July17,2010 wilhelm@math.hu-berlin.de PaulWilhelm GreedyAlgorithmAndEdmondsMatroidIntersectionAlgorithm

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Greedy Algorithm Transversal Matroids Edmonds’ Intersection Algorithm References

Greedy Algorithm And

Edmonds Matroid Intersection Algorithm

Paul Wilhelm wilhelm@math.hu-berlin.de

Institut f¨ ur Mathematik Humboldt-Universit¨ at zu Berlin

July 17, 2010

Paul Wilhelm (Math – HU Berlin) Greedy Algorithm July 17, 2010 1 / 32

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Content I

1 Greedy Algorithm Prerequisites Algorithm

Feasibility In Matroids

Proof ’Greedy ⇒ Matroid’

Proof ’Matroid ⇒ Greedy’

2 Transversal Matroids Job Scheduling

Job Scheduling Problem Mathematical Describtion

Transversal Matroids

Definition Transversal Matroids Proof

Greedy Algorithm In Transversal Matroids

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Content II

Prerequisites

Example Intersection Of Two Matriods

Algorithm

Counterexample Intersection Of Three Matriods Complexity Of Edmonds’ Algorithm

Outlook

Correctnes of the Algorithm

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Remind

Independence System / Matroid (E , I)

∅ ⊆ I (nonempty)

Y ⊆ I , X ⊆ Y ⇒ X ⊆ I (hereditary)

X , Y ∈ I, |X | < |Y | ⇒ ∃y ∈ Y \ X : X ∪ {y } ∈ I (augmentation)

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Prerequisites

Remind

Independence System / Matroid (E , I)

∅ ⊆ I (nonempty)

Y ⊆ I , X ⊆ Y ⇒ X ⊆ I (hereditary)

X , Y ∈ I, |X | < |Y | ⇒ ∃y ∈ Y \ X : X ∪ {y } ∈ I (augmentation)

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Remind

Independence System / Matroid (E , I)

∅ ⊆ I (nonempty)

Y ⊆ I , X ⊆ Y ⇒ X ⊆ I (hereditary)

X , Y ∈ I, |X | < |Y | ⇒ ∃y ∈ Y \ X : X ∪ {y } ∈ I (augmentation)

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Prerequisites

Min/Max - Problem

(due to (Korte and Vygen, 2007, p. 306)) Maximization Problem

Find X ∈ I s.t. the weight w (X ) = P

e∈X

w (e) is maximum

Minimization Problem

Find a basis B s.t. the weight w (X ) = P

e∈X

w (e ) is minimum

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Min/Max - Problem

(due to (Korte and Vygen, 2007, p. 306)) Maximization Problem

Find X ∈ I s.t. the weight w (X ) = P

e∈X

w (e) is maximum

Minimization Problem

Find a basis B s.t. the weight w (X ) = P

e∈X

w (e ) is minimum

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Prerequisites

Min/Max - Problem

(due to (Korte and Vygen, 2007, p. 306)) Maximization Problem

Find X ∈ I s.t. the weight w (X ) = P

e∈X

w (e) is maximum

Minimization Problem

Find a basis B s.t. the weight w (X ) = P

e∈X

w (e ) is minimum

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Examples

Many combinatorial optimization problems can be formulated as min/max-problem (more in (Korte and Vygen, 2007, p. 306)):

maximum weight stable set travelling salesman shortest path

knapsack

minimum spanning tree maximum weight forest steiner tree

maximum weight branching

maximum weight matching

job scheduling

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Prerequisites

Examples

Many combinatorial optimization problems can be formulated as min/max-problem (more in (Korte and Vygen, 2007, p. 306)):

maximum weight stable set travelling salesman shortest path

knapsack

minimum spanning tree maximum weight forest steiner tree

maximum weight branching maximum weight matching job scheduling

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Examples

Many combinatorial optimization problems can be formulated as min/max-problem (more in (Korte and Vygen, 2007, p. 306)):

maximum weight stable set travelling salesman shortest path

knapsack

minimum spanning tree maximum weight forest steiner tree

maximum weight branching

maximum weight matching

job scheduling

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Prerequisites

Examples

Many combinatorial optimization problems can be formulated as min/max-problem (more in (Korte and Vygen, 2007, p. 306)):

maximum weight stable set travelling salesman shortest path

knapsack

minimum spanning tree maximum weight forest steiner tree

maximum weight branching maximum weight matching job scheduling

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Examples

Many combinatorial optimization problems can be formulated as min/max-problem (more in (Korte and Vygen, 2007, p. 306)):

maximum weight stable set travelling salesman shortest path

knapsack

minimum spanning tree maximum weight forest steiner tree

maximum weight branching

maximum weight matching

job scheduling

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Prerequisites

Examples

Many combinatorial optimization problems can be formulated as min/max-problem (more in (Korte and Vygen, 2007, p. 306)):

maximum weight stable set travelling salesman shortest path

knapsack

minimum spanning tree maximum weight forest steiner tree

maximum weight branching maximum weight matching job scheduling

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Examples

Many combinatorial optimization problems can be formulated as min/max-problem (more in (Korte and Vygen, 2007, p. 306)):

maximum weight stable set travelling salesman shortest path

knapsack

minimum spanning tree maximum weight forest steiner tree

maximum weight branching

maximum weight matching

job scheduling

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Prerequisites

Examples

Many combinatorial optimization problems can be formulated as min/max-problem (more in (Korte and Vygen, 2007, p. 306)):

maximum weight stable set travelling salesman shortest path

knapsack

minimum spanning tree maximum weight forest steiner tree

maximum weight branching maximum weight matching job scheduling

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Examples

Many combinatorial optimization problems can be formulated as min/max-problem (more in (Korte and Vygen, 2007, p. 306)):

maximum weight stable set travelling salesman shortest path

knapsack

minimum spanning tree maximum weight forest steiner tree

maximum weight branching

maximum weight matching

job scheduling

(19)

Prerequisites

Examples

Many combinatorial optimization problems can be formulated as min/max-problem (more in (Korte and Vygen, 2007, p. 306)):

maximum weight stable set travelling salesman shortest path

knapsack

minimum spanning tree maximum weight forest steiner tree

maximum weight branching maximum weight matching job scheduling

(20)

Examples

Many combinatorial optimization problems can be formulated as min/max-problem (more in (Korte and Vygen, 2007, p. 306)):

maximum weight stable set travelling salesman shortest path

knapsack

minimum spanning tree maximum weight forest steiner tree

maximum weight branching

maximum weight matching

job scheduling

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Prerequisites

Oracle

The independence system (E, I ) is given by an oracle:

Independence Oracle

An oracle which decides whether F ⊆ E is in I or not.

