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Kernels in bipolar-valued digraphs Enumerating kernels in crisp digraphs Solving the bipolar-valued case Conclusion

On enumerating the kernels in a bipolar valued digraph

Raymond Bisdorff

University of Luxembourg

Paris, Saturday, October 28, 2006

Kernels in bipolar-valued digraphs Enumerating kernels in crisp digraphs Solving the bipolar-valued case Conclusion

Outline

Kernels in bipolar-valued digraphs The bipolar-valued digraph Qualified choices in a digraph Kernels in digraphs Enumerating kernels in crisp digraphs Reducing covering or extending irredundant choices Extending independent choices until dominance /and/or absorbency Solving the bipolar-valued case Bipolar-valued qualified choices Kernel characteristic equations Solving the kernel characteristic equations

Kernels in bipolar-valued digraphs Enumerating kernels in crisp digraphs Solving the bipolar-valued case Conclusion

Bipolar-valued outranking graph

Definition

We consider a binary relation S defined on a set X : {x, y, z, . . .} of decision actions. S is characterised in a rational bipolar credibility domain:

• S : X × X → L = [−1, 1].
• S(x, y) = 1.0 signifies that x S y is certainly true.
• S(x, y) > 0.0 signifies that x S y is rather true than false.
• S(x, y) = 0.0 signifies that x S y may be true or false.
• S(x, y) < 0.0 signifies that x S y is rather false than true.
• S(x, y) = −1.0 signifies that x S y is certainly false.

We call G(X, S) a bipolar-valued digraph.

Kernels in bipolar-valued digraphs Enumerating kernels in crisp digraphs Solving the bipolar-valued case Conclusion

Bipolar-valued outranking graph

Definition

We consider a binary relation S defined on a set X : {x, y, z, . . .} of decision actions. S is characterised in a rational bipolar credibility domain:

• S : X × X → L = [−1, 1].
• S(x, y) = 1.0 signifies that x S y is certainly true.
• S(x, y) > 0.0 signifies that x S y is rather true than false.
• S(x, y) = 0.0 signifies that x S y may be true or false.
• S(x, y) < 0.0 signifies that x S y is rather false than true.
• S(x, y) = −1.0 signifies that x S y is certainly false.

We call G(X, S) a bipolar-valued digraph.

Kernels in bipolar-valued digraphs Enumerating kernels in crisp digraphs Solving the bipolar-valued case Conclusion

Bipolar-valued outranking graph

Definition

We consider a binary relation S defined on a set X : {x, y, z, . . .} of decision actions. S is characterised in a rational bipolar credibility domain:

• S : X × X → L = [−1, 1].
• S(x, y) = 1.0 signifies that x S y is certainly true.
• S(x, y) > 0.0 signifies that x S y is rather true than false.
• S(x, y) = 0.0 signifies that x S y may be true or false.
• S(x, y) < 0.0 signifies that x S y is rather false than true.
• S(x, y) = −1.0 signifies that x S y is certainly false.

We call G(X, S) a bipolar-valued digraph.

Kernels in bipolar-valued digraphs Enumerating kernels in crisp digraphs Solving the bipolar-valued case Conclusion

Bipolar-valued outranking graph

Definition

We consider a binary relation S defined on a set X : {x, y, z, . . .} of decision actions. S is characterised in a rational bipolar credibility domain:

• S : X × X → L = [−1, 1].
• S(x, y) = 1.0 signifies that x S y is certainly true.
• S(x, y) > 0.0 signifies that x S y is rather true than false.
• S(x, y) = 0.0 signifies that x S y may be true or false.
• S(x, y) < 0.0 signifies that x S y is rather false than true.
• S(x, y) = −1.0 signifies that x S y is certainly false.

We call G(X, S) a bipolar-valued digraph.

Kernels in bipolar-valued digraphs Enumerating kernels in crisp digraphs Solving the bipolar-valued case Conclusion

Bipolar-valued outranking graph

Definition

We consider a binary relation S defined on a set X : {x, y, z, . . .} of decision actions. S is characterised in a rational bipolar credibility domain:

• S : X × X → L = [−1, 1].
• S(x, y) = 1.0 signifies that x S y is certainly true.
• S(x, y) > 0.0 signifies that x S y is rather true than false.
• S(x, y) = 0.0 signifies that x S y may be true or false.
• S(x, y) < 0.0 signifies that x S y is rather false than true.
• S(x, y) = −1.0 signifies that x S y is certainly false.

We call G(X, S) a bipolar-valued digraph.

Kernels in bipolar-valued digraphs Enumerating kernels in crisp digraphs Solving the bipolar-valued case Conclusion

Bipolar-valued outranking graph

Definition

We consider a binary relation S defined on a set X : {x, y, z, . . .} of decision actions. S is characterised in a rational bipolar credibility domain:

• S : X × X → L = [−1, 1].
• S(x, y) = 1.0 signifies that x S y is certainly true.
• S(x, y) > 0.0 signifies that x S y is rather true than false.
• S(x, y) = 0.0 signifies that x S y may be true or false.
• S(x, y) < 0.0 signifies that x S y is rather false than true.
• S(x, y) = −1.0 signifies that x S y is certainly false.

We call G(X, S) a bipolar-valued digraph.

Kernels in bipolar-valued digraphs Enumerating kernels in crisp digraphs Solving the bipolar-valued case Conclusion

Bipolar-valued outranking graph

Definition

We consider a binary relation S defined on a set X : {x, y, z, . . .} of decision actions. S is characterised in a rational bipolar credibility domain:

• S : X × X → L = [−1, 1].
• S(x, y) = 1.0 signifies that x S y is certainly true.
• S(x, y) > 0.0 signifies that x S y is rather true than false.
• S(x, y) = 0.0 signifies that x S y may be true or false.
• S(x, y) < 0.0 signifies that x S y is rather false than true.
• S(x, y) = −1.0 signifies that x S y is certainly false.

We call G(X, S) a bipolar-valued digraph.

Kernels in bipolar-valued digraphs Enumerating kernels in crisp digraphs Solving the bipolar-valued case Conclusion

Bipolar-valued outranking graph

Definition

We consider a binary relation S defined on a set X : {x, y, z, . . .} of decision actions. S is characterised in a rational bipolar credibility domain:

• S : X × X → L = [−1, 1].
• S(x, y) = 1.0 signifies that x S y is certainly true.
• S(x, y) > 0.0 signifies that x S y is rather true than false.
• S(x, y) = 0.0 signifies that x S y may be true or false.
• S(x, y) < 0.0 signifies that x S y is rather false than true.
• S(x, y) = −1.0 signifies that x S y is certainly false.

We call G(X, S) a bipolar-valued digraph.

Example: Condorcet ruled social choice

Consider seven voters {1, 2, . . . , 7} on a set X = {a, b, c, d, e} of five alternatives. (From A. Taylor 2005, p.33)

X-voting profile 1 2 3 4 5 6 7 a a a c c b e b d d b d c c c b b d b d d d e e e a a b e c c a e e a Bipolar-valued majority difference e S1 a b c d e a

• 1/7
• 1/7
• 1/7

3/7 b 1/7

• 1/7
• 1/7

5/7 c 1/7

• 1/7
• 3/7

1/7 d 1/7 1/7

• 3/7
• 5/7

e

• 3/7
• 5/7
• 1/7
• 5/7
• b

c d e a

Majority voting preference

Example: Outranking based decision

Consider a set X1 = {a, b, c, d, e} of decision alternatives randomly evaluated on a coherent family of five criteria of equal significance supporting a rational preference scale [0,1] with indifference threshold = 0.1, preference threshold = 0.2, weak veto threshold = 0.6, and veto threshold = 0.8.

Coherent family of criteria X1 1 2 3 4 5 a 0.5 0.8 0.1 1.0 0.1 b 0.9 0.3 0.4 0.8 0.3 c 0.8 0.3 0.6 0.4 0.9 d 0.3 0.6 0.7 0.1 0.1 e 0.2 0.1 0.2 0.9 0.6 Bipolar-valued outranking e S1 a b c d e a 1.0 0.0

• 0.2

0.6 0.4 b 0.4 1.0 0.4 0.2 0.6 c 0.2 0.6 1.0 0.6 0.6 d

• 1.0

0.0

• 0.2

1.0 0.2 e 0.2 0.2

• 0.4

0.2 1.0

b c e d a

Majority concordant outranking

Kernels in bipolar-valued digraphs Enumerating kernels in crisp digraphs Solving the bipolar-valued case Conclusion

Properties of bipolar-valued outranking digraphs

Definition (The associated 0-cut crisp digraph)

Let G(X, S) be a bipolar-valued digraph. We denote G(X, S) its associated strict 0-cut crisp digraph where: S = {(x, y) ∈ X × X| S(x, y) > 0.0}.

Comments (example qualifications)

The order of G(X, S) will be n = |X|. The size of G(X, S) will be m = |S|. The arc density of G(X, S) will be m/n(n − 1).

• G(X,

S) is a weak order iff G(X, S) is a weak order.

Kernels in bipolar-valued digraphs Enumerating kernels in crisp digraphs Solving the bipolar-valued case Conclusion

Properties of bipolar-valued outranking digraphs

Definition (The associated 0-cut crisp digraph)

Let G(X, S) be a bipolar-valued digraph. We denote G(X, S) its associated strict 0-cut crisp digraph where: S = {(x, y) ∈ X × X| S(x, y) > 0.0}.

Comments (example qualifications)

The order of G(X, S) will be n = |X|. The size of G(X, S) will be m = |S|. The arc density of G(X, S) will be m/n(n − 1).

• G(X,

S) is a weak order iff G(X, S) is a weak order.

