Speaker: Yoni RozensheinInstructor: Prof. Zeev Nutov

Talk outlineProblem description and known

approximationsInterpretation in graphs (Independent Set

Problem)The greedy local search method

Approximation ratioPolynomial time implementation

2

Weighted k-set packingGiven a collection of weighted sets of size at

most k,find a maximum weight collection of disjoint sets

Example with k = 3:Set Weight

{ 4, 5 } 16

{ 2, 3, 5 } 6

{ 3, 4, 5 } 12

{ 1, 5 } 5

{ 4 } 14

{ 4, 5 }

{ 2, 3, 5 }{ 4 }

{ 3, 4, 5 }

{ 1, 5 }{ 4 }

Set packing

3

Weight

16

20

12

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Hardness of the problemFor k = 2 we get the Maximum (Weighted)

Matching Problem, which admits a polynomial time algorithm [Edmonds 1965]

The problem is NP-hard for k ≥ 3 [Karp 1972]Reducible to k-SATk = 3: 3-Dimensional Matching

4

Approximation algorithmsOPT(I) – weight of the best solution (on instance

I)ALG(I) – weight of a given algorithm’s solutionAlgorithm’s approximation ratio on instance I:

Algorithm’s approximation ratio:

We seek an algorithm that minimizes the ratio

OPT

ALG

II

I

5

OPTmax : is an instance of the problem

ALG

II

I

The greedy algorithmRepeatedly choose maximum-weight set S

and delete from the family all sets that intersect S

Very fast; performance ratio k:Set Weight

{ 1, 2, 3, …, k } 1 + ε

{ 1 } 1

{ 2 } 1

… …

{ k } 1

Greedy choice

Intersects...

A A, A1, A2, …, An

B B, B1, B2, …, Bm

… …

Z Z, Z1, Z2, …, Zl

Optimal

Greedy Optimal

ik w A w A

k w w

Greedy Optimalk w w

6

Local search heuristicReplace a subset of the solution with a better

subsetRepeat until “locally optimal”

How are the improvements chosen?Polynomial running time?

Performance ratio k – 1 + ε [Arkin and Hassin 1997]

Can we improve on these ratios?7

Interpretation in graphsThe sets’ intersection graph:

Nodes correspond to setsEdges correspond to sets sharing elements

Set packing is a particular case of Maximum Weight Independent Set in intersection graphs

What characterizes intersection graphs arising from k-set packing instances?

Set Weight

A = { 4, 5 } 16

B = { 2, 3, 5 } 6

C = { 3, 4, 5 } 12

D = { 1, 5 } 5

E = { 4 } 14

AB

E D

C

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The characterizationThe graph is k + 1-claw free

From now on, we consider the “Independent Set Problem in k + 1-claw free graphs”

1

2

k + 1

Example with k = 3

9

k + 1-claw

Proof:a. At most k

elements in parent node

b. At least one in each child node

c. Pigeon-hole principle

Approach: Greedy local searchPrevious algorithms combined

How are improvements chosen?Polynomial running time?

GREEDY-LOCAL-SEARCH(G)I ← GREEDY(G)while I is not locally optimal do

I’ ← local improvement of II ← I’

end whileoutput I

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Local improvement schemeImprovement: Pick some v ϵ I, add some of v’s

neighbors, and delete any interfering nodesThe payoff factor is

nodes added

nodes removed

w

w

w Dpayoff factor

w A w B

11

A

B

E D

C

Local ImprovementPick B

Add DDelete A, B

A

B

E D

CPick BAdd D

Delete A, B

A

B

E D

C

First algorithm: ANYIMP

Polynomial running time?For now, we will analyze the approximation

ratio only

ANYIMP(G, α)I ← GREEDY(G)while I is not locally optimal do

I’ ← any improvement of I with payoff factor ≥ αI ← I’

end whileoutput I

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Analysis of the approximation ratioProjection: “Distance” between I and OPTproj(I) and w(I) maintain equilibrium

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proj(I) w(I) Φ(I)≈ x⋅proj(I) + y⋅w(I)

min Φ(I)

: OPT ; :Maximum-weight neighbor of in I I

b b If I f b

b I b I

Projection

AB

E D

CA

B

E D

COPT I

max ,I

I

f D A B

f E E

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OPT

: Ib

proj I w f b

Projection propertiesEquilibrium property:

Local optimum property:

Result:

(for any intermediate I) OPT1 1

kw I proj I

(for the final I) 11 OPTk w I proj I

(for the final I) 2

1

1 1

1OPT 4 2

1 5

k k

w I

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Second algorithm: BESTIMP

BESTIMP(G)I ← GREEDY(G)while I is not locally optimal do

I’ ← an improvement of I with the highest payoff factorI ← I’

end whileoutput I

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Potential function

In ANYIMP’s potential*, d was constant

(Ii, di): Sub-sequence of local improvements

Largest such that di (payoff factor) is descending:

2, 5, 3, 2, 4, 1.5, 3, 2, 1.2, 4, 3, 1.1

1, :

1

kI d w I proj I

d

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1

kw I proj I

*

d1 = 2 d2 = 1.5 d3 = 1.2 d4 = 1.1

(Greedy) I0 I1 I2 I3 I4 (Final)

Potential propertiesEquilibrium property:

Weight evaluation property:

Result:

(for i = 1, 2, …) , OPT

1i

i ii

d h iI d

d

(for i = 1, 2, …) 1

11 1 OPTi

ii i i

h i dk w I

d d d

(for the final I) 1

2

2 1OPT 1 11 2

2 3m m

kk

w I d d

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Running time analysisReminder:

Individual steps run in polynomial timeNumber of improvements?

GREEDY-LOCAL-SEARCH(G)I ← GREEDY(G)while I is not locally optimal do

I’ ← local improvement of II ← I’

end whileoutput I

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Polynomial-time approximation schemeGiven an algorithm with approximation ratio

ρ, produce a polynomial time algorithm with approximation ratio ρ + ε

Well-known example: KnapsackThe running time may depend strongly on ε

(For example: Polynomial in 1/ ε)

Our greedy local search algorithm already runs in pseudo-polynomial time

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PTAS: Weight scaling

Each improvement increases weight by 1Number of improvements at most:

1 1 2 2Input , ; , ; ; ,q qS w S w S w

maximum weightmm

wK w

q

ii

ww

K

2m

m

w qN qw q

K

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Weight scaling – cont’d

Special case:

If we get within r of OPT'', we get within r(1 – ε) of OPT

ii i

ww Kw K

K

1ii i

ww K w K

K

OPTi i mw w Kq w

OPT OPT 1iw

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Questions?