∀F ⊆ E O (F ) :=

1 ,if F ∈ I 0 ,else

Basis-Superset Oracle

An oracle which decides whether B ⊆ E is an basis for I or not.

∀B ⊆ E O (B ) :=

1 ,if B ∈ I and ∄x ∈ E : B ∪ {x} ∈ I 0 ,else

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Oracle

The independence system (E, I ) is given by an oracle:

Independence Oracle

An oracle which decides whether F ⊆ E is in I or not.

∀F ⊆ E O (F ) :=

1 ,if F ∈ I 0 ,else

Basis-Superset Oracle

An oracle which decides whether B ⊆ E is an basis for I or not.

∀B ⊆ E O (B ) :=

1 ,if B ∈ I and ∄x ∈ E : B ∪ {x} ∈ I

0 ,else

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Prerequisites

Oracle

The independence system (E, I ) is given by an oracle:

Independence Oracle

An oracle which decides whether F ⊆ E is in I or not.

∀F ⊆ E O (F ) :=

1 ,if F ∈ I 0 ,else

Basis-Superset Oracle

An oracle which decides whether B ⊆ E is an basis for I or not.

∀B ⊆ E O (B ) :=

1 ,if B ∈ I and ∄x ∈ E : B ∪ {x} ∈ I 0 ,else

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Oracle

The independence system (E, I ) is given by an oracle:

Independence Oracle

An oracle which decides whether F ⊆ E is in I or not.

∀F ⊆ E O (F ) :=

1 ,if F ∈ I 0 ,else

Basis-Superset Oracle

An oracle which decides whether B ⊆ E is an basis for I or not.

∀B ⊆ E O (B ) :=

1 ,if B ∈ I and ∄x ∈ E : B ∪ {x} ∈ I

0 ,else

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Prerequisites

Oracle

The independence system (E, I ) is given by an oracle:

Independence Oracle

An oracle which decides whether F ⊆ E is in I or not.

∀F ⊆ E O (F ) :=

1 ,if F ∈ I 0 ,else

Basis-Superset Oracle

An oracle which decides whether B ⊆ E is an basis for I or not.

∀B ⊆ E O (B ) :=

1 ,if B ∈ I and ∄x ∈ E : B ∪ {x} ∈ I 0 ,else

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Oracle

The independence system (E, I ) is given by an oracle:

Independence Oracle

An oracle which decides whether F ⊆ E is in I or not.

∀F ⊆ E O (F ) :=

1 ,if F ∈ I 0 ,else

Basis-Superset Oracle

An oracle which decides whether B ⊆ E is an basis for I or not.

∀B ⊆ E O (B ) :=

1 ,if B ∈ I and ∄x ∈ E : B ∪ {x} ∈ I

0 ,else

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Prerequisites

Oracle

The independence system (E, I ) is given by an oracle:

Independence Oracle

An oracle which decides whether F ⊆ E is in I or not.

∀F ⊆ E O (F ) :=

1 ,if F ∈ I 0 ,else

Basis-Superset Oracle

An oracle which decides whether B ⊆ E is an basis for I or not.

∀B ⊆ E O (B ) :=

1 ,if B ∈ I and ∄x ∈ E : B ∪ {x} ∈ I 0 ,else

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Algorithm

Best-In-Greedy

Sort E = {e 1 , ..., e n } s.t. w (e ₁ ) ≥ ... ≥ w (e n ) Set F := ∅∈ I

For i:=1 to n do:

If F ∪ {e

i

}∈ I then set F := F ∪ {e

i

} Output: F ∈ I

Sort the elements in E (O(n · log(n)))

Have to ask oracle in each step! Complexity of algorithm depends

mainly on complexity of oracle (O( O )).

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Algorithm

Best-In-Greedy

Sort E = {e 1 , ..., e n } s.t. w (e ₁ ) ≥ ... ≥ w (e n ) Set F := ∅∈ I

For i:=1 to n do:

If F ∪ {e

i

}∈ I then set F := F ∪ {e

i

} Output: F ∈ I

Sort the elements in E (O(n · log(n)))

Have to ask oracle in each step! Complexity of algorithm depends mainly on complexity of oracle (O( O )).

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Algorithm

Best-In-Greedy

Sort E = {e 1 , ..., e n } s.t. w (e ₁ ) ≥ ... ≥ w (e n ) Set F := ∅∈ I

For i:=1 to n do:

If F ∪ {e

i

}∈ I then set F := F ∪ {e

i

} Output: F ∈ I

Sort the elements in E (O(n · log(n)))

Have to ask oracle in each step! Complexity of algorithm depends

mainly on complexity of oracle (O( O )).

(31)

Algorithm

Best-In-Greedy

Sort E = {e 1 , ..., e n } s.t. w (e ₁ ) ≥ ... ≥ w (e n ) Set F := ∅∈ I

For i:=1 to n do:

If F ∪ {e

i

}∈ I then set F := F ∪ {e

i

} Output: F ∈ I

Sort the elements in E (O(n · log(n)))

Have to ask oracle in each step! Complexity of algorithm depends mainly on complexity of oracle (O( O )).

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Algorithm

Best-In-Greedy

Sort E = {e 1 , ..., e n } s.t. w (e ₁ ) ≥ ... ≥ w (e n ) Set F := ∅∈ I

For i:=1 to n do:

If F ∪ {e

i

}∈ I then set F := F ∪ {e

i

} Output: F ∈ I

Sort the elements in E (O(n · log(n)))

Have to ask oracle in each step! Complexity of algorithm depends

mainly on complexity of oracle (O( O )).