Kernels in bipolar-valued digraphs Enumerating kernels in crisp digraphs Solving the bipolar-valued case Conclusion

Properties of bipolar-valued outranking digraphs

Definition (The associated 0-cut crisp digraph)

Let G(X, S) be a bipolar-valued digraph. We denote G(X, S) its associated strict 0-cut crisp digraph where: S = {(x, y) ∈ X × X| S(x, y) > 0.0}.

Comments (example qualifications)

The order of G(X, S) will be n = |X|. The size of G(X, S) will be m = |S|. The arc density of G(X, S) will be m/n(n − 1).

• G(X,

S) is a weak order iff G(X, S) is a weak order.

Kernels in bipolar-valued digraphs Enumerating kernels in crisp digraphs Solving the bipolar-valued case Conclusion

Properties of bipolar-valued outranking digraphs

Definition (The associated 0-cut crisp digraph)

Let G(X, S) be a bipolar-valued digraph. We denote G(X, S) its associated strict 0-cut crisp digraph where: S = {(x, y) ∈ X × X| S(x, y) > 0.0}.

Comments (example qualifications)

The order of G(X, S) will be n = |X|. The size of G(X, S) will be m = |S|. The arc density of G(X, S) will be m/n(n − 1).

• G(X,

S) is a weak order iff G(X, S) is a weak order.

Kernels in bipolar-valued digraphs Enumerating kernels in crisp digraphs Solving the bipolar-valued case Conclusion

Properties of bipolar-valued outranking digraphs

Definition (The associated 0-cut crisp digraph)

Let G(X, S) be a bipolar-valued digraph. We denote G(X, S) its associated strict 0-cut crisp digraph where: S = {(x, y) ∈ X × X| S(x, y) > 0.0}.

Comments (example qualifications)

The order of G(X, S) will be n = |X|. The size of G(X, S) will be m = |S|. The arc density of G(X, S) will be m/n(n − 1).

• G(X,

S) is a weak order iff G(X, S) is a weak order.

Dominant and absorbent choices

Definition

Let G(X, S) be a bipolar valued outranking graph. A choice Y ⊆ X is a dominant choice iff ∀x ∈ X : x ∈ Y ⇒ ∃y ∈ Y : S(y, x) > 0. A choice Y ⊆ X is an absorbent choice iff ∀x ∈ X : x ∈ Y ⇒ ∃y ∈ Y : S(x, y) > 0.

Example(s) (B. Roy, private communication 2005)

Outranking digraph

Dominant and absorbent choices

Definition

Let G(X, S) be a bipolar valued outranking graph. A choice Y ⊆ X is a dominant choice iff ∀x ∈ X : x ∈ Y ⇒ ∃y ∈ Y : S(y, x) > 0. A choice Y ⊆ X is an absorbent choice iff ∀x ∈ X : x ∈ Y ⇒ ∃y ∈ Y : S(x, y) > 0.

Example(s) (B. Roy, private communication 2005)

Outranking digraph

Dominant and absorbent choices

Definition

Let G(X, S) be a bipolar valued outranking graph. A choice Y ⊆ X is a dominant choice iff ∀x ∈ X : x ∈ Y ⇒ ∃y ∈ Y : S(y, x) > 0. A choice Y ⊆ X is an absorbent choice iff ∀x ∈ X : x ∈ Y ⇒ ∃y ∈ Y : S(x, y) > 0.

Example(s) (B. Roy, private communication 2005)

Outranking digraph

Dominant and absorbent choices

Definition

Let G(X, S) be a bipolar valued outranking graph. A choice Y ⊆ X is a dominant choice iff ∀x ∈ X : x ∈ Y ⇒ ∃y ∈ Y : S(y, x) > 0. A choice Y ⊆ X is an absorbent choice iff ∀x ∈ X : x ∈ Y ⇒ ∃y ∈ Y : S(x, y) > 0.

Example(s) (B. Roy, private communication 2005)

Outranking digraph

Dominant and absorbent choices

Definition

Let G(X, S) be a bipolar valued outranking graph. A choice Y ⊆ X is a dominant choice iff ∀x ∈ X : x ∈ Y ⇒ ∃y ∈ Y : S(y, x) > 0. A choice Y ⊆ X is an absorbent choice iff ∀x ∈ X : x ∈ Y ⇒ ∃y ∈ Y : S(x, y) > 0.

Example(s) (B. Roy, private communication 2005)

Outranking digraph

Dominant and absorbent choices

Definition

Let G(X, S) be a bipolar valued outranking graph. A choice Y ⊆ X is a dominant choice iff ∀x ∈ X : x ∈ Y ⇒ ∃y ∈ Y : S(y, x) > 0. A choice Y ⊆ X is an absorbent choice iff ∀x ∈ X : x ∈ Y ⇒ ∃y ∈ Y : S(x, y) > 0.

Example(s) (B. Roy, private communication 2005)

Outranking digraph

Dominant and absorbent choices

Definition

Let G(X, S) be a bipolar valued outranking graph. A choice Y ⊆ X is a dominant choice iff ∀x ∈ X : x ∈ Y ⇒ ∃y ∈ Y : S(y, x) > 0. A choice Y ⊆ X is an absorbent choice iff ∀x ∈ X : x ∈ Y ⇒ ∃y ∈ Y : S(x, y) > 0.

Example(s) (B. Roy, private communication 2005)

a b c d e

Outranking digraph

Dominant and absorbent choices

Definition

Let G(X, S) be a bipolar valued outranking graph. A choice Y ⊆ X is a dominant choice iff ∀x ∈ X : x ∈ Y ⇒ ∃y ∈ Y : S(y, x) > 0. A choice Y ⊆ X is an absorbent choice iff ∀x ∈ X : x ∈ Y ⇒ ∃y ∈ Y : S(x, y) > 0.

Example(s) (B. Roy, private communication 2005)

b a d c e

{a, b, d, e} is a dominant choice in G.

Dominant and absorbent choices

Definition

Let G(X, S) be a bipolar valued outranking graph. A choice Y ⊆ X is a dominant choice iff ∀x ∈ X : x ∈ Y ⇒ ∃y ∈ Y : S(y, x) > 0. A choice Y ⊆ X is an absorbent choice iff ∀x ∈ X : x ∈ Y ⇒ ∃y ∈ Y : S(x, y) > 0.

Example(s) (B. Roy, private communication 2005)

b a d c e

{b, d, e} is an absorbent choice in G.

Kernels in bipolar-valued digraphs Enumerating kernels in crisp digraphs Solving the bipolar-valued case Conclusion

Minimal and maximal choices

Definition

A qualified choice Y in G is minimal with this quality whenever ∀Y ′ ⊆ Y we have Y ′ not equally qualified. A qualified choice Y in G is maximal with this quality whenever ∀Y ′ ⊇ Y we have Y ′ not equally qualified.

Example(s)

is a minimal choice.

Kernels in bipolar-valued digraphs Enumerating kernels in crisp digraphs Solving the bipolar-valued case Conclusion

Minimal and maximal choices

Definition

A qualified choice Y in G is minimal with this quality whenever ∀Y ′ ⊆ Y we have Y ′ not equally qualified. A qualified choice Y in G is maximal with this quality whenever ∀Y ′ ⊇ Y we have Y ′ not equally qualified.

Example(s)

is a minimal choice.

Kernels in bipolar-valued digraphs Enumerating kernels in crisp digraphs Solving the bipolar-valued case Conclusion

Minimal and maximal choices

Definition

A qualified choice Y in G is minimal with this quality whenever ∀Y ′ ⊆ Y we have Y ′ not equally qualified. A qualified choice Y in G is maximal with this quality whenever ∀Y ′ ⊇ Y we have Y ′ not equally qualified.

Example(s)

is a minimal choice.

Kernels in bipolar-valued digraphs Enumerating kernels in crisp digraphs Solving the bipolar-valued case Conclusion

Minimal and maximal choices

Definition

A qualified choice Y in G is minimal with this quality whenever ∀Y ′ ⊆ Y we have Y ′ not equally qualified. A qualified choice Y in G is maximal with this quality whenever ∀Y ′ ⊇ Y we have Y ′ not equally qualified.

Example(s)

b a d c e

{a, c} is a minimal domi- nant choice.

Kernels in bipolar-valued digraphs Enumerating kernels in crisp digraphs Solving the bipolar-valued case Conclusion

Minimal and maximal choices

Definition

A qualified choice Y in G is minimal with this quality whenever ∀Y ′ ⊆ Y we have Y ′ not equally qualified. A qualified choice Y in G is maximal with this quality whenever ∀Y ′ ⊇ Y we have Y ′ not equally qualified.

Example(s)

b a d c e

{b, d, e} is a minimal ab- sorbent choice.

Kernels in bipolar-valued digraphs Enumerating kernels in crisp digraphs Solving the bipolar-valued case Conclusion

Irredundant choices

Definition (Neighbourhoods of nodes and choices)

We denote N±[x] the closed dominated (+), respectively absorbed (-), neighbourhood of a node x ∈ X, i.e. {y ∈ X| S(x, y) > 0} ∪ {x}, resp. {y ∈ X| S(y, x) > 0} ∪ {x}. The dominated (+), resp. absorbed (-), neighbourhood of a choice Y in G is defined as N±[Y ] =

x∈Y N±[x].

Kernels in bipolar-valued digraphs Enumerating kernels in crisp digraphs Solving the bipolar-valued case Conclusion

Irredundant choices

Definition (Neighbourhoods of nodes and choices)

We denote N±[x] the closed dominated (+), respectively absorbed (-), neighbourhood of a node x ∈ X, i.e. {y ∈ X| S(x, y) > 0} ∪ {x}, resp. {y ∈ X| S(y, x) > 0} ∪ {x}. The dominated (+), resp. absorbed (-), neighbourhood of a choice Y in G is defined as N±[Y ] =

x∈Y N±[x].

