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Friedhelm Meyer auf der Heide 1

HEINZ NIXDORF INSTITUTE

University of Paderborn

Algorithms and Complexity

Welcome to the course

Algorithm Design

Summer Term 2011

Friedhelm Meyer auf der Heide

Lecture 6, 20.5.2011

Friedhelm Meyer auf der Heide 2

HEINZ NIXDORF INSTITUTE

University of Paderborn

Algorithms and Complexity Topics

- Divide & conquer

- Dynamic programming

- Greedy Algorithms

- Approximation Algorithms

- Randomized Algorithms

- Online Algorithms

Friedhelm Meyer auf der Heide 3

HEINZ NIXDORF INSTITUTE

University of Paderborn

Algorithms and Complexity

For which problems are Greedy-Algorithms

optimal?

Consider a finite set E and a system U of subsets of E. (E,U) is called a

subset-system, if the following holds:

(i) ? 2 U

(ii) For each B 2 U, also each subset of B is in U.

B2 U is called maximal, if no proper superset of B is in U.

The Optimization problem corresponding to (E,U) is :

Given a weight function w:E Q+ ,compute a maximal set B 2 U with

maximizes w(B) = §e2 B w(e).

(Minimization problems are described analogously.)

Friedhelm Meyer auf der Heide 4

HEINZ NIXDORF INSTITUTE

University of Paderborn

Algorithms and Complexity

Subset-Systems and the

Canonical Greedy-Algorithm

Canonical Greedy ((E,U))

(1) Sort E such that w(e1) ¸ … ¸ w(en).

(2) B ?.

(3) For k=1 to n

if B [ {ek} 2 U then B B [ {ek}

(4) Return B

Friedhelm Meyer auf der Heide 5

HEINZ NIXDORF INSTITUTE

University of Paderborn

Algorithms and Complexity Greedy-Algorithms und Matroids

A system of subsets (E,U) is a matroid, if in addition, the following

exchange property holds:

(iii) For all A,B 2 U with |A|<|B|, there is x2 B-A such that A[{x} 2 U.

Remark: All maximal sets of a matroid have the same size. (homework)

Theorem: Consider a system of subsets (E,U). Canonical Greedy is optimal

for (E,U) for every weight function w, if and only if (E,U) is a matroid.

Friedhelm Meyer auf der Heide 6

HEINZ NIXDORF INSTITUTE

University of Paderborn

Algorithms and Complexity Matroids and the canonical Greedy Algorithm

Theorem: Consider a system of subsets (E,U). Canonical Greedy is optimal

for (E,U) for every weight function w, if and only if (E,U) is a matroid.

Proof:

“(”: Consider a matroid (E,U) and a weight function w.

Let w(e1)¸ …¸ w(en).

- Consider an optimal solution T’={ei1, …, eik

}.

- Assume the solution T={ej1, …, ejk

} found by Canonical Greedy were not

optimal, i.e. w(T’)> w(T).

- Then there is an index p with w(eip) > w(ejp

). Let p be minimal with this

property. Note that ip < jp, because the items are sorted w.r.t weight.

- Apply the exchange property to A = {ej1, …,ejp-1

} and B ={ei1, …, eip

} :

As |A| < |B|, there is eiq 2 B-A with A [ {eiq

} 2 U.

- As w(eiq ) ¸ w(eip

) > w(ejp), Canonical Greedy would have chosen eiq

before

ejp , a contradiction.

Friedhelm Meyer auf der Heide 7

HEINZ NIXDORF INSTITUTE

University of Paderborn

Algorithms and Complexity Matroids and the canonical Greedy Algorithm

Theorem: Consider a system of subsets (E,U). Canonical Greedy is optimal for (E,U) for every weight function w, if and only if (E,U) is a matroid.

Proof:

“)”: Assume that, for some A,B 2 U with |A|<|B|, it holds that A[{e} is not in U, for every e2B-A.

Let |B|=r and consider the weight function w with

- w(e) = r+1 for e2A

- w(e) = r for e2B-A

- w(e) = 0 else

Then, Canonical Greedy would select a solution T with T µ A and T Å (B-A) =?.

As w(T)=|A|(r+1) · (r-1)(r+1)· r2 -1, T is not optimal, because B is a solution with larger weight |B|r = r2.

Friedhelm Meyer auf der Heide 8

HEINZ NIXDORF INSTITUTE

University of Paderborn

Algorithms and Complexity Topics

- Divide & conquer

- Dynamic programming

- Greedy Algorithms

- Approximation Algorithms

- Randomized Algorithms

- Online Algorithms

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Approximation Algorithms

• Greedy Techniques – Load-Balancing Problem

– Center Selection Problem

• Pricing Method – Vertex Cover Problem

• Linear Programming and Rounding – Vertex Cover Problem

– Generalized Load-Balancing Problem

• Polynomial Time Approximation Scheme – Knapsack Problem

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Load Balancing: List Scheduling

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Optimal Schedule

List schedule

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1.

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Approximation Algorithms

• Greedy Techniques – Load-Balancing Problem

– Center Selection Problem

• Pricing Method – Vertex Cover Problem

• Linear Programming and Rounding – Vertex Cover Problem

– Generalized Load-Balancing Problem

• Polynomial Time Approximation Scheme – Knapsack Problem

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Approximation Algorithms

• Greedy Techniques – Load-Balancing Problem

– Center Selection Problem

• Pricing Method – Vertex Cover Problem

• Linear Programming and Rounding – Vertex Cover Problem

– Generalized Load-Balancing Problem

• Polynomial Time Approximation Scheme – Knapsack Problem

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Friedhelm Meyer auf der Heide 46

HEINZ NIXDORF INSTITUTE

University of Paderborn

Algorithms and Complexity

Friedhelm Meyer auf der Heide

Heinz Nixdorf Institute

& Computer Science Department

University of Paderborn

Fürstenallee 11

33102 Paderborn, Germany

Tel.: +49 (0) 52 51/60 64 80

Fax: +49 (0) 52 51/62 64 82

E-Mail: fmadh@upb.de

http://www.upb.de/cs/ag-madh

Thank you for

your attention!