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Algorithm

Best-In-Greedy

Sort E = {e 1 , ..., e n } s.t. w (e ₁ ) ≥ ... ≥ w (e n ) Set F := ∅∈ I

For i:=1 to n do:

If F ∪ {e

i

}∈ I then set F := F ∪ {e

i

} Output: F ∈ I

Sort the elements in E (O(n · log(n)))

Have to ask oracle in each step! Complexity of algorithm depends mainly on complexity of oracle (O( O )).

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Algorithm

Best-In-Greedy

Sort E = {e 1 , ..., e n } s.t. w (e ₁ ) ≥ ... ≥ w (e n ) Set F := ∅∈ I

For i:=1 to n do:

If F ∪ {e

i

}∈ I then set F := F ∪ {e

i

} Output: F ∈ I

Sort the elements in E (O(n · log(n)))

Have to ask oracle in each step! Complexity of algorithm depends

mainly on complexity of oracle (O( O )).

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Algorithm

Best-In-Greedy

Sort E = {e 1 , ..., e n } s.t. w (e ₁ ) ≥ ... ≥ w (e n ) Set F := ∅∈ I

For i:=1 to n do:

If F ∪ {e

i

}∈ I then set F := F ∪ {e

i

} Output: F ∈ I

Sort the elements in E (O(n · log(n)))

Have to ask oracle in each step! Complexity of algorithm depends mainly on complexity of oracle (O( O )).

(36)

Algorithm

Best-In-Greedy

Sort E = {e 1 , ..., e n } s.t. w (e ₁ ) ≥ ... ≥ w (e n ) Set F := ∅∈ I

For i:=1 to n do:

If F ∪ {e

i

}∈ I then set F := F ∪ {e

i

} Output: F ∈ I

Sort the elements in E (O(n · log(n)))

Have to ask oracle in each step! Complexity of algorithm depends

mainly on complexity of oracle (O( O )).

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Feasibility In Matroids

Theorem Feasibility

Asume the greedy algorithm work correctly in the independence system (E, I).

Theorem

An independence system is a matroid iff the best-in-greedy algorithm finds for all weight functions an optimum solution for the maximization problem.

If not a matroid:

find only locally optimum

cardinallity don’t have to be maximum, just maximal (i.e. there exists no superset)

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Theorem Feasibility

Asume the greedy algorithm work correctly in the independence system (E, I).

Theorem

An independence system is a matroid iff the best-in-greedy algorithm finds for all weight functions an optimum solution for the maximization problem.

If not a matroid:

find only locally optimum

cardinallity don’t have to be maximum, just maximal (i.e. there exists

no superset)

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Theorem Feasibility

Asume the greedy algorithm work correctly in the independence system (E, I).

Theorem

An independence system is a matroid iff the best-in-greedy algorithm finds for all weight functions an optimum solution for the maximization problem.

If not a matroid:

find only locally optimum

cardinallity don’t have to be maximum, just maximal (i.e. there exists no superset)

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Theorem Feasibility

Asume the greedy algorithm work correctly in the independence system (E, I).

Theorem

An independence system is a matroid iff the best-in-greedy algorithm finds for all weight functions an optimum solution for the maximization problem.

If not a matroid:

find only locally optimum

cardinallity don’t have to be maximum, just maximal (i.e. there exists

no superset)

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Theorem Feasibility

Asume the greedy algorithm work correctly in the independence system (E, I).

Theorem

An independence system is a matroid iff the best-in-greedy algorithm finds for all weight functions an optimum solution for the maximization problem.

If not a matroid:

find only locally optimum

cardinallity don’t have to be maximum, just maximal (i.e. there exists no superset)

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Proof ’Greedy ⇒ Matroid’ By Contradiction

Assumption: (E , I) Is No Matroid:

i.e. ∃ X , Y ∈ I with |X | < |Y | s.t. ∀e ∈ Y \ X : X ∪ {e} ∈ I / Define Weight Function:

w (e) :=







1 + ε , if e ∈ X 1 , if e ∈ Y \ X 0 , if e ∈ E \ {X ∪ Y }

⇒ greedy outputs F with w (F ) = |X | · (1 + ε) + 0

⇒ w (F ) = |X |(1 + ε) < w (Y ) = |Y | for ε < ^|Y _|X ^| _| − 1

^to ^w ^(F ^{) maximum} ^{for all} weight functions.

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Proof ’Greedy ⇒ Matroid’ By Contradiction

Assumption: (E , I) Is No Matroid:

i.e. ∃ X , Y ∈ I with |X | < |Y | s.t. ∀e ∈ Y \ X : X ∪ {e} ∈ I / Define Weight Function:

w (e) :=







1 + ε , if e ∈ X 1 , if e ∈ Y \ X 0 , if e ∈ E \ {X ∪ Y }

⇒ greedy outputs F with w (F ) = |X | · (1 + ε) + 0

⇒ w (F ) = |X |(1 + ε) < w (Y ) = |Y | for ε < ^|Y _|X ^| _| − 1

^to ^w ^(F ^{) maximum} ^{for all} weight functions.

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Proof ’Greedy ⇒ Matroid’ By Contradiction

Assumption: (E , I) Is No Matroid:

i.e. ∃ X , Y ∈ I with |X | < |Y | s.t. ∀e ∈ Y \ X : X ∪ {e} ∈ I / Define Weight Function:

w (e) :=







1 + ε , if e ∈ X 1 , if e ∈ Y \ X 0 , if e ∈ E \ {X ∪ Y }

⇒ greedy outputs F with w (F ) = |X | · (1 + ε) + 0

⇒ w (F ) = |X |(1 + ε) < w (Y ) = |Y | for ε < ^|Y _|X ^| _| − 1

^to ^w ^(F ^{) maximum} ^{for all} weight functions.