Kernels in bipolar-valued digraphs Enumerating kernels in crisp digraphs Solving the bipolar-valued case Conclusion

Irredundant choices

Definition (Neighbourhoods of nodes and choices)

We denote N±[x] the closed dominated (+), respectively absorbed (-), neighbourhood of a node x ∈ X, i.e. {y ∈ X| S(x, y) > 0} ∪ {x}, resp. {y ∈ X| S(y, x) > 0} ∪ {x}. The dominated (+), resp. absorbed (-), neighbourhood of a choice Y in G is defined as N±[Y ] =

x∈Y N±[x].

Kernels in bipolar-valued digraphs Enumerating kernels in crisp digraphs Solving the bipolar-valued case Conclusion

Irredundant choices (continue)

Definition (Private neighbourhoods)

The (closed) private dominated (+), resp. absorbed (-), neighbourhood N±

Y [x] of a node x in a choice Y is defined as

follows: N±

Y [x] = N±[x] − N±[Y − {x}].

Example(s)

Kernels in bipolar-valued digraphs Enumerating kernels in crisp digraphs Solving the bipolar-valued case Conclusion

Irredundant choices (continue)

Definition (Private neighbourhoods)

The (closed) private dominated (+), resp. absorbed (-), neighbourhood N±

Y [x] of a node x in a choice Y is defined as

follows: N±

Y [x] = N±[x] − N±[Y − {x}].

Example(s)

Kernels in bipolar-valued digraphs Enumerating kernels in crisp digraphs Solving the bipolar-valued case Conclusion

Irredundant choices (continue)

Definition (Private neighbourhoods)

The (closed) private dominated (+), resp. absorbed (-), neighbourhood N±

Y [x] of a node x in a choice Y is defined as

follows: N±

Y [x] = N±[x] − N±[Y − {x}].

Example(s)

Kernels in bipolar-valued digraphs Enumerating kernels in crisp digraphs Solving the bipolar-valued case Conclusion

Irredundant choices (continue)

Definition (Private neighbourhoods)

The (closed) private dominated (+), resp. absorbed (-), neighbourhood N±

Y [x] of a node x in a choice Y is defined as

follows: N±

Y [x] = N±[x] − N±[Y − {x}].

Example(s)

b a d c e

Y = {a, b, d, e} N+

Y [a] = {}; N+ Y [b] = {c};

N+

Y [d] = {}; N+ Y [e] = {e}

Kernels in bipolar-valued digraphs Enumerating kernels in crisp digraphs Solving the bipolar-valued case Conclusion

Irredundant choices (continue)

Definition (Private neighbourhoods)

The (closed) private dominated (+), resp. absorbed (-), neighbourhood N±

Y [x] of a node x in a choice Y is defined as

follows: N±

Y [x] = N±[x] − N±[Y − {x}].

Example(s)

b a d c e

Y = {b, d, e} N−

Y [b] = {a, b}

N−

Y [d] = {c}

N−

Y [e] = {e}

Kernels in bipolar-valued digraphs Enumerating kernels in crisp digraphs Solving the bipolar-valued case Conclusion

Irredundant choices (continue)

Definition

A choice Y in G is called +irredundant (resp.-irredundant) iff all its elements have a non empty private dominated neighborhood, i.e. ∀x ∈ Y : N±

Y [x] = ∅.

Example(s)

Kernels in bipolar-valued digraphs Enumerating kernels in crisp digraphs Solving the bipolar-valued case Conclusion

Irredundant choices (continue)

Definition

A choice Y in G is called +irredundant (resp.-irredundant) iff all its elements have a non empty private dominated neighborhood, i.e. ∀x ∈ Y : N±

Y [x] = ∅.

Example(s)

Kernels in bipolar-valued digraphs Enumerating kernels in crisp digraphs Solving the bipolar-valued case Conclusion

Irredundant choices (continue)

Definition

A choice Y in G is called +irredundant (resp.-irredundant) iff all its elements have a non empty private dominated neighborhood, i.e. ∀x ∈ Y : N±

Y [x] = ∅.

Example(s)

Kernels in bipolar-valued digraphs Enumerating kernels in crisp digraphs Solving the bipolar-valued case Conclusion

Irredundant choices (continue)

Definition

A choice Y in G is called +irredundant (resp.-irredundant) iff all its elements have a non empty private dominated neighborhood, i.e. ∀x ∈ Y : N±

Y [x] = ∅.

Example(s)

b a d c e

N+

{a,c}[a] = {a, b}

N+

{a,c}[c] = {c, d, e}

Kernels in bipolar-valued digraphs Enumerating kernels in crisp digraphs Solving the bipolar-valued case Conclusion

Irredundant choices (continue)

Definition

A choice Y in G is called +irredundant (resp.-irredundant) iff all its elements have a non empty private dominated neighborhood, i.e. ∀x ∈ Y : N±

Y [x] = ∅.

Example(s)

b a d c e

N−

{b,d,e}[b] = {a, b}

N−

{b,d,e}[d] = {c}

N−

{b,d,e}[e] = {e}

Kernels in bipolar-valued digraphs Enumerating kernels in crisp digraphs Solving the bipolar-valued case Conclusion

Minimal dominant (absorbent) and maximal ±irredundant choices

Theorem (Cockayne, Hedetniemi, Miller 1978)

A choice Y in G is minimal dominant (resp. absorbent) iff it is dominant (resp. absorbent) and ±irredundant.

Theorem (Bollob´ as, Cockayne, 1979)

Every minimal dominant (resp. absorbent) choice Y in G is maximal ±irredundant (but not vice versa).

Kernels in bipolar-valued digraphs Enumerating kernels in crisp digraphs Solving the bipolar-valued case Conclusion

Minimal dominant (absorbent) and maximal ±irredundant choices

Theorem (Cockayne, Hedetniemi, Miller 1978)

A choice Y in G is minimal dominant (resp. absorbent) iff it is dominant (resp. absorbent) and ±irredundant.

Theorem (Bollob´ as, Cockayne, 1979)

Every minimal dominant (resp. absorbent) choice Y in G is maximal ±irredundant (but not vice versa).

Kernels in bipolar-valued digraphs Enumerating kernels in crisp digraphs Solving the bipolar-valued case Conclusion

Minimal dominant (absorbent) and maximal ±irredundant choices

Theorem (Cockayne, Hedetniemi, Miller 1978)

A choice Y in G is minimal dominant (resp. absorbent) iff it is dominant (resp. absorbent) and ±irredundant.

Theorem (Bollob´ as, Cockayne, 1979)

Every minimal dominant (resp. absorbent) choice Y in G is maximal ±irredundant (but not vice versa).

Kernels in bipolar-valued digraphs Enumerating kernels in crisp digraphs Solving the bipolar-valued case Conclusion

Minimal dominant (absorbent) and maximal ±irredundant choices

Theorem (Cockayne, Hedetniemi, Miller 1978)

A choice Y in G is minimal dominant (resp. absorbent) iff it is dominant (resp. absorbent) and ±irredundant.

Theorem (Bollob´ as, Cockayne, 1979)

Every minimal dominant (resp. absorbent) choice Y in G is maximal ±irredundant (but not vice versa).

Kernels in bipolar-valued digraphs Enumerating kernels in crisp digraphs Solving the bipolar-valued case Conclusion

Independent choices

Let G(X, S) be a bipolar valued outranking graph.

Definition

A choice Y in G is called independent if and only if ∀x, y ∈ Y : S(x, y) < 0. A dominant (resp. absorbent) and independent choice Y in G is called a dominant (resp. absorbent) kernel of the graph G.

Comments

Also called independent semi-dominating sets, in– and out–kernels,

• etc. Origins: Grundy (1939), Von Neumann (1944), Berge (1958).

Kernels in bipolar-valued digraphs Enumerating kernels in crisp digraphs Solving the bipolar-valued case Conclusion

Independent choices

Let G(X, S) be a bipolar valued outranking graph.

Definition

A choice Y in G is called independent if and only if ∀x, y ∈ Y : S(x, y) < 0. A dominant (resp. absorbent) and independent choice Y in G is called a dominant (resp. absorbent) kernel of the graph G.

Comments

Also called independent semi-dominating sets, in– and out–kernels,

• etc. Origins: Grundy (1939), Von Neumann (1944), Berge (1958).

Kernels in bipolar-valued digraphs Enumerating kernels in crisp digraphs Solving the bipolar-valued case Conclusion

Independent choices

Let G(X, S) be a bipolar valued outranking graph.

Definition

A choice Y in G is called independent if and only if ∀x, y ∈ Y : S(x, y) < 0. A dominant (resp. absorbent) and independent choice Y in G is called a dominant (resp. absorbent) kernel of the graph G.

Comments

Also called independent semi-dominating sets, in– and out–kernels,

• etc. Origins: Grundy (1939), Von Neumann (1944), Berge (1958).

Kernels in bipolar-valued digraphs Enumerating kernels in crisp digraphs Solving the bipolar-valued case Conclusion

Independent choices

Let G(X, S) be a bipolar valued outranking graph.

Definition

A choice Y in G is called independent if and only if ∀x, y ∈ Y : S(x, y) < 0. A dominant (resp. absorbent) and independent choice Y in G is called a dominant (resp. absorbent) kernel of the graph G.

Comments

Also called independent semi-dominating sets, in– and out–kernels,

• etc. Origins: Grundy (1939), Von Neumann (1944), Berge (1958).

Kernels in bipolar-valued digraphs Enumerating kernels in crisp digraphs Solving the bipolar-valued case Conclusion

Kernels in digraphs

Proposition (Berge, 1958)

Every kernel is a minimal dominant (resp. absorbent) choice. Every minimal dominant (resp. absorbent) and independent choice is maximal independent.

Example(s) (Not all minimal dominant (resp. absorbent) choices are independent, i.e. kernels !)

