capacitated vertex cover - rutgers universitycrab.rutgers.edu/~rajivg/studenttalks/cvc.pdf · 2009....

The ProblemIntegrality Gap

A Rounding AlgorithmPrimal-Dual Algorithm

Capacitated Vertex Cover

James Davis

Department of Computer ScienceRutgers University–Camden

[email protected]

April 29, 2009

James Davis Capacitated Vertex Cover 1 / 60



The Problem

Input:Graph G = (V ,E)

For each vertex, v :A weight, wv , assigned to each copy of vA capacity, kv , assigned to each copy of v .

Objective:Find a multiset of vertices, C, of minimum weight s.t. novertex copy is assigned more than kv edges.




The Problem

4,416,2

12,7

20,1

16,18,2

13,1

16,3 4,1

1

1

11

1

Cost = 8 · 1 + 16 · 1 + 4 · 1 + 4 · 1 + 16 · 1 = 48James Davis Capacitated Vertex Cover 3 / 60



The Problem

4,416,2

12,7

20,1

16,18,2

13,1

16,3 4,1

2

2

1 1

1

1

Cost = 12 · 1 + 16 · 1 + 13 · 2 + 16 · 1 + 20 · 2 = 110James Davis Capacitated Vertex Cover 4 / 60



Approximation Algorithms

CVC is NP-hard.No optimal poly-time algorithms are known

β-approximation alg. for a minimization problem PPoly-time algorithm.For every instance, I, of P, the alg. produces solution ofcost at most β ·OPT (I)We don’t know OPT (I)...




Approximation Algorithms

Compute a lower bound on OPT .Compare cost of our solution with the lower bound.

LB Our Cost

Approximation Guarantee

OPT




IP Formulation

xv : the number of copies of v in C.γev : given the value 1 if edge e is assigned to v , 0 o.w.

min∑v∈V

wv · xv

γeu + γev ≥ 1, ∀e = (u, v)

kv · xv −∑

e∈E(v)

γev ≥ 0, ∀v

xv ∈ Nγev ∈ {0,1}




The Relaxation

min∑v∈V

wv · xv

γeu + γev ≥ 1, ∀e = (u, v)

kv · xv −∑

e∈(v)

γev ≥ 0, ∀v

xv ≥ 0

γev ≥ 0




Why Do This?

LP can be solved in polynomial timeEvery solution to the IP is also a solution to the LPHence:

OPTLP ≤ OPTIP

We can use OPTLP as a lower bound on OPTIP




Integrality Gap

γeu + γev ≥ 1

kv · xv −∑

e∈E(v)

γev ≥ 0

1,k 1,k1

IP Cost = 1

1,k 1,k1/k

LP Cost = 1/k

1k can be arbitrarily small: there is an unbounded gap.OPTLP is a poor lower bound.




Integrality Gap

LP cost

Integrality Gap

OPT

Our Cost

β, our appx. guarantee, would be unboundedly large.




New IP

Add a new constraint: xv ≥ γev ; the IP still represents CVC.

min∑v∈V

wv · xv

γeu + γev ≥ 1, ∀e = (u, v)

kv · xv −∑

e∈E(v)

γev ≥ 0, ∀v

xv ≥ γev , ∀v∀e ∈ E(v)

xv ∈ Nγev ∈ {0,1}




New LP

min∑v∈V

wv · xv

γeu + γev ≥ 1, ∀e = (u, v)

kv · xv −∑

e∈E(v)

γev ≥ 0, ∀v


xv ≥ 0

γev ≥ 0




The Integrality Gap Fixed

γeu + γev ≥ 1

kv · xv −∑

e∈E(v)

γev ≥ 0

xv ≥ γev

1,k 1,k1

New IP Cost = 1

New LP

1,k 1,k1

New LP Cost = 1We must choose a vertex

once.




A Rounding Algorithm

Solve the LP.Set γ̂ev = 1 whenever γ∗ev ≥ 1

2 and set γ̂ev = 0 otherwise.

Set x̂v = dnumber of edges assigned to vkv

e = d∑

e∈E(v) γ̂ev

kve.




Feasibility

min∑v∈V

wv · xv

γeu + γev ≥ 1, ∀e = (u, v)

kv · xv −∑

e∈E(v)

γev ≥ 0, ∀v


xv ∈ Nγev ∈ {0,1}




Feasibility

1 γeu + γev ≥ 1, ∀e = (u, v)

γ∗eu ≥ 12 or γ∗ev ≥ 1

2 → γ̂eu = 1 or γ̂ev = 1.