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Proof ’Greedy ⇒ Matroid’ By Contradiction

Assumption: (E , I) Is No Matroid:

i.e. ∃ X , Y ∈ I with |X | < |Y | s.t. ∀e ∈ Y \ X : X ∪ {e} ∈ I / Define Weight Function:

w (e) :=







1 + ε , if e ∈ X choose first |X | steps 1 , if e ∈ Y \ X

0 , if e ∈ E \ {X ∪ Y }

⇒ greedy outputs F with w (F ) = |X | · (1 + ε) + 0

⇒ w (F ) = |X |(1 + ε) < w (Y ) = |Y | for ε < ^|Y _|X ^| _| − 1

^to ^w ^(F ^{) maximum} ^{for all} weight functions.

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Proof ’Greedy ⇒ Matroid’ By Contradiction

Assumption: (E , I) Is No Matroid:

i.e. ∃ X , Y ∈ I with |X | < |Y | s.t. ∀e ∈ Y \ X : X ∪ {e} ∈ I / Define Weight Function:

w (e) :=







1 + ε , if e ∈ X choose first |X | steps

1 , if e ∈ Y \ X can’t choose

0 , if e ∈ E \ {X ∪ Y }

⇒ greedy outputs F with w (F ) = |X | · (1 + ε) + 0

⇒ w (F ) = |X |(1 + ε) < w (Y ) = |Y | for ε < ^|Y _|X ^| _| − 1

^to ^w ^(F ^{) maximum} ^{for all} weight functions.

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Proof ’Greedy ⇒ Matroid’ By Contradiction

Assumption: (E , I) Is No Matroid:

i.e. ∃ X , Y ∈ I with |X | < |Y | s.t. ∀e ∈ Y \ X : X ∪ {e} ∈ I / Define Weight Function:

w (e) :=







1 + ε , if e ∈ X choose first |X | steps

1 , if e ∈ Y \ X can’t choose

0 , if e ∈ E \ {X ∪ Y } don’t change the weight

⇒ greedy outputs F with w (F ) = |X | · (1 + ε) + 0

⇒ w (F ) = |X |(1 + ε) < w (Y ) = |Y | for ε < ^|Y _|X ^| _| − 1

^to ^w ^(F ^{) maximum} ^{for all} weight functions.

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Proof ’Greedy ⇒ Matroid’ By Contradiction

Assumption: (E , I) Is No Matroid:

i.e. ∃ X , Y ∈ I with |X | < |Y | s.t. ∀e ∈ Y \ X : X ∪ {e} ∈ I / Define Weight Function:

w (e) :=







1 + ε , if e ∈ X choose first |X | steps

1 , if e ∈ Y \ X can’t choose

0 , if e ∈ E \ {X ∪ Y } don’t change the weight

⇒ greedy outputs F with w (F ) = |X | · (1 + ε) + 0

⇒ w (F ) = |X |(1 + ε) < w (Y ) = |Y | for ε < ^|Y _|X ^| _| − 1

^to ^w ^(F ^{) maximum} ^{for all} weight functions.

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Proof ’Greedy ⇒ Matroid’ By Contradiction

Assumption: (E , I) Is No Matroid:

i.e. ∃ X , Y ∈ I with |X | < |Y | s.t. ∀e ∈ Y \ X : X ∪ {e} ∈ I / Define Weight Function:

w (e) :=







1 + ε , if e ∈ X choose first |X | steps

1 , if e ∈ Y \ X can’t choose

0 , if e ∈ E \ {X ∪ Y } don’t change the weight

⇒ greedy outputs F with w (F ) = |X | · (1 + ε) + 0

⇒ w (F ) = |X |(1 + ε) < w (Y ) = |Y | for ε < ^|Y _|X ^| _| − 1

^to ^w ^(F ^{) maximum} ^{for all} weight functions.

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Proof ’Greedy ⇒ Matroid’ By Contradiction

Assumption: (E , I) Is No Matroid:

i.e. ∃ X , Y ∈ I with |X | < |Y | s.t. ∀e ∈ Y \ X : X ∪ {e} ∈ I / Define Weight Function:

w (e) :=







1 + ε , if e ∈ X choose first |X | steps

1 , if e ∈ Y \ X can’t choose

0 , if e ∈ E \ {X ∪ Y } don’t change the weight

⇒ greedy outputs F with w (F ) = |X | · (1 + ε) + 0

⇒ w (F ) = |X |(1 + ε) < w (Y ) = |Y | for ε < ^|Y _|X ^| _| − 1

^to ^w ^(F ^{) maximum} ^{for all} weight functions.

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Proof ’Matroid ⇒ Greedy’

let w be an abitrary weight function

let F = {f 1 , ..., f r } the output of the greedy algorithm w.l.o.g. w (f 1 ) ≥ ... ≥ w (f r ) (if not change numbering) first prove that F has maximum cardinality

then prove by contradiction that w (F ) is maximum

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Proof ’Matroid ⇒ Greedy’

let w be an abitrary weight function

let F = {f 1 , ..., f r } the output of the greedy algorithm w.l.o.g. w (f 1 ) ≥ ... ≥ w (f r ) (if not change numbering) first prove that F has maximum cardinality

then prove by contradiction that w (F ) is maximum

(53)

Proof ’Matroid ⇒ Greedy’

let w be an abitrary weight function

let F = {f 1 , ..., f r } the output of the greedy algorithm w.l.o.g. w (f 1 ) ≥ ... ≥ w (f r ) (if not change numbering) first prove that F has maximum cardinality

then prove by contradiction that w (F ) is maximum

(54)

Proof ’Matroid ⇒ Greedy’

let w be an abitrary weight function

let F = {f 1 , ..., f r } the output of the greedy algorithm w.l.o.g. w (f 1 ) ≥ ... ≥ w (f r ) (if not change numbering) first prove that F has maximum cardinality

then prove by contradiction that w (F ) is maximum

(55)

Proof ’Matroid ⇒ Greedy’

let w be an abitrary weight function

let F = {f 1 , ..., f r } the output of the greedy algorithm w.l.o.g. w (f 1 ) ≥ ... ≥ w (f r ) (if not change numbering) first prove that F has maximum cardinality

then prove by contradiction that w (F ) is maximum

(56)

