1 Combinatorial Dominance Analysis The Knapsack Problem Keywords: Combinatorial Dominance (CD)...

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Combinatorial Dominance Analysis

The Knapsack Problem

Keywords:Combinatorial Dominance (CD)Domination number/ratio (domn, domr)

Knapsack (KP)

Incremental Insertion (II)Local Exchange (LE)PTAS Optimal Head - Greedy Tail (GRT)

Presented by:

Yochai Twitto

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Overview Background

On approximations and approximation ratio. Combinatorial Dominance

What is it ? Definitions & Notations.

The Knapsack Problem simple Algorithms & Analysis

Incremental Insertion Local Exchange PTASing “Optimal head - greedy tail” algorithm

Summary

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Overview Background





Summary

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Background NP complexity class.

AA and quality of approximations.

The classical approximation ratio analysis.

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NP

If P ≠ NP, then finding the optimum of NP-hard problem is difficult.

If P = NP, P would encompass the NP and NP-Complete areas.

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Approximations

So we are satisfied with an approximate solution.

Question: How can we measure

the solution quality ?Solutions

quality line

OPT

Infeasible

Near optimal

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Solution Quality

Most of the time, naturally derived from the problem definition.

If not, it should be given as external information.

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The classical Approximation Ratio(For maximization problem)

Assume 0 ≤ β ≤ 1. A.r. ≥ β if

the solution quality is greater than β·OPT

Solutions quality line

OPT

Infeasible

Near optimal

½OPT

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Overview Background





Summary

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Combinatorial Dominance

What is a “combinatorial dominance guarantee” ?

Why do we need such guarantees ?

Definitions and notations.

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What is a

“combinatorial dominance guarantee” ?

A letter of reference: “She is half as good as I am, but I am the best

in the world…” “she finished first in my class of 75 students…”

The former is akin to an approximation ratio.

The latter to combinatorial dominance guarantee.

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What is a

“combinatorial dominance guarantee” ? (cont.)

We can ask:

Is the returned solution

guaranteed to be always

in the top O(n) bestsolutions ?

Solutions quality line

OPT

Infeasible

Near optimaltop

O(n)

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Why do we need that ?

Assume an problem for which all solutions are at least a half as good as optimal solution.

Then, 2-factor approximating the problem is meaningless.

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Corollary

The approximation ratio analysis gives us only a partial insight of the performance of the algorithm.

Dominance analysis makes the picture fuller.

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Definitions & Notations

Domination number: domn

Domination ratio: domr

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Domination Number: domn Let P be a CO problem. Let A be an approximation for P .

For an instance I of P, the domination number domn(I, A) of A on I is the number of feasible solutions of I that are not better than the solution found by A.

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domn (example)

STSP on 5 vertices. There exist 12 tours

If A returns a tour of length 7 then domn(I, A) = 8

4, 5, 5, 6, 7, 9, 9, 11, 11, 12, 14, 14

(tours lengths)

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Domination Number: domn Let P be a CO problem. Let A be an approximation for P .

For any size n of P, the domination number domn(P, n, A) of an approximation A for P is the minimumminimum of domn(I, A) over all instances I of P of size n.

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Domination Ratio: domr Let P be a CO problem. Let A be an approximation for P . Denote by sol(sol(I I )) the number of all

feasiblefeasible solutions of I.

For any size n of P, the domination ratio domn(P, n, A) of an approximation A for P is the minimumminimum of domn(I, A) / sol(I ) taken over all instances I of P of size n.

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Overview Background





Summary

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The Knapsack Problem Instance:

Multiset of integers

Capacity

Find:

},,,{ 21 nwwwS T

)'(maxarg

)'('

SW

TSWSS

')'( where

SwwSW

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Simple

Algorithms & Analysis Incremental Insertion (II)

Arbitrary order Increasing order Decreasing order (Greedy)

Local Exchange (LE)

PTASing

“Optimal Head – Greedy Tail” (GRT)

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II – Arbitrary Order

Go over the elements (arbitrary order)

Insert an element if the capacity not exceeded

Theorem: 12),,domn( 1 nnIIKP

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Proof

Suppose the weights are Let be any locally optimal

solution

We may assume Otherwise, is optimal

12),,domn( 1 nnIIKP

nwww 21

1S

Twn

ii

1

1SSS 1

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Proof (cont.)