Kernels in bipolar-valued digraphs Enumerating kernels in crisp digraphs Solving the bipolar-valued case Conclusion

Kernels in digraphs

Proposition (Berge, 1958)

Every kernel is a minimal dominant (resp. absorbent) choice. Every minimal dominant (resp. absorbent) and independent choice is maximal independent.

Example(s) (Not all minimal dominant (resp. absorbent) choices are independent, i.e. kernels !)

Kernels in bipolar-valued digraphs Enumerating kernels in crisp digraphs Solving the bipolar-valued case Conclusion

Kernels in digraphs

Proposition (Berge, 1958)

Every kernel is a minimal dominant (resp. absorbent) choice. Every minimal dominant (resp. absorbent) and independent choice is maximal independent.

Example(s) (Not all minimal dominant (resp. absorbent) choices are independent, i.e. kernels !)

Kernels in bipolar-valued digraphs Enumerating kernels in crisp digraphs Solving the bipolar-valued case Conclusion

Kernels in digraphs

Proposition (Berge, 1958)

Every kernel is a minimal dominant (resp. absorbent) choice. Every minimal dominant (resp. absorbent) and independent choice is maximal independent.

Example(s) (Not all minimal dominant (resp. absorbent) choices are independent, i.e. kernels !)

b a d c e b a d c e

Kernels in bipolar-valued digraphs Enumerating kernels in crisp digraphs Solving the bipolar-valued case Conclusion

Kernels in digraphs

Proposition (Berge, 1958)

Every kernel is a minimal dominant (resp. absorbent) choice. Every minimal dominant (resp. absorbent) and independent choice is maximal independent.

Example(s) (Not all minimal dominant (resp. absorbent) choices are independent, i.e. kernels !)

b a d c e b a d c e

Kernels in bipolar-valued digraphs Enumerating kernels in crisp digraphs Solving the bipolar-valued case Conclusion

Kernels in digraphs

Proposition

Let G(X, S) be a transitive digraph: A choice Y in G is a dominant (resp. absorbent) kernel if and

• nly if Y verifies one of the following conditions:

1

Y is minimal dominant (resp. absorbent);

2

Y is dominant (resp. absorbent) and (maximal) independent;

3

Y is dominant (resp. absorbent) and (maximal) +irredundant (resp. -irredundant).

Kernels in bipolar-valued digraphs Enumerating kernels in crisp digraphs Solving the bipolar-valued case Conclusion

Kernels in digraphs

Proposition

Let G(X, S) be a transitive digraph: A choice Y in G is a dominant (resp. absorbent) kernel if and

• nly if Y verifies one of the following conditions:

1

Y is minimal dominant (resp. absorbent);

2

Y is dominant (resp. absorbent) and (maximal) independent;

3

Y is dominant (resp. absorbent) and (maximal) +irredundant (resp. -irredundant).

Kernels in bipolar-valued digraphs Enumerating kernels in crisp digraphs Solving the bipolar-valued case Conclusion

Kernels in digraphs

Proposition

Let G(X, S) be a transitive digraph: A choice Y in G is a dominant (resp. absorbent) kernel if and

• nly if Y verifies one of the following conditions:

1

Y is minimal dominant (resp. absorbent);

2

Y is dominant (resp. absorbent) and (maximal) independent;

3

Y is dominant (resp. absorbent) and (maximal) +irredundant (resp. -irredundant).

Kernels in bipolar-valued digraphs Enumerating kernels in crisp digraphs Solving the bipolar-valued case Conclusion

Kernels in digraphs

Proposition

Let G(X, S) be a transitive digraph: A choice Y in G is a dominant (resp. absorbent) kernel if and

• nly if Y verifies one of the following conditions:

1

Y is minimal dominant (resp. absorbent);

2

Y is dominant (resp. absorbent) and (maximal) independent;

3

Y is dominant (resp. absorbent) and (maximal) +irredundant (resp. -irredundant).

Kernels in bipolar-valued digraphs Enumerating kernels in crisp digraphs Solving the bipolar-valued case Conclusion

Kernels in digraphs

Proposition

Let G(X, S) be a transitive digraph: A choice Y in G is a dominant (resp. absorbent) kernel if and

• nly if Y verifies one of the following conditions:

1

Y is minimal dominant (resp. absorbent);

2

Y is dominant (resp. absorbent) and (maximal) independent;

3

Y is dominant (resp. absorbent) and (maximal) +irredundant (resp. -irredundant).

Kernels in bipolar-valued digraphs Enumerating kernels in crisp digraphs Solving the bipolar-valued case Conclusion

Kernels in digraphs

Proposition

Let G(X, S) be a transitive digraph: A choice Y in G is a dominant (resp. absorbent) kernel if and

• nly if Y verifies one of the following conditions:

1

Y is minimal dominant (resp. absorbent);

2

Y is dominant (resp. absorbent) and (maximal) independent;

3

Y is dominant (resp. absorbent) and (maximal) +irredundant (resp. -irredundant).

Existence of kernels

Proposition (Existence results)

1

Every digraph supports minimal dominant (resp. absorbent) choices.

2

A transitive digraph always supports a dominant (resp. absorbent) kernel and all its kernels are of same cardinality (K¨

• nig, 1950).

3

A symmetric digraph always supports a conjointly dominant and absorbent kernel (Berge, 1958).

4

An acyclic digraph always supports a unique dominant (resp. absorbent) digraph (Von Neumann, 1944).

5

If a digraph has no odd asymmetric circuits, it supports a dominant (resp. absorbent) kernel (Richardson, 1953).

6

The probability that a random digraph of order n possesses a kernel tends to 1 when n → ∞ (De la Vega, 1990).

7

Almost every random digraph of order n contains only kernels K such that Cn − 1.43 ≤ |K| ≤ Cn + 2.11 where Cn = ln(n) − ln(ln(n)) (Tomescu 1990).

Existence of kernels

Proposition (Existence results)

1

Every digraph supports minimal dominant (resp. absorbent) choices.

2

A transitive digraph always supports a dominant (resp. absorbent) kernel and all its kernels are of same cardinality (K¨

• nig, 1950).

3

A symmetric digraph always supports a conjointly dominant and absorbent kernel (Berge, 1958).

4

An acyclic digraph always supports a unique dominant (resp. absorbent) digraph (Von Neumann, 1944).

5

If a digraph has no odd asymmetric circuits, it supports a dominant (resp. absorbent) kernel (Richardson, 1953).

6

The probability that a random digraph of order n possesses a kernel tends to 1 when n → ∞ (De la Vega, 1990).

7

Almost every random digraph of order n contains only kernels K such that Cn − 1.43 ≤ |K| ≤ Cn + 2.11 where Cn = ln(n) − ln(ln(n)) (Tomescu 1990).

Existence of kernels

Proposition (Existence results)

1

Every digraph supports minimal dominant (resp. absorbent) choices.

2

A transitive digraph always supports a dominant (resp. absorbent) kernel and all its kernels are of same cardinality (K¨

• nig, 1950).

3

A symmetric digraph always supports a conjointly dominant and absorbent kernel (Berge, 1958).

4

An acyclic digraph always supports a unique dominant (resp. absorbent) digraph (Von Neumann, 1944).

5

If a digraph has no odd asymmetric circuits, it supports a dominant (resp. absorbent) kernel (Richardson, 1953).

6

The probability that a random digraph of order n possesses a kernel tends to 1 when n → ∞ (De la Vega, 1990).

7

Almost every random digraph of order n contains only kernels K such that Cn − 1.43 ≤ |K| ≤ Cn + 2.11 where Cn = ln(n) − ln(ln(n)) (Tomescu 1990).

Existence of kernels

Proposition (Existence results)

1

Every digraph supports minimal dominant (resp. absorbent) choices.

2

A transitive digraph always supports a dominant (resp. absorbent) kernel and all its kernels are of same cardinality (K¨

• nig, 1950).

3

A symmetric digraph always supports a conjointly dominant and absorbent kernel (Berge, 1958).

4

An acyclic digraph always supports a unique dominant (resp. absorbent) digraph (Von Neumann, 1944).

5

If a digraph has no odd asymmetric circuits, it supports a dominant (resp. absorbent) kernel (Richardson, 1953).

6

The probability that a random digraph of order n possesses a kernel tends to 1 when n → ∞ (De la Vega, 1990).

7

Almost every random digraph of order n contains only kernels K such that Cn − 1.43 ≤ |K| ≤ Cn + 2.11 where Cn = ln(n) − ln(ln(n)) (Tomescu 1990).

Existence of kernels

Proposition (Existence results)

1

Every digraph supports minimal dominant (resp. absorbent) choices.

2

A transitive digraph always supports a dominant (resp. absorbent) kernel and all its kernels are of same cardinality (K¨

• nig, 1950).

3

A symmetric digraph always supports a conjointly dominant and absorbent kernel (Berge, 1958).

4

An acyclic digraph always supports a unique dominant (resp. absorbent) digraph (Von Neumann, 1944).

5

If a digraph has no odd asymmetric circuits, it supports a dominant (resp. absorbent) kernel (Richardson, 1953).

6

The probability that a random digraph of order n possesses a kernel tends to 1 when n → ∞ (De la Vega, 1990).

7

Almost every random digraph of order n contains only kernels K such that Cn − 1.43 ≤ |K| ≤ Cn + 2.11 where Cn = ln(n) − ln(ln(n)) (Tomescu 1990).

Existence of kernels

Proposition (Existence results)

1

Every digraph supports minimal dominant (resp. absorbent) choices.

2

A transitive digraph always supports a dominant (resp. absorbent) kernel and all its kernels are of same cardinality (K¨

• nig, 1950).

3

A symmetric digraph always supports a conjointly dominant and absorbent kernel (Berge, 1958).

4

An acyclic digraph always supports a unique dominant (resp. absorbent) digraph (Von Neumann, 1944).

5

If a digraph has no odd asymmetric circuits, it supports a dominant (resp. absorbent) kernel (Richardson, 1953).

