:e hits v INF INF

INF

10

INF 6 11 15

INF

Vh

INF 15 INF 4

4

4 12

COST = ∞ Joshua Wetzel

Partial Vertex Cover

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Primal-Dual Algorithm: Example X

ue ≤ wv

e:e hits v INF INF

INF

INF

INF

10

Vh

INF INF

INF INF INF

INF

INF INF INF INF

COST = ∞ Joshua Wetzel

Partial Vertex Cover

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Primal-Dual Algorithm: Example X

ue ≤ wv

e:e hits v 12 INF

INF

INF

12 12 15

18

Vh 3

15

17 3

3

3 12

COST = 60 Joshua Wetzel

Partial Vertex Cover

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Primal-Dual Algorithm: Example X

ue ≤ wv

e:e hits v

24

INF

Vh 10

12 7 12 15

18 12 3

15

17 3

3

3 12

COST = 76 Joshua Wetzel

Partial Vertex Cover

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Primal-Dual Algorithm: Example X

ue ≤ wv

e:e hits v

Vh 24

50

10

6 6 15

18 6 3

15

17 3

3

3 12

COST = 80 Joshua Wetzel

Partial Vertex Cover

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Primal-Dual Algorithm: Example Heaviest Vertex

Cost

24

50

10

15

18

15

17

12

15 12 17 15 10 18 24 50

∞ ∞ 69 ∞ ∞ 60 76 80

Return cover with cost of 60

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Analysis: Cost of OPT Ih : inst. in which we correctly guess heaviest vertex in OPT 12 24

50

INF

INF 10

10

12 12

15

15

18

Vh 3

15

15 17

17 3

3

3 12

12

OPT

Ih OPT = OPT (Ih ) + w (vh )

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Analysis: Our Cost

OPT = OPT (Ih ) + w (vh ) 12 24

50

INF

INF 10

10

12 12

15

15

18

18

Vh 15

Vh 3

15 17

17 3

3

3 12

12

OPT (Ih ) + w (vh ) = 45

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COST (Ih ) + w (vh ) = 60

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Analysis: What is z? Z INF

INF

12 INF

INF

10

Z 10

12

Z 15

12 15

18

Vl

Vh

18

Vl

3

15

Vh

17

3

15

3

17 3

3

3 3

12

3 12

ue ≤ z, ∀ e

Ih

z = 12 Num of edges for which ue = z is at least m′ − k ′

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Analysis: Our Cost OPT = OPT (Ih ) + w (vh ) Z

Our Cost ≤ Cost(Ih ) + w (vh )

INF

INF

10

Z Z 15

≤ w (RedVert) + 2w (vh ) X ≤ 2 ue + 2w (vh )

18

Vl

Vh

3

15

e hits red

17 3

= w (RedVert) + w (vl ) + w (vh )

3

3 12

X ≤ 2[ ue − z(m′ − k ′ )] + 2w (vh ) e

≤ 2DFS(Ih ) + 2w (vh )

Ih

≤ 2OPT (Ih ) + 2w (vh ) ≤ 2OPT

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Partial Vertex Cover

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Primal LP and Dual LP

Primal LP: min

X

Dual LP:

wv xv

max

v ∈V

X

ue − z(m − k)

e∈E

s.t.

s.t. ye + xa + xb ≥ 1, ∀e X ye ≤ m − k

X

ue ≤ wv , ∀v ∈ V

e : e hits v

e∈E

ue ≤ z, ∀ e ∈ E

xv ≥ 0, ∀ v ∈ V

ue ≥ 0, ∀ e ∈ E

ye ≥ 0, ∀ e ∈ E

z ≥0

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Partial Vertex Cover

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Reference

K. L. Clarkson. A modification of the greedy algorithm for the vertex cover. Information Processing Letters 16:23-25, 1983. R. Gandhi, S. Khuller, and A. Srinivasan. Approximation Algorithms for Partial Covering Problems. Journal of Algorithms, 53(1):55-84, October 2004.

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Thank You.

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