Proof ’Matroid ⇒ Greedy’

let w be an abitrary weight function

let F = {f 1 , ..., f r } the output of the greedy algorithm w.l.o.g. w (f 1 ) ≥ ... ≥ w (f r ) (if not change numbering) first prove that F has maximum cardinality

then prove by contradiction that w (F ) is maximum

(57)

Proof ’Matroid ⇒ Greedy’

let w be an abitrary weight function

let F = {f 1 , ..., f r } the output of the greedy algorithm w.l.o.g. w (f 1 ) ≥ ... ≥ w (f r ) (if not change numbering) first prove that F has maximum cardinality

then prove by contradiction that w (F ) is maximum Proof Maximum Cardinality Of F By Contradiction Assume ∃ G ∈ I s.t. |F | < |G |

Augmentation property ⇒ ∃g ∈ G \ F s.t. F ∪ {g }

⇒ ∃t s.t. {f 1 , ..., f _t , g , f _t+1 , ..., f _s } = F ∪ {g } ∈ I with w (f t ) ≥ w (g ) ≥ w (f _t+1 )

⇒ {f 1 , ..., f t } ⊆ {f 1 , ..., f t , g } ∈ I

g should been choosen in step t + 1 of greedy algorithm

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Proof ’Matroid ⇒ Greedy’

let w be an abitrary weight function

let F = {f 1 , ..., f r } the output of the greedy algorithm w.l.o.g. w (f 1 ) ≥ ... ≥ w (f r ) (if not change numbering) first prove that F has maximum cardinality

then prove by contradiction that w (F ) is maximum Proof Maximum Cardinality Of F By Contradiction Assume ∃ G ∈ I s.t. |F | < |G |

Augmentation property ⇒ ∃g ∈ G \ F s.t. F ∪ {g }

⇒ ∃t s.t. {f 1 , ..., f _t , g , f _t+1 , ..., f _s } = F ∪ {g } ∈ I with w (f t ) ≥ w (g ) ≥ w (f _t+1 )

⇒ {f 1 , ..., f t } ⊆ {f 1 , ..., f t , g } ∈ I

g should been choosen in step t + 1 of greedy algorithm

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Proof ’Matroid ⇒ Greedy’

let w be an abitrary weight function

let F = {f 1 , ..., f r } the output of the greedy algorithm w.l.o.g. w (f 1 ) ≥ ... ≥ w (f r ) (if not change numbering) first prove that F has maximum cardinality

then prove by contradiction that w (F ) is maximum Proof Maximum Cardinality Of F By Contradiction Assume ∃ G ∈ I s.t. |F | < |G |

Augmentation property ⇒ ∃g ∈ G \ F s.t. F ∪ {g }

⇒ ∃t s.t. {f 1 , ..., f _t , g , f _t+1 , ..., f _s } = F ∪ {g } ∈ I with w (f t ) ≥ w (g ) ≥ w (f _t+1 )

⇒ {f 1 , ..., f t } ⊆ {f 1 , ..., f t , g } ∈ I

g should been choosen in step t + 1 of greedy algorithm

(60)

Proof ’Matroid ⇒ Greedy’

let w be an abitrary weight function

let F = {f 1 , ..., f r } the output of the greedy algorithm w.l.o.g. w (f 1 ) ≥ ... ≥ w (f r ) (if not change numbering) first prove that F has maximum cardinality

then prove by contradiction that w (F ) is maximum Proof Maximum Cardinality Of F By Contradiction Assume ∃ G ∈ I s.t. |F | < |G |

Augmentation property ⇒ ∃g ∈ G \ F s.t. F ∪ {g }

⇒ ∃t s.t. {f 1 , ..., f _t , g , f _t+1 , ..., f _s } = F ∪ {g } ∈ I with w (f t ) ≥ w (g ) ≥ w (f _t+1 )

⇒ {f 1 , ..., f t } ⊆ {f 1 , ..., f t , g } ∈ I

g should been choosen in step t + 1 of greedy algorithm

(61)

Proof ’Matroid ⇒ Greedy’

let w be an abitrary weight function

let F = {f 1 , ..., f r } the output of the greedy algorithm w.l.o.g. w (f 1 ) ≥ ... ≥ w (f r ) (if not change numbering) first prove that F has maximum cardinality

then prove by contradiction that w (F ) is maximum Proof Maximum Cardinality Of F By Contradiction Assume ∃ G ∈ I s.t. |F | < |G |

Augmentation property ⇒ ∃g ∈ G \ F s.t. F ∪ {g }

⇒ ∃t s.t. {f 1 , ..., f _t , g , f _t+1 , ..., f _s } = F ∪ {g } ∈ I with w (f t ) ≥ w (g ) ≥ w (f _t+1 )

⇒ {f 1 , ..., f t } ⊆ {f 1 , ..., f t , g } ∈ I

g should been choosen in step t + 1 of greedy algorithm

(62)

Proof ’Matroid ⇒ Greedy’

let w be an abitrary weight function

let F = {f 1 , ..., f r } the output of the greedy algorithm w.l.o.g. w (f 1 ) ≥ ... ≥ w (f r ) (if not change numbering) first prove that F has maximum cardinality

then prove by contradiction that w (F ) is maximum Proof Maximum Cardinality Of F By Contradiction Assume ∃ G ∈ I s.t. |F | < |G |

Augmentation property ⇒ ∃g ∈ G \ F s.t. F ∪ {g }

⇒ ∃t s.t. {f 1 , ..., f _t , g , f _t+1 , ..., f _s } = F ∪ {g } ∈ I with w (f t ) ≥ w (g ) ≥ w (f _t+1 )

⇒ {f 1 , ..., f t } ⊆ {f 1 , ..., f t , g } ∈ I

g should been choosen in step t + 1 of greedy algorithm

(63)

Proof ’Matroid ⇒ Greedy’

Proof w (F ) is maximum in (E , I ) by contradiction

Assume ∃G = {g 1 , ..., g r } ∈ I s.t. w (G ) > w (F ) and w(g i ) ≥ w (g _i+1 )

⇒ P

g

i

∈G

w (g _i ) > P

f

i

∈F

w (f _i )

⇒ ∃ k : w (g _k ) > w (f _k ) (because |G | ≤ |F |)

take X := {f 1 , ..., f _k−1 } (= ∅ if k = 1) and Y := {g 1 , ..., g k }

|X | < |Y | (augmentation property) ⇒ ∃ g t ∈ Y \ X with t ≤ k s.t.