6

The probability that a random digraph of order n possesses a kernel tends to 1 when n → ∞ (De la Vega, 1990).

7

Almost every random digraph of order n contains only kernels K such that Cn − 1.43 ≤ |K| ≤ Cn + 2.11 where Cn = ln(n) − ln(ln(n)) (Tomescu 1990).

Existence of kernels

Proposition (Existence results)

1

Every digraph supports minimal dominant (resp. absorbent) choices.

2

A transitive digraph always supports a dominant (resp. absorbent) kernel and all its kernels are of same cardinality (K¨

• nig, 1950).

3

A symmetric digraph always supports a conjointly dominant and absorbent kernel (Berge, 1958).

4

An acyclic digraph always supports a unique dominant (resp. absorbent) digraph (Von Neumann, 1944).

5

If a digraph has no odd asymmetric circuits, it supports a dominant (resp. absorbent) kernel (Richardson, 1953).

6

The probability that a random digraph of order n possesses a kernel tends to 1 when n → ∞ (De la Vega, 1990).

7

Almost every random digraph of order n contains only kernels K such that Cn − 1.43 ≤ |K| ≤ Cn + 2.11 where Cn = ln(n) − ln(ln(n)) (Tomescu 1990).

Existence of kernels

Proposition (Existence results)

1

Every digraph supports minimal dominant (resp. absorbent) choices.

2

A transitive digraph always supports a dominant (resp. absorbent) kernel and all its kernels are of same cardinality (K¨

• nig, 1950).

3

A symmetric digraph always supports a conjointly dominant and absorbent kernel (Berge, 1958).

4

An acyclic digraph always supports a unique dominant (resp. absorbent) digraph (Von Neumann, 1944).

5

If a digraph has no odd asymmetric circuits, it supports a dominant (resp. absorbent) kernel (Richardson, 1953).

6

The probability that a random digraph of order n possesses a kernel tends to 1 when n → ∞ (De la Vega, 1990).

7

Almost every random digraph of order n contains only kernels K such that Cn − 1.43 ≤ |K| ≤ Cn + 2.11 where Cn = ln(n) − ln(ln(n)) (Tomescu 1990).

Kernels in bipolar-valued digraphs Enumerating kernels in crisp digraphs Solving the bipolar-valued case Conclusion

Kernels in bipolar-valued digraphs The bipolar-valued digraph Qualified choices in a digraph Kernels in digraphs Enumerating kernels in crisp digraphs Reducing covering or extending irredundant choices Extending independent choices until dominance /and/or absorbency Solving the bipolar-valued case Bipolar-valued qualified choices Kernel characteristic equations Solving the kernel characteristic equations

Kernels in bipolar-valued digraphs Enumerating kernels in crisp digraphs Solving the bipolar-valued case Conclusion

Enumerating minimal dominant or absorbent choices

Super-heredity of dominance and absorbency and heredity of irredundancy gives two possible approaches: Reducing redundant dominance (resp. absorbency) until minimality, Extending ±-irredundancy until dominance (absorbency).

Random filled graphs of order 15

Kernels in bipolar-valued digraphs Enumerating kernels in crisp digraphs Solving the bipolar-valued case Conclusion

Enumerating dominant and absorbent kernels

Heredity of Independence gives a possible approach for enumerating both kind of kernels in a same run: Extending independence until dominance and/or absorbency.

Kernels in bipolar-valued digraphs Enumerating kernels in crisp digraphs Solving the bipolar-valued case Conclusion

Dominant and absorbent kernels in the same run

K + ←= ∅ # initialize the dominant result K − ←= ∅ # initialize the absorbent result for x ∈ X: # each singleton is an initial independent choice Y ← {x} (K +, K −) ← (K +, K −) ∪ AllKernels(Y , (K +, K −)) def AllKernels(In: Y independent, (K +

0 , K − 0 ); Out: (K +, K −)):

if (Y − X) − N+(Y ) = ∅: K + ← K +

0 ∪ Y # Y is dominant

else if N−(Y ) − (Y − X) = ∅: K − ← K −

0 ∪ Y # Y is absorbent

else: # try adding all independent singletons (K +, K −) ← (K +

0 , K − 0 )

for [x ∈ notN(Y )]: Y1 = Y ∪ {x} (K +, K −) ← (K +, K −) ∪ AllKernels(Y1, (K +, K −)) return (K +, K −)

Kernels in bipolar-valued digraphs Enumerating kernels in crisp digraphs Solving the bipolar-valued case Conclusion

Dominant and absorbent kernels in the same run

K + ←= ∅ # initialize the dominant result K − ←= ∅ # initialize the absorbent result for x ∈ X: # each singleton is an initial independent choice Y ← {x} (K +, K −) ← (K +, K −) ∪ AllKernels(Y , (K +, K −)) def AllKernels(In: Y independent, (K +

0 , K − 0 ); Out: (K +, K −)):

if (Y − X) − N+(Y ) = ∅: K + ← K +

0 ∪ Y # Y is dominant

else if N−(Y ) − (Y − X) = ∅: K − ← K −

0 ∪ Y # Y is absorbent

else: # try adding all independent singletons (K +, K −) ← (K +

0 , K − 0 )

for [x ∈ notN(Y )]: Y1 = Y ∪ {x} (K +, K −) ← (K +, K −) ∪ AllKernels(Y1, (K +, K −)) return (K +, K −)

Kernels in bipolar-valued digraphs Enumerating kernels in crisp digraphs Solving the bipolar-valued case Conclusion

Dominant and absorbent kernels in the same run

K + ←= ∅ # initialize the dominant result K − ←= ∅ # initialize the absorbent result for x ∈ X: # each singleton is an initial independent choice Y ← {x} (K +, K −) ← (K +, K −) ∪ AllKernels(Y , (K +, K −)) def AllKernels(In: Y independent, (K +

0 , K − 0 ); Out: (K +, K −)):

if (Y − X) − N+(Y ) = ∅: K + ← K +

0 ∪ Y # Y is dominant

else if N−(Y ) − (Y − X) = ∅: K − ← K −

0 ∪ Y # Y is absorbent

else: # try adding all independent singletons (K +, K −) ← (K +

0 , K − 0 )

for [x ∈ notN(Y )]: Y1 = Y ∪ {x} (K +, K −) ← (K +, K −) ∪ AllKernels(Y1, (K +, K −)) return (K +, K −)

Kernels in bipolar-valued digraphs Enumerating kernels in crisp digraphs Solving the bipolar-valued case Conclusion

Dominant and absorbent kernels in the same run

K + ←= ∅ # initialize the dominant result K − ←= ∅ # initialize the absorbent result for x ∈ X: # each singleton is an initial independent choice Y ← {x} (K +, K −) ← (K +, K −) ∪ AllKernels(Y , (K +, K −)) def AllKernels(In: Y independent, (K +

0 , K − 0 ); Out: (K +, K −)):

if (Y − X) − N+(Y ) = ∅: K + ← K +

0 ∪ Y # Y is dominant

else if N−(Y ) − (Y − X) = ∅: K − ← K −

0 ∪ Y # Y is absorbent

else: # try adding all independent singletons (K +, K −) ← (K +

0 , K − 0 )

for [x ∈ notN(Y )]: Y1 = Y ∪ {x} (K +, K −) ← (K +, K −) ∪ AllKernels(Y1, (K +, K −)) return (K +, K −)

Kernels in bipolar-valued digraphs Enumerating kernels in crisp digraphs Solving the bipolar-valued case Conclusion

Dominant and absorbent kernels in the same run

K + ←= ∅ # initialize the dominant result K − ←= ∅ # initialize the absorbent result for x ∈ X: # each singleton is an initial independent choice Y ← {x} (K +, K −) ← (K +, K −) ∪ AllKernels(Y , (K +, K −)) def AllKernels(In: Y independent, (K +

0 , K − 0 ); Out: (K +, K −)):

if (Y − X) − N+(Y ) = ∅: K + ← K +

0 ∪ Y # Y is dominant

else if N−(Y ) − (Y − X) = ∅: K − ← K −

0 ∪ Y # Y is absorbent

else: # try adding all independent singletons (K +, K −) ← (K +

0 , K − 0 )

for [x ∈ notN(Y )]: Y1 = Y ∪ {x} (K +, K −) ← (K +, K −) ∪ AllKernels(Y1, (K +, K −)) return (K +, K −)

Run time statistics for AllKernels

Run time statistics for AllKernels

Kernels in bipolar-valued digraphs Enumerating kernels in crisp digraphs Solving the bipolar-valued case Conclusion

Kernels in bipolar-valued digraphs The bipolar-valued digraph Qualified choices in a digraph Kernels in digraphs Enumerating kernels in crisp digraphs Reducing covering or extending irredundant choices Extending independent choices until dominance /and/or absorbency Solving the bipolar-valued case Bipolar-valued qualified choices Kernel characteristic equations Solving the kernel characteristic equations

Kernels in bipolar-valued digraphs Enumerating kernels in crisp digraphs Solving the bipolar-valued case Conclusion

Definition (Bipolar-valued qualification of choices)

The degree of independence of a choice Y in G: ∆ind(Y ) =

y=x

min

y∈Y min x∈Y

• −

S(x, y)

• .

The degree of dominance of a choice Y in G: ∆dom(Y ) = min

x∈Y max y∈Y

• S(y, x)
• .

The degree of absorbency of a choice Y in G: ∆abs(Y ) = min

x∈Y max y∈Y

• S(x, y)
• .

Kernels in bipolar-valued digraphs Enumerating kernels in crisp digraphs Solving the bipolar-valued case Conclusion

Definition (Bipolar-valued qualification of choices)

The degree of independence of a choice Y in G: ∆ind(Y ) =

y=x

min

y∈Y min x∈Y

• −

S(x, y)

• .

