On enumerating the kernels in a bipolar valued digraph Raymond - PowerPoint PPT Presentation

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) b a c e d 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 c e d { 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 c e d { 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) b { a , c } is a minimal domi- a c e nant choice. d

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

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 G is defined as N ± [ Y ] = � choice Y in � 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) b Y = { a , b , d , e } a e N + Y [ a ] = {} ; N + c Y [ b ] = { c } ; N + Y [ d ] = {} ; N + Y [ e ] = { e } d

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 Y = { b , d , e } N − Y [ b ] = { a , b } a e c N − Y [ d ] = { c } N − d 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) b N + { a , c } [ a ] = { a , b } a c e N + { a , c } [ c ] = { c , d , e } d

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) N − { b , d , e } [ b ] = { a , b } b N − { b , d , e } [ d ] = { c } a c e N − { b , d , e } [ e ] = { e } d

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 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 b a c e a c e d d

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 b a a c e c e d d

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 only if Y verifies one of the following conditions: Y is minimal dominant (resp. absorbent); 1 Y is dominant (resp. absorbent) and (maximal) independent; 2 Y is dominant (resp. absorbent) and (maximal) +irredundant 3 (resp. -irredundant).

Existence of kernels Proposition (Existence results) Every digraph supports minimal dominant (resp. absorbent) choices. 1 A transitive digraph always supports a dominant (resp. absorbent) 2 kernel and all its kernels are of same cardinality (K¨ onig, 1950). A symmetric digraph always supports a conjointly dominant and 3 absorbent kernel (Berge, 1958). An acyclic digraph always supports a unique dominant (resp. 4 absorbent) digraph (Von Neumann, 1944). If a digraph has no odd asymmetric circuits, it supports a dominant 5 (resp. absorbent) kernel (Richardson, 1953). The probability that a random digraph of order n possesses a kernel 6 tends to 1 when n → ∞ (De la Vega, 1990). Almost every random digraph of order n contains only kernels K 7 such that C n − 1 . 43 ≤ | K | ≤ C n + 2 . 11 where C n = 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 )]: Y 1 = Y ∪ { x } ( K + , K − ) ← ( K + , K − ) ∪ AllKernels ( Y 1 , ( K + , K − )) return ( K + , K − )

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 : � � y � = x − � ∆ ind ( Y ) = y ∈ Y min min S( x , y ) . x ∈ Y The degree of dominance of a choice Y in � G : �� ∆ dom ( Y ) = min x �∈ Y max S( y , x ) . y ∈ Y The degree of absorbency of a choice Y in � G : �� ∆ abs ( Y ) = min x �∈ Y max S( x , y ) . y ∈ 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 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 � = x ( � Y ◦ � Y ( y ) , � = − � S)( x ) = y ∈ X min max S( y , x ) Y ( x ) , ∀ x ∈ X ; The absorbent kernels in � G are characterised with bipolar-valued choices � Y which verify: �� y � = x ( � S ◦ � S( x , y ) , � = − � Y ( x ) , ∀ x ∈ X . Y )( x ) = y ∈ X min max Y ( y )

On enumerating the kernels in a bipolar valued digraph Raymond - PowerPoint PPT Presentation

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,

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