{f 1 , ..., f _k−1 , g t } = X ∪ {g t } ∈ I

Because w (g t ) ≥ w (g k ) > w (f k ) g t should been choosen before step k of the greedy algorithm

to correctness of greedy algorithm

(64)

Proof ’Matroid ⇒ Greedy’

Proof w (F ) is maximum in (E , I ) by contradiction

Assume ∃G = {g 1 , ..., g r } ∈ I s.t. w (G ) > w (F ) and w(g i ) ≥ w (g _i+1 )

⇒ P

g

i

∈G

w (g _i ) > P

f

i

∈F

w (f _i )

⇒ ∃ k : w (g _k ) > w (f _k ) (because |G | ≤ |F |)

take X := {f 1 , ..., f _k−1 } (= ∅ if k = 1) and Y := {g 1 , ..., g k }

|X | < |Y | (augmentation property) ⇒ ∃ g t ∈ Y \ X with t ≤ k s.t.

{f 1 , ..., f _k−1 , g t } = X ∪ {g t } ∈ I

Because w (g t ) ≥ w (g k ) > w (f k ) g t should been choosen before step k of the greedy algorithm

to correctness of greedy algorithm

(65)

Proof ’Matroid ⇒ Greedy’

Proof w (F ) is maximum in (E , I ) by contradiction

Assume ∃G = {g 1 , ..., g r } ∈ I s.t. w (G ) > w (F ) and w(g i ) ≥ w (g _i+1 )

⇒ P

g

i

∈G

w (g _i ) > P

f

i

∈F

w (f _i )

⇒ ∃ k : w (g _k ) > w (f _k ) (because |G | ≤ |F |)

take X := {f 1 , ..., f _k−1 } (= ∅ if k = 1) and Y := {g 1 , ..., g k }

|X | < |Y | (augmentation property) ⇒ ∃ g t ∈ Y \ X with t ≤ k s.t.

{f 1 , ..., f _k−1 , g t } = X ∪ {g t } ∈ I

Because w (g t ) ≥ w (g k ) > w (f k ) g t should been choosen before step k of the greedy algorithm

to correctness of greedy algorithm

(66)

Proof ’Matroid ⇒ Greedy’

Proof w (F ) is maximum in (E , I ) by contradiction

Assume ∃G = {g 1 , ..., g r } ∈ I s.t. w (G ) > w (F ) and w(g i ) ≥ w (g _i+1 )

⇒ P

g

i

∈G

w (g _i ) > P

f

i

∈F

w (f _i )

⇒ ∃ k : w (g _k ) > w (f _k ) (because |G | ≤ |F |)

take X := {f 1 , ..., f _k−1 } (= ∅ if k = 1) and Y := {g 1 , ..., g k }

|X | < |Y | (augmentation property) ⇒ ∃ g t ∈ Y \ X with t ≤ k s.t.

{f 1 , ..., f _k−1 , g t } = X ∪ {g t } ∈ I

Because w (g t ) ≥ w (g k ) > w (f k ) g t should been choosen before step k of the greedy algorithm

to correctness of greedy algorithm

(67)

Proof ’Matroid ⇒ Greedy’

Proof w (F ) is maximum in (E , I ) by contradiction

Assume ∃G = {g 1 , ..., g r } ∈ I s.t. w (G ) > w (F ) and w(g i ) ≥ w (g _i+1 )

⇒ P

g

i

∈G

w (g _i ) > P

f

i

∈F

w (f _i )

⇒ ∃ k : w (g _k ) > w (f _k ) (because |G | ≤ |F |)

take X := {f 1 , ..., f _k−1 } (= ∅ if k = 1) and Y := {g 1 , ..., g k }

|X | < |Y | (augmentation property) ⇒ ∃ g t ∈ Y \ X with t ≤ k s.t.

{f 1 , ..., f _k−1 , g t } = X ∪ {g t } ∈ I

Because w (g t ) ≥ w (g k ) > w (f k ) g t should been choosen before step k of the greedy algorithm

to correctness of greedy algorithm

(68)

Proof ’Matroid ⇒ Greedy’

Proof w (F ) is maximum in (E , I ) by contradiction

Assume ∃G = {g 1 , ..., g r } ∈ I s.t. w (G ) > w (F ) and w(g i ) ≥ w (g _i+1 )

⇒ P

g

i

∈G

w (g _i ) > P

f

i

∈F

w (f _i )

⇒ ∃ k : w (g _k ) > w (f _k ) (because |G | ≤ |F |)

take X := {f 1 , ..., f _k−1 } (= ∅ if k = 1) and Y := {g 1 , ..., g k }

|X | < |Y | (augmentation property) ⇒ ∃ g t ∈ Y \ X with t ≤ k s.t.

{f 1 , ..., f _k−1 , g t } = X ∪ {g t } ∈ I

Because w (g t ) ≥ w (g k ) > w (f k ) g t should been choosen before step k of the greedy algorithm

to correctness of greedy algorithm

(69)

Proof ’Matroid ⇒ Greedy’

Proof w (F ) is maximum in (E , I ) by contradiction

Assume ∃G = {g 1 , ..., g r } ∈ I s.t. w (G ) > w (F ) and w(g i ) ≥ w (g _i+1 )

⇒ P

g

i

∈G

w (g _i ) > P

f

i

∈F

w (f _i )

⇒ ∃ k : w (g _k ) > w (f _k ) (because |G | ≤ |F |)

take X := {f 1 , ..., f _k−1 } (= ∅ if k = 1) and Y := {g 1 , ..., g k }

|X | < |Y | (augmentation property) ⇒ ∃ g t ∈ Y \ X with t ≤ k s.t.