The degree of dominance of a choice Y in G: ∆dom(Y ) = min

x∈Y max y∈Y

• S(y, x)
• .

The degree of absorbency of a choice Y in G: ∆abs(Y ) = min

x∈Y max y∈Y

• S(x, y)
• .

Kernels in bipolar-valued digraphs Enumerating kernels in crisp digraphs Solving the bipolar-valued case Conclusion

Definition (Bipolar-valued qualification of choices)

The degree of independence of a choice Y in G: ∆ind(Y ) =

y=x

min

y∈Y min x∈Y

• −

S(x, y)

• .

The degree of dominance of a choice Y in G: ∆dom(Y ) = min

x∈Y max y∈Y

• S(y, x)
• .

The degree of absorbency of a choice Y in G: ∆abs(Y ) = min

x∈Y max y∈Y

• S(x, y)
• .

Kernels in bipolar-valued digraphs Enumerating kernels in crisp digraphs Solving the bipolar-valued case Conclusion

Definition (Bipolar-valued qualification of choices)

The degree of independence of a choice Y in G: ∆ind(Y ) =

y=x

min

y∈Y min x∈Y

• −

S(x, y)

• .

The degree of dominance of a choice Y in G: ∆dom(Y ) = min

x∈Y max y∈Y

• S(y, x)
• .

The degree of absorbency of a choice Y in G: ∆abs(Y ) = min

x∈Y max y∈Y

• S(x, y)
• .

Kernels in bipolar-valued digraphs Enumerating kernels in crisp digraphs Solving the bipolar-valued case Conclusion

Comments

By convention a singleton choice Y is certainly independent (∆ind(Y ) = m) and the greedy choice is certainly dominant and absorbent (∆dom(Y ) = m, resp. ∆abs(Y ) = m).

Proposition (Kitainik (1993), Bisdorff, Pirlot, Roubens (2006))

Y in G is a dominant kernel in G iff ∆ind(Y ) > 0 and ∆dom(Y ) > 0. Y in G is an absorbent kernel in G iff ∆ind(Y ) > 0 and ∆abs(Y ) > 0.

Kernels in bipolar-valued digraphs Enumerating kernels in crisp digraphs Solving the bipolar-valued case Conclusion

Comments

By convention a singleton choice Y is certainly independent (∆ind(Y ) = m) and the greedy choice is certainly dominant and absorbent (∆dom(Y ) = m, resp. ∆abs(Y ) = m).

Proposition (Kitainik (1993), Bisdorff, Pirlot, Roubens (2006))

Y in G is a dominant kernel in G iff ∆ind(Y ) > 0 and ∆dom(Y ) > 0. Y in G is an absorbent kernel in G iff ∆ind(Y ) > 0 and ∆abs(Y ) > 0.

Kernels in bipolar-valued digraphs Enumerating kernels in crisp digraphs Solving the bipolar-valued case Conclusion

Comments

By convention a singleton choice Y is certainly independent (∆ind(Y ) = m) and the greedy choice is certainly dominant and absorbent (∆dom(Y ) = m, resp. ∆abs(Y ) = m).

Proposition (Kitainik (1993), Bisdorff, Pirlot, Roubens (2006))

Y in G is a dominant kernel in G iff ∆ind(Y ) > 0 and ∆dom(Y ) > 0. Y in G is an absorbent kernel in G iff ∆ind(Y ) > 0 and ∆abs(Y ) > 0.

Kernels in bipolar-valued digraphs Enumerating kernels in crisp digraphs Solving the bipolar-valued case Conclusion

Comments

By convention a singleton choice Y is certainly independent (∆ind(Y ) = m) and the greedy choice is certainly dominant and absorbent (∆dom(Y ) = m, resp. ∆abs(Y ) = m).

Proposition (Kitainik (1993), Bisdorff, Pirlot, Roubens (2006))

Y in G is a dominant kernel in G iff ∆ind(Y ) > 0 and ∆dom(Y ) > 0. Y in G is an absorbent kernel in G iff ∆ind(Y ) > 0 and ∆abs(Y ) > 0.

Kernels in bipolar-valued digraphs Enumerating kernels in crisp digraphs Solving the bipolar-valued case Conclusion

Bipolar-valued choice characterisation

Definition

Let G(X, S) be a bipolar valued outranking graph. We may characterise a choice Y in G with the help of a bipolar-valued function Y : X → [−1, 1]: where x ∈ Y ⇔ Y (x) > 0, ∀x ∈ X.

Example(s)

• Y (a) = −0.6,

Y (b) = 0.6, Y (c) = −0.6, Y (d) = 1.0,

• Y (e) = 0.9 characterises the choice Y = {b, d, e}
• Y (a) = 0.6,

Y (b) = −0.6, Y (c) = 0.6, Y (d) = −1.0,

• Y (e) = −0.9 characterises the choice Y = {a, c}

Kernels in bipolar-valued digraphs Enumerating kernels in crisp digraphs Solving the bipolar-valued case Conclusion

Bipolar-valued choice characterisation

Definition

Let G(X, S) be a bipolar valued outranking graph. We may characterise a choice Y in G with the help of a bipolar-valued function Y : X → [−1, 1]: where x ∈ Y ⇔ Y (x) > 0, ∀x ∈ X.

Example(s)

• Y (a) = −0.6,

Y (b) = 0.6, Y (c) = −0.6, Y (d) = 1.0,

• Y (e) = 0.9 characterises the choice Y = {b, d, e}
• Y (a) = 0.6,

Y (b) = −0.6, Y (c) = 0.6, Y (d) = −1.0,

• Y (e) = −0.9 characterises the choice Y = {a, c}

Kernels in bipolar-valued digraphs Enumerating kernels in crisp digraphs Solving the bipolar-valued case Conclusion

Bipolar-valued choice characterisation

Definition

Let G(X, S) be a bipolar valued outranking graph. We may characterise a choice Y in G with the help of a bipolar-valued function Y : X → [−1, 1]: where x ∈ Y ⇔ Y (x) > 0, ∀x ∈ X.

Example(s)

• Y (a) = −0.6,

Y (b) = 0.6, Y (c) = −0.6, Y (d) = 1.0,

• Y (e) = 0.9 characterises the choice Y = {b, d, e}
• Y (a) = 0.6,

Y (b) = −0.6, Y (c) = 0.6, Y (d) = −1.0,

• Y (e) = −0.9 characterises the choice Y = {a, c}

Kernels in bipolar-valued digraphs Enumerating kernels in crisp digraphs Solving the bipolar-valued case Conclusion

Bipolar-valued choice characterisation

Definition

Let G(X, S) be a bipolar valued outranking graph. We may characterise a choice Y in G with the help of a bipolar-valued function Y : X → [−1, 1]: where x ∈ Y ⇔ Y (x) > 0, ∀x ∈ X.

Example(s)

• Y (a) = −0.6,

Y (b) = 0.6, Y (c) = −0.6, Y (d) = 1.0,

• Y (e) = 0.9 characterises the choice Y = {b, d, e}
• Y (a) = 0.6,

Y (b) = −0.6, Y (c) = 0.6, Y (d) = −1.0,

• Y (e) = −0.9 characterises the choice Y = {a, c}

Kernels in bipolar-valued digraphs Enumerating kernels in crisp digraphs Solving the bipolar-valued case Conclusion

Bipolar-valued choice characterisation

Definition

Let G(X, S) be a bipolar valued outranking graph. We may characterise a choice Y in G with the help of a bipolar-valued function Y : X → [−1, 1]: where x ∈ Y ⇔ Y (x) > 0, ∀x ∈ X.

Example(s)

• Y (a) = −0.6,

Y (b) = 0.6, Y (c) = −0.6, Y (d) = 1.0,

• Y (e) = 0.9 characterises the choice Y = {b, d, e}
• Y (a) = 0.6,

Y (b) = −0.6, Y (c) = 0.6, Y (d) = −1.0,

• Y (e) = −0.9 characterises the choice Y = {a, c}

Kernels in bipolar-valued digraphs Enumerating kernels in crisp digraphs Solving the bipolar-valued case Conclusion

Bipolar-valued choice characterisation

Definition

Let G(X, S) be a bipolar valued outranking graph. We may characterise a choice Y in G with the help of a bipolar-valued function Y : X → [−1, 1]: where x ∈ Y ⇔ Y (x) > 0, ∀x ∈ X.

Example(s)

• Y (a) = −0.6,

Y (b) = 0.6, Y (c) = −0.6, Y (d) = 1.0,

• Y (e) = 0.9 characterises the choice Y = {b, d, e}
• Y (a) = 0.6,

Y (b) = −0.6, Y (c) = 0.6, Y (d) = −1.0,

• Y (e) = −0.9 characterises the choice Y = {a, c}

Kernels in bipolar-valued digraphs Enumerating kernels in crisp digraphs Solving the bipolar-valued case Conclusion

Bipolar-valued choice characterisation

Definition

Let G(X, S) be a bipolar valued outranking graph. We may characterise a choice Y in G with the help of a bipolar-valued function Y : X → [−1, 1]: where x ∈ Y ⇔ Y (x) > 0, ∀x ∈ X.

Example(s)

• Y (a) = −0.6,

Y (b) = 0.6, Y (c) = −0.6, Y (d) = 1.0,

• Y (e) = 0.9 characterises the choice Y = {b, d, e}
• Y (a) = 0.6,

Y (b) = −0.6, Y (c) = 0.6, Y (d) = −1.0,

• Y (e) = −0.9 characterises the choice Y = {a, c}

Kernels in bipolar-valued digraphs Enumerating kernels in crisp digraphs Solving the bipolar-valued case Conclusion

The kernel characteristic equations

Theorem (Berge 1958, BPR 2006)

Let G(X, S) be a bipolar-valued digraph. The dominant kernels in G are characterised with bipolar-valued choices Y which verify: ( Y ◦ S)(x) =

y=x

max

y∈X min

• Y (y),

S(y, x)

• = −

Y (x), ∀x ∈ X; The absorbent kernels in G are characterised with bipolar-valued choices Y which verify: ( S ◦ Y )(x) =

y=x

max

y∈X min

• S(x, y),

Y (y)

• = −

Y (x), ∀x ∈ X.

