Www.cs.technion.ac.il/~reuven 1 New Developments in the Local Ratio Technique Reuven Bar-Yehuda...
-
date post
21-Dec-2015 -
Category
Documents
-
view
216 -
download
0
Transcript of Www.cs.technion.ac.il/~reuven 1 New Developments in the Local Ratio Technique Reuven Bar-Yehuda...
www.cs.technion.ac.il/~reuven 1
New Developments in the New Developments in the Local Ratio TechniqueLocal Ratio Technique
Reuven Bar-Yehuda
www.cs.technion.ac.il/~reuven
www.cs.technion.ac.il/~reuven 2
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
www.cs.technion.ac.il/~reuven 3
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|
www.cs.technion.ac.il/~reuven 4
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
www.cs.technion.ac.il/~reuven 6
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
www.cs.technion.ac.il/~reuven 7
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}.
www.cs.technion.ac.il/~reuven 8
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|
www.cs.technion.ac.il/~reuven 9
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
www.cs.technion.ac.il/~reuven 10
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’;
www.cs.technion.ac.il/~reuven 15
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|
www.cs.technion.ac.il/~reuven 16
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
www.cs.technion.ac.il/~reuven 20
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
www.cs.technion.ac.il/~reuven 21
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
www.cs.technion.ac.il/~reuven 22
4-approximation for2 Dimentional Interval graphs