Sur quelques g´ en´ eralisations polynomiales de la d´ ecomposition modulaire. Vincent Limouzy LIAFA – Universit´ e Paris Diderot December 3, 2008 1 / 45
Outline of the Thesis Part I. Generalizations of Modular Decomposition • Homogeneous relations and modular decomposition. • Umodular decomposition: a new point of view. Part II. Efficient Algorithms • Overlap Components. • NLC-2 graphs recognition algorithm. 2 / 45
Outline 1 A brief Introduction to Homogeneous Relations First encounter Modular decomposition Results 2 Umodules Arbitrary relations Local congruence 2 Self complemented families Undirected graphs Tournaments 3 Overlap components 4 Perspectives Homogeneous relations Overlap components NLC-width 3 / 45
Basic definitions Modules and Modular decomposition Module 4 / 45
Basic definitions Modules and Modular decomposition Module 4 / 45
Basic definitions Modules and Modular decomposition Module Substitution / Contraction Contraction Substitution 4 / 45
Generalizing Why and How ? Modular decomposition Known generalizations • Social sciences, Role coloring:Everett & • Bioinformatics, Borgatti’91 proven NP-complete by Fiala & • Computer science Paulusma’05 that this problem • ... Desired properties of the generalizations • Polynomial computation • Compact encoding of the family • Good structural properties • ... • Decomposition tree 5 / 45
Summary Module A module is a set of vertices which have the same neighborhood outside. 6 / 45
Summary Module A module is a set of vertices which have the same neighborhood outside. Role A “role” in a graph is a set of vertices which plays the same role. 6 / 45
Summary Module A module is a set of vertices which have the same neighborhood outside. Homogeneous Relations Homogeneous relation is something in between... Role A “role” in a graph is a set of vertices which plays the same role. 6 / 45
Homogeneous Relations 7 / 45
Homogeneous Relations Definition Let X be a finite set. A Homogeneous Relation is a collection of triples on X , noted H ( a | b , c ) fullfiling the following properties: 1 Reflexivity : H ( a | x , x ) , 2 Symmetry : H ( a | x , y ) ≡ H ( a | y , x ) and 3 Transitivity : H ( a | x , y ) and H ( a | y , z ) ⇒ H ( a | x , z ) 8 / 45
Homogeneous Relations Definition Let X be a finite set. A Homogeneous Relation is a collection of triples on X , noted H ( a | b , c ) fullfiling the following properties: 1 Reflexivity : H ( a | x , x ) , 2 Symmetry : H ( a | x , y ) ≡ H ( a | y , x ) and 3 Transitivity : H ( a | x , y ) and H ( a | y , z ) ⇒ H ( a | x , z ) H ( a | b , c ) a is said to be homogeneous with respect to b and c , or a does not distinguish b from c . 8 / 45
An example X = { a , b , c , d } Let H be defined as follows: H ( a | c , d ) , H ( a | b , b ) , H ( b | a , c ) , H ( b | c , d ) , H ( b | a , d ) , H ( c | a , a ) , H ( c | b , b ) , H ( c | d , d ) , H ( d | b , c ) , H ( d | a , a ) . Homogeneous relation ∼ Equivalence relations To each element x of X , thanks to the transitivity property we can associate an equivalence relation H x defined on X \ { x } 9 / 45
Homogeneous Relations: Representation Matrix representation Equivalence relation a b c d = { b } , { c , d } H a a 0 1 2 2 = { a , c , d } H b b 1 0 1 1 = { a } , { b } , { d } H c c 1 2 0 3 H d = { a } , { b , c } d 1 2 2 0 10 / 45
Graphic Homogeneous Relations Graphic A homogeneous relations H is graphic if there exists a graph G s.t. ∀ v of V ( G ) , H v = N ( v ) , N ( v ) Theorem A homogeneous relation H is graphic iff ∀ x , y , z ∈ X , H does not contain: 1 H ( x | y , z ) ∧ H ( y | x , z ) ∧ H ( z | x , y ) 2 H ( x | y , z ) ∧ H ( y | x , z ) ∧ H ( z | x , y ) 11 / 45
Graphic Homogeneous Relations Graphic A homogeneous relations H is graphic if there exists a graph G s.t. ∀ v of V ( G ) , H v = N ( v ) , N ( v ) Theorem A homogeneous relation H is graphic iff ∀ x , y , z ∈ X , H does not contain: 1 H ( x | y , z ) ∧ H ( y | x , z ) ∧ H ( z | x , y ) 2 H ( x | y , z ) ∧ H ( y | x , z ) ∧ H ( z | x , y ) a = { b , c } , { d } H a = { a , c } , { d } H b = { a , b , d } H c c d = { a , b } , { c } b H d 11 / 45
Homogeneous Relations Properties Local Congruence Maximum number of classes associated to an element. Example H a = { b } , { c , d } H b = { a , c , d } H c = { a } , { b } , { c } H d = { a } , { b , c } 12 / 45