Kernels in bipolar-valued digraphs Enumerating kernels in crisp digraphs Solving the bipolar-valued case Conclusion

The kernel characteristic equations

Theorem (Berge 1958, BPR 2006)

Let G(X, S) be a bipolar-valued digraph. The dominant kernels in G are characterised with bipolar-valued choices Y which verify: ( Y ◦ S)(x) =

y=x

max

y∈X min

• Y (y),

S(y, x)

• = −

Y (x), ∀x ∈ X; The absorbent kernels in G are characterised with bipolar-valued choices Y which verify: ( S ◦ Y )(x) =

y=x

max

y∈X min

• S(x, y),

Y (y)

• = −

Y (x), ∀x ∈ X.

Kernels in bipolar-valued digraphs Enumerating kernels in crisp digraphs Solving the bipolar-valued case Conclusion

The kernel characteristic equations

Theorem (Berge 1958, BPR 2006)

Let G(X, S) be a bipolar-valued digraph. The dominant kernels in G are characterised with bipolar-valued choices Y which verify: ( Y ◦ S)(x) =

y=x

max

y∈X min

• Y (y),

S(y, x)

• = −

Y (x), ∀x ∈ X; The absorbent kernels in G are characterised with bipolar-valued choices Y which verify: ( S ◦ Y )(x) =

y=x

max

y∈X min

• S(x, y),

Y (y)

• = −

Y (x), ∀x ∈ X.

Kernels in bipolar-valued digraphs Enumerating kernels in crisp digraphs Solving the bipolar-valued case Conclusion

Sharpness of bipolar-valued terms

Definition

1 Let

Y represent a bipolar-valued characterisation of a choice Y in

• G. We call

Y determined if Y (x) = 0 for all x ∈ X.

2 Let ˜

Y1 and ˜ Y2 represent two bipolar-valued characterisations

• f a choice in
• G. We say that ˜

Y1 is sharper than ˜ Y2 iff, for all x ∈ X, either ˜ Y1(x) ≤ ˜ Y2(x) ≤ 0, or 0 ≤ ˜ Y2(x) ≤ ˜ Y1(x).

Kernels in bipolar-valued digraphs Enumerating kernels in crisp digraphs Solving the bipolar-valued case Conclusion

Sharpness of bipolar-valued terms

Definition

1 Let

Y represent a bipolar-valued characterisation of a choice Y in

• G. We call

Y determined if Y (x) = 0 for all x ∈ X.

2 Let ˜

Y1 and ˜ Y2 represent two bipolar-valued characterisations

• f a choice in
• G. We say that ˜

Y1 is sharper than ˜ Y2 iff, for all x ∈ X, either ˜ Y1(x) ≤ ˜ Y2(x) ≤ 0, or 0 ≤ ˜ Y2(x) ≤ ˜ Y1(x).

Kernels in bipolar-valued digraphs Enumerating kernels in crisp digraphs Solving the bipolar-valued case Conclusion

Sharpness of bipolar-valued terms

Definition

1 Let

Y represent a bipolar-valued characterisation of a choice Y in

• G. We call

Y determined if Y (x) = 0 for all x ∈ X.

2 Let ˜

Y1 and ˜ Y2 represent two bipolar-valued characterisations

• f a choice in
• G. We say that ˜

Y1 is sharper than ˜ Y2 iff, for all x ∈ X, either ˜ Y1(x) ≤ ˜ Y2(x) ≤ 0, or 0 ≤ ˜ Y2(x) ≤ ˜ Y1(x).

Kernels in bipolar-valued digraphs Enumerating kernels in crisp digraphs Solving the bipolar-valued case Conclusion

General bipolar-valued case

Theorem (Bisdorff-Pirlot-Roubens 2006)

Let G(X, S) be a bipolar-valued digraph. A choice Y in G is a dominant (resp. absorbent) kernel in G iff there exists an associated Y which is a maximal sharp and determined solution of the corresponding bipolar-valued kernel equation system.

Comments

Based on an original fixpoint result (due to Pirlot), the proof of the bijection is fully constructive.

Kernels in bipolar-valued digraphs Enumerating kernels in crisp digraphs Solving the bipolar-valued case Conclusion

General bipolar-valued case

Theorem (Bisdorff-Pirlot-Roubens 2006)

Let G(X, S) be a bipolar-valued digraph. A choice Y in G is a dominant (resp. absorbent) kernel in G iff there exists an associated Y which is a maximal sharp and determined solution of the corresponding bipolar-valued kernel equation system.

Comments

Based on an original fixpoint result (due to Pirlot), the proof of the bijection is fully constructive.

Kernels in bipolar-valued digraphs Enumerating kernels in crisp digraphs Solving the bipolar-valued case Conclusion

General bipolar-valued case

Theorem (Bisdorff-Pirlot-Roubens 2006)

Let G(X, S) be a bipolar-valued digraph. A choice Y in G is a dominant (resp. absorbent) kernel in G iff there exists an associated Y which is a maximal sharp and determined solution of the corresponding bipolar-valued kernel equation system.

Comments

Based on an original fixpoint result (due to Pirlot), the proof of the bijection is fully constructive.

Kernels in bipolar-valued digraphs Enumerating kernels in crisp digraphs Solving the bipolar-valued case Conclusion

General bipolar-valued case

Theorem (Bisdorff-Pirlot-Roubens 2006)

Let G(X, S) be a bipolar-valued digraph. A choice Y in G is a dominant (resp. absorbent) kernel in G iff there exists an associated Y which is a maximal sharp and determined solution of the corresponding bipolar-valued kernel equation system.

Comments

Based on an original fixpoint result (due to Pirlot), the proof of the bijection is fully constructive.

Kernels in bipolar-valued digraphs Enumerating kernels in crisp digraphs Solving the bipolar-valued case Conclusion

Using finite domain solver

Comments

A finite domain solver implementation of the kernel characteristic equations in CHIP or GNU-prolog for instance allows to efficiently enumerate all possible choices with smart propagation of the constraints provided by the characteristic equations.

Run time statistics for the GNU-prolog FD-solver

Kernels in bipolar-valued digraphs Enumerating kernels in crisp digraphs Solving the bipolar-valued case Conclusion

Pirlot’s fixpoint for determined kernels

Algorithm (Pirlot 2006)

Let G(X, S) be a general bipolar-valued digraph.

1 Extract with AllKernels all determined dominant and

absorbent kernels K1, K2, . . . , Kj from the associated 0-cut graph G(X, S). Let T : Y → Y be the following transformation of bipolar-valued choice characterisations in G: T ( Y ) = −( Y ◦ S).

2 For each Kj: With

Y0(x) = 1.0 for all x ∈ Kj and

• Y0(x) = −1.0 for all x ∈ Kj, the iteration

Yi = T ( Yi−1) for i = 1, 2, . . . converges to a fixpoint ˜ Kj = T ( ˜ Kj), which is the corresponding maximal sharp bipolar-valued kernel characterisation.

Kernels in bipolar-valued digraphs Enumerating kernels in crisp digraphs Solving the bipolar-valued case Conclusion

Pirlot’s fixpoint for determined kernels

Algorithm (Pirlot 2006)

Let G(X, S) be a general bipolar-valued digraph.

1 Extract with AllKernels all determined dominant and

absorbent kernels K1, K2, . . . , Kj from the associated 0-cut graph G(X, S). Let T : Y → Y be the following transformation of bipolar-valued choice characterisations in G: T ( Y ) = −( Y ◦ S).

2 For each Kj: With

Y0(x) = 1.0 for all x ∈ Kj and

• Y0(x) = −1.0 for all x ∈ Kj, the iteration

Yi = T ( Yi−1) for i = 1, 2, . . . converges to a fixpoint ˜ Kj = T ( ˜ Kj), which is the corresponding maximal sharp bipolar-valued kernel characterisation.

Kernels in bipolar-valued digraphs Enumerating kernels in crisp digraphs Solving the bipolar-valued case Conclusion

Pirlot’s fixpoint for determined kernels

Algorithm (Pirlot 2006)

Let G(X, S) be a general bipolar-valued digraph.

1 Extract with AllKernels all determined dominant and

absorbent kernels K1, K2, . . . , Kj from the associated 0-cut graph G(X, S). Let T : Y → Y be the following transformation of bipolar-valued choice characterisations in G: T ( Y ) = −( Y ◦ S).

2 For each Kj: With

Y0(x) = 1.0 for all x ∈ Kj and

• Y0(x) = −1.0 for all x ∈ Kj, the iteration

Yi = T ( Yi−1) for i = 1, 2, . . . converges to a fixpoint ˜ Kj = T ( ˜ Kj), which is the corresponding maximal sharp bipolar-valued kernel characterisation.

Kernels in bipolar-valued digraphs Enumerating kernels in crisp digraphs Solving the bipolar-valued case Conclusion

Pirlot’s fixpoint for determined kernels

Algorithm (Pirlot 2006)

Let G(X, S) be a general bipolar-valued digraph.

1 Extract with AllKernels all determined dominant and

absorbent kernels K1, K2, . . . , Kj from the associated 0-cut graph G(X, S). Let T : Y → Y be the following transformation of bipolar-valued choice characterisations in G: T ( Y ) = −( Y ◦ S).

2 For each Kj: With

Y0(x) = 1.0 for all x ∈ Kj and

• Y0(x) = −1.0 for all x ∈ Kj, the iteration

Yi = T ( Yi−1) for i = 1, 2, . . . converges to a fixpoint ˜ Kj = T ( ˜ Kj), which is the corresponding maximal sharp bipolar-valued kernel characterisation.