{f 1 , ..., f _k−1 , g t } = X ∪ {g t } ∈ I

Because w (g t ) ≥ w (g k ) > w (f k ) g t should been choosen before step k of the greedy algorithm

to correctness of greedy algorithm

(70)

Proof ’Matroid ⇒ Greedy’

Proof w (F ) is maximum in (E , I ) by contradiction

Assume ∃G = {g 1 , ..., g r } ∈ I s.t. w (G ) > w (F ) and w(g i ) ≥ w (g _i+1 )

⇒ P

g

i

∈G

w (g _i ) > P

f

i

∈F

w (f _i )

⇒ ∃ k : w (g _k ) > w (f _k ) (because |G | ≤ |F |)

take X := {f 1 , ..., f _k−1 } (= ∅ if k = 1) and Y := {g 1 , ..., g k }

|X | < |Y | (augmentation property) ⇒ ∃ g t ∈ Y \ X with t ≤ k s.t.

{f 1 , ..., f _k−1 , g t } = X ∪ {g t } ∈ I

Because w (g t ) ≥ w (g k ) > w (f k ) g t should been choosen before step k of the greedy algorithm

to correctness of greedy algorithm

(71)

Proof ’Matroid ⇒ Greedy’

Proof w (F ) is maximum in (E , I ) by contradiction

Assume ∃G = {g 1 , ..., g r } ∈ I s.t. w (G ) > w (F ) and w(g i ) ≥ w (g _i+1 )

⇒ P

g

i

∈G

w (g _i ) > P

f

i

∈F

w (f _i )

⇒ ∃ k : w (g _k ) > w (f _k ) (because |G | ≤ |F |)

take X := {f 1 , ..., f _k−1 } (= ∅ if k = 1) and Y := {g 1 , ..., g k }

|X | < |Y | (augmentation property) ⇒ ∃ g t ∈ Y \ X with t ≤ k s.t.

{f 1 , ..., f _k−1 , g t } = X ∪ {g t } ∈ I

Because w (g t ) ≥ w (g k ) > w (f k ) g t should been choosen before step k of the greedy algorithm

to correctness of greedy algorithm

(72)

Example Job Scheduling

Job Scheduling Problem Set of one worker jobs

Arranged in order of importance Pool of workers

Each is qualified to do a subset of jobs

Jobs have to be done simultaneously, so every worker can do at least one job

if not all jobs can be done, try optimal relative to the priority

(73)

Job Scheduling

Example Job Scheduling

Job Scheduling Problem Set of one worker jobs

Arranged in order of importance Pool of workers

Each is qualified to do a subset of jobs

Jobs have to be done simultaneously, so every worker can do at least one job

if not all jobs can be done, try optimal relative to the priority

(74)

Example Job Scheduling

Job Scheduling Problem Set of one worker jobs

Arranged in order of importance Pool of workers

Each is qualified to do a subset of jobs

Jobs have to be done simultaneously, so every worker can do at least one job

if not all jobs can be done, try optimal relative to the priority

(75)

Job Scheduling

Example Job Scheduling

Job Scheduling Problem Set of one worker jobs

Arranged in order of importance Pool of workers

Each is qualified to do a subset of jobs

Jobs have to be done simultaneously, so every worker can do at least one job

if not all jobs can be done, try optimal relative to the priority

(76)

Example Job Scheduling

Job Scheduling Problem Set of one worker jobs

Arranged in order of importance Pool of workers

Each is qualified to do a subset of jobs

Jobs have to be done simultaneously, so every worker can do at least one job

if not all jobs can be done, try optimal relative to the priority

(77)

Job Scheduling

Example Job Scheduling

Job Scheduling Problem Set of one worker jobs

Arranged in order of importance Pool of workers

Each is qualified to do a subset of jobs

Jobs have to be done simultaneously, so every worker can do at least one job

if not all jobs can be done, try optimal relative to the priority

(78)

Example Job Scheduling

Job Scheduling Problem Set of one worker jobs

Arranged in order of importance Pool of workers

Each is qualified to do a subset of jobs

Jobs have to be done simultaneously, so every worker can do at least one job

if not all jobs can be done, try optimal relative to the priority

(79)

Job Scheduling

Mathematical Describtion

Figure 1.19 from (Oxley, 2006, p. 47)

J-set of jobs in order of priority W-set of workers

W _i is the set of jobs worker i can do W := {W i |i ∈ I , W i ⊆ J} (family of subsets)

Vertex set J ∪ W and the edge set

{jW i |j ∈ J, W i ∈ W} define the graph △[W]

(80)

Mathematical Describtion

J j ₁ ◦ j 2 ◦ j ₃ ◦ j ₄ ◦ j 5 ◦

J-set of jobs in order of priority W-set of workers

W i is the set of jobs worker i can do W := {W i |i ∈ I , W i ⊆ J} (family of subsets)

Vertex set J ∪ W and the edge set

{jW i |j ∈ J, W i ∈ W} define the graph △[W]

(81)

Job Scheduling

Mathematical Describtion

J W

j ₁ ◦

j 2 ◦ ◦W 1

j ₃ ◦ ◦W ₂

j ₄ ◦ ◦W 3

j 5 ◦ ◦W 4

j ₆ ◦

J-set of jobs in order of priority W-set of workers

W i is the set of jobs worker i can do W := {W i |i ∈ I , W i ⊆ J} (family of subsets)

Vertex set J ∪ W and the edge set

{jW i |j ∈ J, W i ∈ W} define the graph △[W]

(82)

Mathematical Describtion

J W

j ₁ ◦

j 2 ◦ ◦W 1

j ₃ ◦ ◦W ₂

j ₄ ◦ ◦W 3

j 5 ◦ ◦W 4

J-set of jobs in order of priority W-set of workers

W i is the set of jobs worker i can do W := {W i |i ∈ I , W i ⊆ J} (family of subsets)

Vertex set J ∪ W and the edge set

{jW i |j ∈ J, W i ∈ W} define the graph △[W]

(83)

