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New Developments in the New Developments in the Local Ratio TechniqueLocal Ratio Technique

Reuven Bar-Yehuda

www.cs.technion.ac.il/~reuven

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General framework:General framework:Given a weight vector w.

Minimize [Maximize] w·x

Subject to: feasibility constraints F(x)

x is an r-approximation if F(x) and w·x rw·x*

[w·x rw·x* ]

An algorithm is an r-approximation if for any w, F

it returns an r-approximation

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The minimum vertex cover problemThe minimum vertex cover problem

Minimize w·x

Subject to: xu + xv 1 e=(u,v) E

x {0,1}|V|

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15

Min 5xBisli+8xTea+12xWater+10xBamba+20xShampoo+15xPopcorn+6xChocolate

s.t. xShampoo + xWater 1

5

812

20

6

10

www.cs.technion.ac.il/~reuven 5 Movie:Movie:1 4 the price of 21 4 the price of 2

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2-Approx 2-Approx VC(G,w)VC(G,w)If G= return If v V w(v)=0 return {v}+GVC(G-E(v)-v, w)

Let {u,v} E and = min {w(u), w(v)}.

if i{u,v}

w11(i) =

0 else

Notice:w1 x 2 w1 x for Good(x)

REC= VC(G, VC(G, w2= w- w-ww11))

Return RECReturn REC

Induction hyp is: w2REC 2 w2x

so if Good(REC): w1REC 2 w1x we are done

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2-Approx 2-Approx VC VC (Bar-Yehuda Even 81)(Bar-Yehuda Even 81)

1. For each edge {u,v} do:

2. Let = min {w(u), w(v)}.

3. w(u) w(u) - .

4. w(v) w(v) - .

5. Return {v | w(v) = 0}.

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The generalized vertex cover problemThe generalized vertex cover problem

Minimize w·x

Subject to: xu + xv + xe 1 e={u,v} E

x {0,1}|V|+|E|

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15

Min 5xBisli+8xTea+12xWater+10xBamba+20xShampoo+15xPopcorn+6xChocolate

+$4xWaterShampoo+ • • •

s.t. xShampoo + xWater + xWaterShampoo 1

5

812

20

6

10

$4

$1

$3

$1

$1

$2

$1$1

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2-Approx 2-Approx GVC(G,w)GVC(G,w)

If E= return If e E w(e)=0 return {e}+GVC(G-e, w)

If v V w(v)=0 return {v}+GVC(G-E(v), w)

Let e={u,v} E s.t = min {w(u), w(v), w(e)}>0.

if x{u,v,e}w11(x) =

0 else

Notice:w1 x 2 w1 x for Good(x)

REC= GVC(G, VC(G, w2= w- w-ww11))

Induction hyp is: w2REC 2 w2x

so if Good(REC): w1REC 2 w1x we are done

If REC-e is a cover thenREC=REC-eIf REC-e is a cover thenREC=REC-e

Return RECReturn REC

www.cs.technion.ac.il/~reuven 11

““2 integral for the price of 1 fractional”: 2 integral for the price of 1 fractional”: The local ratio technique for roundingThe local ratio technique for rounding

Let x be the the fractional solution

Minimize w·x

Subject to: xu + xv + xe 1 e=(u,v) E

x [0,1]|V|+|E|

www.cs.technion.ac.il/~reuven 12 ““d d integral for the price of integral for the price of ½(d+1) fractional”: fractional”: 2-2/(2-2/(ΔΔ+1)-Approx +1)-Approx GVC(G,w)GVC(G,w)If E= return If e E w(e)=0 return {e}+GVC(G-e, w)

If v V w(v)=0 return {v}+GVC(G-E(v)-v, w)

Let v V s.t xv is minum and

Let =min(w(i) : i N[v]}

if i N[v]w11(i) =

0 else

Claim:w1 x rΔ w1 x for Good(x)

REC= GVC(G, VC(G, w2= w- w-ww11))

Induction hyp is: w2REC rΔ w2x

so if Good(REC): w1REC rΔ w1x we are done

If REC is not a minimal cover then make REC minimalIf REC is not a minimal cover then make REC minimal

Return RECReturn REC

Min xv

www.cs.technion.ac.il/~reuven 13 ““d d integral for the price of integral for the price of ½(d+1) fractional”: fractional”: Claim: w1 x rΔ w1 x for Good(x)