Let be the largest index of a weight not belonging to

Since is locally optimal

k

1S

kwTSW )( 1

1S

12),,domn( 1 nnIIKP

9 8 7 6 3 2 :

12 11 10 5 4 1:

1

1

CS

S

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Proof (cont.)

Denote by the interval For any solution not containing

Either Or

That is, the number of solutions with total weight in is at most

12),,domn( 1 nnIIKP

L kwSWSW )(),( 11

2S kwLSW )( 2

LwSW k }){( 2

12 1 nL

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Proof (cont.)

Solutions of weight at leastare infeasible.

Solution weighted not more than are not better than

12),,domn( 1 nnIIKP

kwSW )( 1

)( 1SW

1S

12),,domn( 1 nnIIKP

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Proof (cont.)

Blackball instance:

II can lead to Which is locally optimal

12),,domn( 1 nnIIKP

22

132

1

nT

www

nw

n

},,,{ 32 nwww

blackba

ll

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Proof (cont.)

Taking the first item and omitting at least one of the rest is better.

Hence

And we finished...

12),,domn( 1 nnIIKP

12),,domn( 1 nnIIKP

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II – Increasing Order

No Gain!

That was our blackball… In the previous proof.

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II - Decreasing Order (Greedy)

No drastic gain! Blackball instance B:blackba

ll

45

1

2

3

65

432

1

nT

www

nwww

nw

n

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Greedy(B) Weight:

Any solution taking Exactly two elements from Any of the last elements

is better!

},,,,{ 651 nwwww 44 n

432 ,, www4n

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4232),,domn( nnnGreeyKP

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Simple



Local Exchange (LE)

PTASing


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Local Exchange (LE)

Assume is a solution Allowed operations:

Insert a new element x to Exchange x by y

x belongs to y not belongs to x < y

'S

'S

'S'S

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Local Exchange

Theorem:

nnLEKP n 12),,domn(

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Proof

Suppose the weights are Let be any locally optimal

solution

We may assume Otherwise, is optimal

nwww 21

1S

Twn

ii

1

1SSS 1

nnLEKP n 12),,domn(

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Proof (cont.)

Let be the largest index of a weight not belonging to

Since is locally optimal

k

1S

kwTSW )( 1

1S

nnLEKP n 12),,domn(

9 8 7 6 3 2 :

12 11 10 5 4 1:

1

1

CS

S

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Proof (cont.)

Denote by the interval For any solution not containing

Either Or

That is, the number of solutions with total weight in is at most

And there are at least outside

L kwSWSW )(),( 11

kwLSW )( 2

LwSW k }){( 2

12 1 n

nnLEKP n 12),,domn(

9 8 7 6 3 2 :

12 11 10 5 4 1:

1

1

CS

S

12 1 nL

2S

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Proof (cont.)

Let be the number of items belonging to among the first k -1 items

Let be the number of items not belonging to among the first k -1 items

How many solution pairs are of weight not belonging to ?

nnLEKP n 12),,domn(

9 8 7 6 3 2 :

12 11 10 5 4 1:

1

1

CS

S

h1Sl1S

L

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Proof (cont.)

We saw that

All solutions obtained by dispensing of some items fromAnd the one obtained from them by adjoining the ’th item not belong to the interval

12domn 1 n

12 h

nnLEKP n 12),,domn(

9 8 7 6 3 2 :

12 11 10 5 4 1:

1

1

CS

S

h

k

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Proof (cont.)

So

For each of the solutions obtained from by adjoining one of the itemsof

Both the obtained solution And the one obtain by adjoining it the ’th

item

not belong to the interval

1212domn 1 hn

nnLEKP n 12),,domn(

9 8 7 6 3 2 :

12 11 10 5 4 1:

1

1

CS

S

l 1S

l

k

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Proof (cont.)

So

Since our solution can not be improved by local exchange

Each of the n-k solutions obtained by removing one of the last n-k items not belong to the interval

Adding each of them the ’th item we get infeasible solutions

lhn 1212domn 1

nnLEKP n 12),,domn(

9 8 7 6 3 2 :

12 11 10 5 4 1:

1

1

CS

S

k

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Proof (cont.)