2 kv · xv −∑

e∈E(v) γev ≥ 0, ∀v

x̂v = d number of edges assigned to vkv

e = d∑

e∈E(v) γ̂ev

kve ≥

∑e∈E(v) γ̂ev

kvforces this to be true.

3 xv ≥ γev , ∀v∀e ∈ E(v)

x̂v = d number of edges assigned to vkv

e ≥ 1 whenever γ̂ev = 1 forsome e hitting v .




Analysis

1 kv · x∗v ≥∑

e∈E(v) γ∗ev

2 γ̂ev ≤ 2 · γ∗ev3 If x̂v 6= 0 then x∗v ≥ 1

2 .

x̂v 6= 0→ γ̂ev = 1

→ γ∗ev ≥12

→ x∗v ≥12

(x∗v ≥ γ∗ev )




Primal-Dual Method

DualFeasible ≤ DualOPT = PrimalOPT ≤ PrimalFeasible

Construct the dual LPConstruct an algorithm that manually tightens dualconstraints to obtain a “maximal” dual solution




The Primal

min∑v∈V

wv · xv

γeu + γev ≥ 1, ∀e = (u, v) αe

kv · xv −∑

e∈E(v)

γev ≥ 0, ∀v qv

xv ≥ γev , ∀v∀e ∈ E(v) lev

xv ≥ 0γev ≥ 0




The Dual

max∑e∈E

αe

kv qv +∑

e∈E(v)

lev ≤ wv , ∀v

αe ≤ qv + lev , ∀v∀e ∈ E(v)

qv ≥ 0lev ≥ 0αe ≥ 0




The Dual

The dual to the CVC problem is a packing problem.αe: the weight packed into edge e.qv : v ’s global ability to absorb edge weight.lev : v ’s ability to absorb edge weight from edge e.




The Dual

kv qv +∑

e∈E(v)

lev ≤ wv = 17

αe1 = 4 ≤ qv + le1v

αe2 = 7 ≤ qv + le2v

αe3 = 3 ≤ qv + le3v

4

7

3

e1

e2

e3

v

1 Set le1v = 4, le2v = 7, le3v = 3.2 Set qv = 7.3 Set le1v = 1, le2v = 4, le3v = 0, qv = 3.




Primal-Dual Algorithm

Maximize∑

e∈E αe by raising all αe simultaneously whilemaking sure the constraints are satisfied.Put v in our cover when

kv qv +∑

e∈δ(v) lev ≤ wv

becomes tight.Delete v and δ(v) from the graph.Repeat until all edges are removed from the graph.





max∑e∈E

αe

kv qv +∑

e∈δ(v)

lev ≤ wv , ∀v

αe ≤ qv + lev , ∀v∀e ∈ δ(v)

As we raise αe we need to raise either qv or lev . Which?We want kv qv +

∑lev to approach wv slowly.

Lower kv qv +∑

lev values let us raise αe values higher.




Raising Variables

kv qv +∑

e∈E(v)

lev ≤ wv = 17

αe1 = 4 ≤ qv + le1v

αe2 = 7 ≤ qv + le2v

αe3 = 3 ≤ qv + le3v

4

7

3

e1

e2

e3

17,5

v

le1v = 4, le2v = 7, le3v = 3kv qv +

∑lev = 14.

qv = 7kv qv +

∑lev = 35.




Raising Variables

kv qv +∑

e∈E(v)

lev ≤ wv = 17

αe1 = 4 ≤ qv + le1v

αe2 = 7 ≤ qv + le2v

αe3 = 3 ≤ qv + le3v

4

7

3

e1

e2

e3

17,1

v

le1v = 4, le2v = 7, le3v = 3kv qv +

∑lev = 14.

qv = 7kv qv +

∑lev = 7.




Low and High Degree Vertices

A low degree vertex is one where |δ(v)| ≤ kv .Raise lev for all edges in δ(v).

A high degree vertex is one where |δ(v)| > kv .Raise qv .