Von Neumann’s dual fixpoint for acyclic digraphs

Theorem (v. Neumann 1944, Bisdorff 1997)

Let G(X, S) be a bipolar-valued digraph supporting a unique dominant kernel K, and let T 2 : Y → Y be the following dual transformation: T 2( Y ) = −

• − (

Y ◦ S) ◦ S

• .

With Y ∧

0 (x) = −1.0 and

Y ∨

0 (x) = 1.0 for all x ∈ X, the dual

transformation converges to K ∧ and K ∨, a low and a high fixpoint such that K = K ∧ ⊕ K ∨ renders the maximal sharp bipolar-valued characterisation of K. For an absorbent kernel, S is replaced with S

−1 in T 2.

Von Neumann’s dual fixpoint for acyclic digraphs

Theorem (v. Neumann 1944, Bisdorff 1997)

Let G(X, S) be a bipolar-valued digraph supporting a unique dominant kernel K, and let T 2 : Y → Y be the following dual transformation: T 2( Y ) = −

• − (

Y ◦ S) ◦ S

• .

With Y ∧

0 (x) = −1.0 and

Y ∨

0 (x) = 1.0 for all x ∈ X, the dual

transformation converges to K ∧ and K ∨, a low and a high fixpoint such that K = K ∧ ⊕ K ∨ renders the maximal sharp bipolar-valued characterisation of K. For an absorbent kernel, S is replaced with S

−1 in T 2.

Von Neumann’s dual fixpoint for acyclic digraphs

Theorem (v. Neumann 1944, Bisdorff 1997)

Let G(X, S) be a bipolar-valued digraph supporting a unique dominant kernel K, and let T 2 : Y → Y be the following dual transformation: T 2( Y ) = −

• − (

Y ◦ S) ◦ S

• .

With Y ∧

0 (x) = −1.0 and

Y ∨

0 (x) = 1.0 for all x ∈ X, the dual

transformation converges to K ∧ and K ∨, a low and a high fixpoint such that K = K ∧ ⊕ K ∨ renders the maximal sharp bipolar-valued characterisation of K. For an absorbent kernel, S is replaced with S

−1 in T 2.

Von Neumann’s dual fixpoint for acyclic digraphs

Theorem (v. Neumann 1944, Bisdorff 1997)

Let G(X, S) be a bipolar-valued digraph supporting a unique dominant kernel K, and let T 2 : Y → Y be the following dual transformation: T 2( Y ) = −

• − (

Y ◦ S) ◦ S

• .

With Y ∧

0 (x) = −1.0 and

Y ∨

0 (x) = 1.0 for all x ∈ X, the dual

transformation converges to K ∧ and K ∨, a low and a high fixpoint such that K = K ∧ ⊕ K ∨ renders the maximal sharp bipolar-valued characterisation of K. For an absorbent kernel, S is replaced with S

−1 in T 2.

Kernels in bipolar-valued digraphs Enumerating kernels in crisp digraphs Solving the bipolar-valued case Conclusion

Extension to the general bipolar-valued case

Algorithm (Bisdorff 1997)

Let G(X, S) be a general bipolar-valued digraph.

1 Extract with AllKernels all dominant and absorbent kernels

K1, K2, . . . , Kj from the associated 0-cut graph G(X, S).

2 Associate to each Kj a partially defined, bipartite digraph

• G/Kj(X,

S/Kj) supporting exactly the unique kernel Kj.

3 Use the v. Neumann dual fixpoint iteration T 2 for computing

in turn ˜ Kj in each partial graph G/Kj.

Kernels in bipolar-valued digraphs Enumerating kernels in crisp digraphs Solving the bipolar-valued case Conclusion

Extension to the general bipolar-valued case

Algorithm (Bisdorff 1997)

Let G(X, S) be a general bipolar-valued digraph.

1 Extract with AllKernels all dominant and absorbent kernels

K1, K2, . . . , Kj from the associated 0-cut graph G(X, S).

2 Associate to each Kj a partially defined, bipartite digraph

• G/Kj(X,

S/Kj) supporting exactly the unique kernel Kj.

3 Use the v. Neumann dual fixpoint iteration T 2 for computing

in turn ˜ Kj in each partial graph G/Kj.

Kernels in bipolar-valued digraphs Enumerating kernels in crisp digraphs Solving the bipolar-valued case Conclusion

Extension to the general bipolar-valued case

Algorithm (Bisdorff 1997)

Let G(X, S) be a general bipolar-valued digraph.

1 Extract with AllKernels all dominant and absorbent kernels

K1, K2, . . . , Kj from the associated 0-cut graph G(X, S).

2 Associate to each Kj a partially defined, bipartite digraph

• G/Kj(X,

S/Kj) supporting exactly the unique kernel Kj.

3 Use the v. Neumann dual fixpoint iteration T 2 for computing

in turn ˜ Kj in each partial graph G/Kj.

Kernels in bipolar-valued digraphs Enumerating kernels in crisp digraphs Solving the bipolar-valued case Conclusion

Extension to the general bipolar-valued case

Algorithm (Bisdorff 1997)

Let G(X, S) be a general bipolar-valued digraph.

1 Extract with AllKernels all dominant and absorbent kernels

K1, K2, . . . , Kj from the associated 0-cut graph G(X, S).

2 Associate to each Kj a partially defined, bipartite digraph

• G/Kj(X,

S/Kj) supporting exactly the unique kernel Kj.

3 Use the v. Neumann dual fixpoint iteration T 2 for computing

in turn ˜ Kj in each partial graph G/Kj.

Example: Condorcet ruled social choice

Reconsider the seven voters {1, 2, . . . , 7} on a set X = {a, b, c, d, e} of five alternatives. (From A. Taylor 2005, p.33)

b c d e a

Absorbent kernel {e} Pirlot fixpoint for kernel {e} iter. a b c d e

• 7/7
• 7/7
• 7/7
• 7/7

7/7 1

• 3/7
• 5/7
• 1/7
• 5/7

7/7 2

• 3/7
• 5/7
• 1/7
• 5/7

1/7 3

• 1/7
• 1/7
• 1/7
• 1/7

1/7

Example: Condorcet ruled social choice

Reconsider the seven voters {1, 2, . . . , 7} on a set X = {a, b, c, d, e} of five alternatives. (From A. Taylor 2005, p.33)

b c d e a

Absorbent kernel {e}

• v. Neumann fixpoint for kernel {e}

it. fp a b c d e e Y ∧

• 7/7
• 7/7
• 7/7
• 7/7
• 7/7

e Y ∨ 7/7 7/7 7/7 7/7 7/7 1 e Y ∧

• 3/7
• 5/7
• 1/7
• 5/7

7/7 e Y ∨ 7/7 7/7 7/7 7/7 7/7 2 e Y ∧

• 3/7
• 5/7
• 1/7
• 5/7

1/7 e Y ∨

• 1/7
• 1/7
• 1/7
• 1/7

1/7 3 e Y ∧

• 1/7
• 1/7
• 1/7
• 1/7

1/7 e Y ∨

• 1/7
• 1/7
• 1/7
• 1/7

1/7

Example: Outranking based decision

Reconsider the set X1 = {a, b, c, d, e} of decision alternatives evaluated on a coherent family of five criteria of equal significance. We observe two dominant – {b} and {c} –, and two absorbent kernels – {d} and {e} –.

b c e d a

The dominant and absorbent kernels Pirlot fixpoint for absorbent kernel {e} iteration a b c d e

• 1.0
• 1.0
• 1.0
• 1.0

1.0 1

• 0.4
• 0.6
• 0.6
• 0.2

1.0 2

• 0.4
• 0.6
• 0.6
• 0.2

0.2 3

• 0.2
• 0.2
• 0.2
• 0.2

0.2

Example: Outranking based decision

Reconsider the set X1 = {a, b, c, d, e} of decision alternatives evaluated on a coherent family of five criteria of equal significance. We observe two dominant – {b} and {c} –, and two absorbent kernels – {d} and {e} –.

b c e d a

The dominant and absorbent kernels

• v. Neumann fixpoint for absorbent kernel {e}

it. fp a b c d e e Y ∧

• 1.0
• 1.0
• 1.0
• 1.0
• 1.0

e Y ∨ 1.0 1.0 1.0 1.0 1.0 1 e Y ∧

• 0.4
• 0.6
• 0.6
• 0.2

0.0 e Y ∨ 0.0 0.0 0.0 0.0 0.2 2 e Y ∧

• 0.2
• 0.2
• 0.2
• 0.2

0.0 e Y ∨ 0.0 0.0 0.0 0.0 0.2

Kernels in bipolar-valued digraphs Enumerating kernels in crisp digraphs Solving the bipolar-valued case Conclusion

Concluding Remarks

In this communication we have presented: minimal dominant or absorbent choices in a bipolar valued

• utranking graph;

dominant and absorbent kernels in the same graph; algorithms for computing minimal choices and kernels; the algebraic characterisation of kernels via bipolar valued kernel equation systems; fixpoint based algorithms for computing bipolar valued kernel characterisations.

Kernels in bipolar-valued digraphs Enumerating kernels in crisp digraphs Solving the bipolar-valued case Conclusion

References

• R. Bisdorff

The Python digraph implementation for RuBy: User Manual. University of Luxembourg, http://sma.uni.lu/bisdorff/Digraph, 2006.

• R. Bisdorff, M. Pirlot, M. Roubens

Choices and kernels in bipolar valued digraphs. European Journal of Operational Research, 175(1), November 2006, 155–170.

• R. Bisdorff, P. Meyer, M. Roubens

RuBy: a bipolar-valued outranking method for the best choice decision problem. SMA working paper (2006), University of Luxembourg, submittted