Min xv

If Min xv ≥ ½

Then x(N[v]) ≥ ½(d+1)

Else x(N[v]) ≥ ½(d+1)

Thus w1 x ≥ ½(d+1)

But w1 x d

Hence: w1 x/ w1 x 2-2/(d+1)

2-2/(ΔΔ +1) = rΔ

www.cs.technion.ac.il/~reuven 14 A Generalized Local-Ratio Schema for A Generalized Local-Ratio Schema for

M Minimizationinimization [ [MMaximization] problems:aximization] problems:Let x be any “fisible?” vector (e.g. an optimal solution)

Algorithm r-ApproxMin [Max](Set, w)

If Set = then return ;

If v G w(v) = 0 then return {v} r-ApproxMin(Set-{v},w ) ;

[If v G w(v) 0 then return r-ApproxMax(Set-{v},w ) ;]

Define “good” w1 ; i.e. Good(x): w1 x [] r w1 x

REC = r-ApproxMin [Max](Set, w2 ) ;

Induction hyp is: w2REC [] r w2x

so if Good(REC): w1REC [] r w1x we are done,

otherwise “fix it”; return REC’;

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The maximum independent set problemThe maximum independent set problem

Maximize w·x

Subject to: xu + xv ≤ 1 e=(u,v) E

x {0,1}|V|

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The maximum independent set problemThe maximum independent set problem “1 integral for the gain of 2 fractional”: “1 integral for the gain of 2 fractional”:

Let x be the the fractional solution

Maximize w·x

Subject to: xu + xv ≤ 1 e=(u,v) E

x [0,1]|V|

www.cs.technion.ac.il/~reuven 17 Gain Gain 11 integral, lose integral, lose ½(d+1) fractional fractional

2/(2/(ΔΔ+1)-Approx +1)-Approx IS(G,w)IS(G,w)If v V w(v) 0 return IS(G-v, w)

If E= return V

Let v V s.t xv is maximum and

Let = w(v)

if i N[v]w11(i) =

0 else

Claim:w1 x ≥rΔ w1 x for Good(x)

REC= IS(G, (G, w2= w- w-ww11))

Induction hyp is: w2REC ≥ rΔ w2x

so if Good(REC): w1REC ≥ rΔ w1x we are done

If REC+v is an independent set then REC=REC+vIf REC+v is an independent set then REC=REC+v

Return RECReturn REC

Max xv

www.cs.technion.ac.il/~reuven 18 Gain Gain 11 integral, lose integral, lose ½(d+1) fractional fractional Claim: w1 x ≥ rΔ w1 x for Good(x)

Max xv

If Max xv ≤ ½

Then x(N[v]) ≤ ½(d+1)

Else x(N[v]) ≤ ½(d+1)

Thus w1 x ≤ ½(d+1)

But w1 x ≥ d

Hens: w1 x/ w1 x ≥ 2-2/(d+1)

≥ 2-2/(ΔΔ +1) = rΔ

www.cs.technion.ac.il/~reuven 19 Single Machine Scheduling :

Activity9Activity8Activity7Activity6Activity5Activity4Activity3Activity2Activity1 ????????????? time

Maximize s.t. For each instance I:

For each time t:

For each activity A:

I

IxIp )( }1,0{Ix

)()(:

1)(IetIsI

IxIw

1AI

Ix

Bar-Noy, Guha, Naor and Schieber STOC 99: 1/2 LP

Berman, DasGupta, STOC 00: 1/2

This Talk, STOC 00(Independent)      1/2

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ÎÎ, and the weight decomposition:, and the weight decomposition:

• Let Î be the interval which ends first.

I in conflict with Î ,

• Define w1(I) = w2= w-w1

0 otherwise,

w1= w1= w1= w1= w1=

w1= w1=

w1= w1=

w1= 0

w1= 0

w1= 0w1= 0

w1= 0w1 = 0

w1= 0w1= 0

w1= 0 w1= 0

w1= 0

time

Activity9Activity8Activity7Activity6Activity5Activity4Activity3Activity2 Activity1

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½ -Approx IS(G,w):

1. Delete all instances with non-positive weight.

2. If G=, return .

3. Select Î which end first, and let = w (Î ).

I in conflict with Î,

4. Define w1(I) =

0 otherwise,

5. REC IS(G, w2= w-w1)

6. If REC{Î } is a feasible schedule, return REC{Î }

Otherwise, return REC

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4-approximation for2 Dimentional Interval graphs