Maximize∑

e∈E αe by raising all αe simultaneously whilemaking sure the constraints are satisfied.Put v in our cover when

kv qv +∑

e∈δ(v) lev ≤ wv

becomes tight.Delete v and δ(v) from the graph.Repeat until all edges are removed from the graph.




The Algorithm

4,416,2

12,7

20,1

16,18,2

13,1

16,3 4,1

00

0

0

0

0

0

0

0

0

kv qv +∑

e∈E(v)

lev ≤ wv

αe ≤ qv + lev




The Algorithm

4,416,2

12,7

20,1

16,18,2

13,1

16,3 4,1

11

1

1

1

1

1

1

1

1

kv qv +∑

e∈E(v)

lev ≤ wv

αe ≤ qv + lev




The Algorithm

kv qv +∑

e∈E(v)

lev ≤ wv

αe ≤ qv + lev




The Algorithm: Cost = 56

kv qv +∑

e∈E(v)

lev ≤ wv

αe ≤ qv + lev




Analysis

We want to relate our cost to the dual feasible solution.When v is put in our cover

kv qv +∑

e∈δ(v) lev = wvqv + lev = αe

We’ve raised qv and lev variables differently for differentvertices.




3 Types of Vertices

A vertex is always low-degree: “LD” type vertices.A vertex is always high-degree: “HD” type vertices.A vertex changes from high-degree to low-degree: “T” typevertices.




High-Degree to Low-Degree Vertex

High Degree

16,2

Low Degree




Low-Degree Vertices

v

5,6

1 11

1

1

We need 1 copy.lev = 1qv = 0

kv qv +∑

e∈E(v)

lev ≤ wv

αe ≤ qv + lev




Low-Degree Vertices

We always need 1 copy.αe = lev∑

lev = wv

We pay wv =∑

αe

kv qv +∑

e∈E(v)

lev ≤ wv

αe ≤ qv + lev




High-Degree Vertices

v

3,3

1 11

1

1

We need 2 copies.lev = 0qv = 1

kv qv +∑

e∈E(v)

lev ≤ wv

αe ≤ qv + lev




High-Degree Vertices

We need d |δ(v)|kve copies.

d |δ(v)|kve ≤ 2 · |δ(v)|

kv

αe = qv

kv · qv = wv

kv qv +∑

e∈E(v)

lev ≤ wv

αe ≤ qv + lev

We pay d |δ(v)|kvewv ≤ 2 |δ(v)|

kvwv = 2 |δ(v)|

kvkv qv = 2

∑αe




Transitional Vertices when |δ(v)| = kv

We need 1 copy.lev = 2qv = 1

kv qv +∑

e∈E(v)

lev ≤ wv

αe ≤ qv + lev




Transitional Vertices when |δ(v)| = kv

We always need 1 copy.|δ(v)| = kv

αe = lev + qv

kv qv +∑

lev = wv

kv qv +∑

e∈E(v)

lev ≤ wv

αe ≤ qv + lev

We pay wv = kv qv +∑

lev = kv (qv + lev ) =∑

αe




Transitional Vertices: The General Case

To generalize we need more notation:Let D(v) be the edges incident on v at the time v wentfrom a HD vertex to a LD vertex.|D(v)| = kv




Transitional Vertices: The General Case

We pay wv · 1 = qv kv +∑

e∈δ(v)

lev

≤ qv kv +∑

e∈D(v)

lev

=∑

e∈D(v)

lev + qv

=∑

e∈D(v)

αe




Transitional Vertices

Every edge in D(v), not δ(v), is charged.Edges in D(v)− δ(v) are charged from both ends.




Charging Edges

Type 1: u is LD, v is T

u v1 Charge

u v1 Charge

(u, v) is charged 2αe.

Type 2: u is HD, v is T

2 Charges

u v

1 Charge

u v

(u, v) is charged 3αe.




Charging Edges

Each edge is charged at most 3αe

Our Cost ≤ 3∑

e αe.We have a 3 approximation.




Improvement

Our Cover




Improvement

Improved Cover

4,416,2

12,7

20,1

16,18,2

13,1

16,3 4,1

1

1

11

1




References

S. Guha, R. Hassin, S. Khuller, and E. Or. Capacitated VertexCovering with Applications, Symposium on Discrete Algorithms(SODA) 2002.




Thank You!


capacitated vertex cover - rutgers universitycrab.rutgers.edu/~rajivg/studenttalks/cvc.pdf · 2009....

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