Modules Definition A Module in a Homogeneous relation H is a set M such that: ∀ m , m ′ ∈ M and ∀ x ∈ X \ M we have: H ( x | mm ′ ) Family of modules M H : family of modules. Example H a = { b } , { c , d } ; H b = { a , c , d } ; H c = { a } , { b } , { d } ; H d = { a } , { b , c } . The modules are { a } , { b } , { c } , { d } , { a , b , c , d } and { c , d } . 13 / 45
Basic Properties Definition (Overlap) Let A and B be subsets of X . A overlaps B if: � B ≡ A \ B � = ∅ and B \ A � = ∅ and A ∩ B � = ∅ A � A B Proposition (Intersecting family) Let H be a homogeneous relation on X , and let M and M ′ modules of H � M ′ then: s.t. M � M ∩ M ′ ∈ M H and M ∪ M ′ ∈ M H Theorem (Gabow’95) M H can be stored in space O ( n 2 ) 14 / 45
Results on Homogeneous Relations Modular Decomposition On Arbitrary Homogeneous relations: O ( n 2 ) Primality O ( n 3 ) Decomposition algorithm: On good Homogeneous relations O ( n 2 ) Primality O ( n 2 ) Decomposition algorithm: Where n is the cardinality of the ground set X . Good Homogeneous Relations The modules family on good homogeneous relations forms a weakly partitive family. 15 / 45
Umodules 16 / 45
Umodules Definition Let H be a homogeneous relation defined on X , a Umodule U is a set such that: ∀ u , u ′ ∈ U and ∀ x , x ′ ∈ X \ U : H ( u | xx ′ ) ⇐ ⇒ H ( u ′ | xx ′ ) 17 / 45
Umodules Definition Let H be a homogeneous relation defined on X , a Umodule U is a set such that: ∀ u , u ′ ∈ U and ∀ x , x ′ ∈ X \ U : H ( u | xx ′ ) ⇐ ⇒ H ( u ′ | xx ′ ) H m ={x},{x'} m x m x m' x' m' x' H m' ={x},{x'} U U We have H ( m | xx ′ ) and H ( m ′ | xx ′ ) 17 / 45
Basic properties U H is the family of umodules. Proposition (Union closed) Let U and U ′ be two umodules of H such that U � � U ′ then: U ∪ U ′ ∈ U H 18 / 45
Crossing families Definition (Cross) Let A and B be two subsets of X . A crosses B if: • � B ≡ A � A � � B and A ∪ B � = X Definition (Crossing family) Let X be a finite set and F be a family of subset. F is said to be crossing if: • � B ∀ A , B ∈ F such that A � A ∪ B and A ∩ B belong to F . 19 / 45
Homogeneous relations of Local Congruence 2 ( LC 2 ) Proposition Let H be a homogeneous relation of Local Congruence 2 ( LC 2 )and : U H is a crossing family. 20 / 45
Homogeneous relations of Local Congruence 2 ( LC 2 ) Proposition Let H be a homogeneous relation of Local Congruence 2 ( LC 2 )and : U H is a crossing family. Sketch of Proof ∪ : from the previous proposition. B ∩ : Let A and B be two umodules. By hypothesis we have: b H ( a | x , b ) ⇐ ⇒ H ( y | x , b ) ⇐ ⇒ H ( z | x , b ) x H ( b | x , a ) ⇐ ⇒ H ( y | x , a ) ⇐ ⇒ H ( z | x , a ) y z we obtain: a H ( y | a , b ) ⇐ ⇒ H ( z | a , b ) A � 20 / 45
Homogeneous relations of Local Congruence 2 ( LC 2 ) Proposition Let H be a homogeneous relation of Local Congruence 2 ( LC 2 )and : U H is a crossing family. Sketch of Proof ∪ : from the previous proposition. B ∩ : Let A and B be two umodules. By hypothesis we have: b H ( a | x , b ) ⇐ ⇒ H ( y | x , b ) ⇐ ⇒ H ( z | x , b ) x H ( b | x , a ) ⇐ ⇒ H ( y | x , a ) ⇐ ⇒ H ( z | x , a ) y z we obtain: a H ( y | a , b ) ⇐ ⇒ H ( z | a , b ) A � Theorem (Gabow’95 & Bernath’04) Crossing families defined on a ground set X can be stored in O ( n 2 ) space. 20 / 45
Bipartitive families Let X be a finite set, and let B = {{ B 1 1 , B 2 1 } , . . . , { B 1 l , B 2 l }} be a set of bipartitions of X . Definition (Bipartitive families – Cunningham & Edmonds’80) B is a bipartitive family if for all overlapping bipartitions { B 1 k , B 2 k } and { B 1 j , B 2 j } we have: { B 1 k ∪ B 1 j , B 2 k ∩ B 2 { B 1 k ∪ B 2 j , B 2 k ∩ B 1 j } , j } ∈ B { B 2 k ∪ B 1 j , B 1 k ∩ B 2 { B 2 k ∪ B 2 j , B 1 k ∩ B 1 j } , j } B k 1 B k 2 B j 1 X X B j 2 1 U B k 1 ∩ B k 1 U B k 1 ∩ B k B j 1 B j 2 B j 2 B j 1 2 U B k 2 U B k 2 ∩ B k 1 2 ∩ B k B j 2 B j 2 B j B j 2 21 / 45
Bipartitive families Theorem (Cunningham & Edmonds’80) Let B be a bipartitive family defined on X There exists a unique unrooted tree encoding B . Its size is O ( n ) . 22 / 45
Self complemented families Definition Let H be a Homogeneous Relation defined on X . H is said to be self-complemented iff: ∀ U ∈ U H , X \ U belongs to U H Theorem Let U H be self-complemented then U H form a bipartitive family. 23 / 45
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