Job Scheduling

Mathematical Describtion

J W

j ₁ ◦

j 2 ◦ ◦W 1

j ₃ ◦ ◦W ₂

j ₄ ◦ ◦W 3

j 5 ◦ ◦W 4

j ₆ ◦

J-set of jobs in order of priority W-set of workers

W i is the set of jobs worker i can do W := {W i |i ∈ I , W i ⊆ J} (family of subsets)

Vertex set J ∪ W and the edge set

{jW i |j ∈ J, W i ∈ W} define the graph △[W]

(84)

Mathematical Describtion

J W

j ₁ ◦

j 2 ◦ ◦W 1

j ₃ ◦ ◦W ₂

j ₄ ◦ ◦W 3

j 5 ◦ ◦W 4

J-set of jobs in order of priority W-set of workers

W i is the set of jobs worker i can do W := {W i |i ∈ I , W i ⊆ J} (family of subsets)

Vertex set J ∪ W and the edge set

{jW i |j ∈ J, W i ∈ W} define the graph △[W]

(85)

Job Scheduling

Mathematical Describtion

J W

j ₁ ◦

j 2 ◦ ◦W 1

j ₃ ◦ ◦W ₂

j ₄ ◦ ◦W 3

j 5 ◦ ◦W 4

j ₆ ◦

J-set of jobs in order of priority W-set of workers

W i is the set of jobs worker i can do W := {W i |i ∈ I , W i ⊆ J} (family of subsets)

Vertex set J ∪ W and the edge set

{jW i |j ∈ J, W i ∈ W} define the graph △[W]

(86)

Mathematical Describtion

J W

j ₁ ◦

j 2 ◦ ◦W 1

j ₃ ◦ ◦W ₂

j ₄ ◦ ◦W 3

j 5 ◦ ◦W 4

J-set of jobs in order of priority W-set of workers

W i is the set of jobs worker i can do W := {W i |i ∈ I , W i ⊆ J} (family of subsets)

Vertex set J ∪ W and the edge set

{jW i |j ∈ J, W i ∈ W} define the graph △[W]

(87)

Job Scheduling

Mathematical Describtion

J W

j ₁ ◦

j 2 ◦ ◦W 1

j ₃ ◦ ◦W ₂

j ₄ ◦ ◦W 3

j 5 ◦ ◦W 4

j ₆ ◦

J-set of jobs in order of priority W-set of workers

W i is the set of jobs worker i can do W := {W i |i ∈ I , W i ⊆ J} (family of subsets)

Vertex set J ∪ W and the edge set

{jW i |j ∈ J, W i ∈ W} define the graph △[W]

(88)

Mathematical Describtion

J W

j ₁ ◦

j 2 ◦ ◦W 1

j ₃ ◦ ◦W ₂

j ₄ ◦ ◦W 3

j 5 ◦ ◦W 4

J-set of jobs in order of priority W-set of workers

W i is the set of jobs worker i can do W := {W i |i ∈ I , W i ⊆ J} (family of subsets)

Vertex set J ∪ W and the edge set

{jW i |j ∈ J, W i ∈ W} define the graph △[W]

(89)

Job Scheduling

Mathematical Describtion

J W

j ₁ ◦

j 2 ◦ ◦W 1

j ₃ ◦ ◦W ₂

j ₄ ◦ ◦W 3

j 5 ◦ ◦W 4

j ₆ ◦

W := {W i |i ∈ I , W i ⊆ J}

T ⊆ J partial transversal : ⇔ ∃ injective map ψ : T ։ I s.t. t ∈ W _ψ(t) ∀t ∈ T i.e. matching for T in △[W ]

transversal iff |T | = |W|

The cardinality of the biggest (partial) transversal is the maximum number of jobs that can be done simultaneously in the example.

(90)

Mathematical Describtion

J W

j ₁ ◦

j 2 ◦ ◦W 1

j ₃ ◦ ◦W ₂

j ₄ ◦ ◦W 3

j 5 ◦ ◦W 4

W := {W i |i ∈ I , W i ⊆ J}

T ⊆ J partial transversal : ⇔ ∃ injective map ψ : T ։ I s.t. t ∈ W _ψ(t) ∀t ∈ T i.e. matching for T in △[W ]

transversal iff |T | = |W|

The cardinality of the biggest (partial)

transversal is the maximum number of jobs

that can be done simultaneously in the

example.

(91)

Job Scheduling

Mathematical Describtion

J W

j ₁ ◦

j 2 ◦ ◦W 1

j ₃ ◦ ◦W ₂

j ₄ ◦ ◦W 3

j 5 ◦ ◦W 4

j ₆ ◦

W := {W i |i ∈ I , W i ⊆ J}

T ⊆ J partial transversal : ⇔ ∃ injective map ψ : T ։ I s.t. t ∈ W _ψ(t) ∀t ∈ T i.e. matching for T in △[W ]

transversal iff |T | = |W|

The cardinality of the biggest (partial) transversal is the maximum number of jobs that can be done simultaneously in the example.

(92)

Mathematical Describtion

J W

j ₁ ◦

j 2 ◦ ◦W 1

j ₃ ◦ ◦W ₂

j ₄ ◦ ◦W 3

j 5 ◦ ◦W 4

W := {W i |i ∈ I , W i ⊆ J}

T ⊆ J partial transversal : ⇔ ∃ injective map ψ : T ։ I s.t. t ∈ W _ψ(t) ∀t ∈ T i.e. matching for T in △[W ]

transversal iff |T | = |W|

The cardinality of the biggest (partial)

transversal is the maximum number of jobs

that can be done simultaneously in the

example.

(93)

Job Scheduling

Mathematical Describtion

J W

j ₁ ◦

j 2 ◦ ◦W 1

j ₃ ◦ ◦W ₂

j ₄ ◦ ◦W 3

j 5 ◦ ◦W 4

j ₆ ◦

W := {W i |i ∈ I , W i ⊆ J}

T ⊆ J partial transversal : ⇔ ∃ injective map ψ : T ։ I s.t. t ∈ W _ψ(t) ∀t ∈ T i.e. matching for T in △[W ]

transversal iff |T | = |W|

The cardinality of the biggest (partial) transversal is the maximum number of jobs that can be done simultaneously in the example.