So

n

knlh

knl

nLEKP

n

n

hn

1

1

1

2

1112

1212

),,domn(

nnLEKP n 12),,domn(

9 8 7 6 3 2 :

12 11 10 5 4 1:

1

1

CS

S

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Proof (cont.)

Blackball instance:

LE can lead to Which is locally optimal

12),,domn( 1 nnIIKP

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132

1

nT

www

nw

n

},,,{ 32 nwww

blackba

ll

nnLEKP n 12),,domn(

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Proof (cont.)

Taking the first item and omitting at least two of the rest is better.

Hence:

And we finished...

12),,domn( 1 nnIIKP nnLEKP n 12),,domn(

n

nnLEKPn

nn

1

1

2

)]1(12[2),,domn(

b(n)

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Simple



Local Exchange (LE)

PTASing


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PTASing

There exist a PTAS for Knapsack That is, it is possible to approximate

the optimal solution to within any factor c >1

In time polynomial in n and 1/(c -1)

We’ll see

n

c

cnPTASKP

n

ccccn /

1),,domn(12

)1/()1/(

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Theorem 1

Let be an instance of KP Denote the weight of optimal

solution by Assume H is a factor-c

approximation for KP Then ),,domn(12 )1/( nPTASKP c

cn

S

)(SOPT

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Proof Assume that the elements of

optimal solution are labeled such that

Let ’ be the smallest integer such that

),,domn(12 )1/( nPTASKP ccn

'

1

1'

1

)( k

ii

k

ii w

c

SOPTw

},,,{ 21 kopt wwwS 1 ii ww

k

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Proof (cont.)

Let

Observe that


)(\

\)(

},,,{

3

12

'211

SOPTSC

CSOPTC

wwwC k

321 CCCn

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Proof (cont.)

Also note that Since

The weight of every element of is not more than the weight of any element of

is a c -approximated solution to


12 )1( CcC

1C

1C2C

21 CCSopt

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Proof (cont.)

Let is minimized for

Since


31 ,max CCD )1/(1 cnC

1213 CcnCCnC

D

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Proof (cont.)

Note that our solution dominate united with any of the

non-empty subsets of Since they are not feasible

Since is optimal


21 CCSopt 12 3 C

3C

optS

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Proof (cont.)

Note that our solution dominate all subset of

Since the weight of each is not more than

And our solution weight is at least


112 C },,,{ 1'21 kwww

cSOPT /)(

cSOPT /)(

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Proof (cont.)

Summing both terms, the number of solution dominated is

Minimizing the left-hand term we get the result.


113 212 CC

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Theorem 2 For every c >1 there exist a KP

instance and a solution thereof of total weight dominating only

solution.

nc

cn

cc/

1)1/(

S1S

cSOPTSW /)()( 1

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Proof Blackball instance:

can return a solution consisting of items

)1/(

1321

ccnT

wwww nblackba

ll

nc

cnPTASKP

n

ccc /1

),,domn()1/(

cPTAS )1/( cn

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Proof (cont.)

Such solution dominates all solutions consisting of up to item

It also dominates all infeasible solution i.e: solution consisting of more than items.

Those are the only solution it dominates

nc

cnPTASKP

n

ccc /1

),,domn()1/(

)1/( cn

T

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Proof (cont.)

Hence

Employing Stirling’s formula we obtain the theorem

nc

cnPTASKP

n

ccc /1

),,domn()1/(

n

ccnj

cn

jc j

n

j

nnPTASKP

)1/(1

)1/(

0

),,domn(

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Simple



Local Exchange (LE)

PTASing


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Optimal Head – Greedy Tail

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Summary

12),,domn( 1 nnIIKP

4232),,domn( nnnGreeyKP

nnLEKP n 12),,domn(

n

c

cnPTASKP

n

ccccn /

1),,domn(12

)1/()1/(

nnGRTKP n log/112),,domn(

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Combinatorial Dominance Analysis

The Knapsack Problem

1 Combinatorial Dominance Analysis The Knapsack Problem Keywords: Combinatorial Dominance (CD)...

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