The Rank Width of Directed graphs

The Rank Width of Directed graphs
Mamadou Moustapha
KANTÉ
LaBRI, Université Bordeaux 1, CNRS.
November 08 2007
JGA 2007 (Paris)
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Introduction
Introduction
Clique-width based under algebraic expressions, is defined for directed
graphs as well as undirected graphs.
Many algorithms that output a tree-decomposition of width at most k if the
tree-width of the graph is at most k (k is fixed).
For k ≥ 4, no known algorithm that outputs an expression of width k if the
clique-width of the graph is at most k .
Oum defines the notion of rank-width of an undirected graph and give an
O (n3 ) time algorithm that outputs a layout of witdth k of G if rank-width of
G at most k .
We extend the notion of rank-width to directed graphs
=⇒ Polynomial time approximation algorithm for clique-width of directed
graphs.
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Introduction
We have two notions of rank-width of directed graphs:
One based on ranks of GF (2)-matrices.
Another based on ranks of GF (4)-matrices.
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Introduction
Plan
1
Preliminaries
2
Rank-width of directed graphs
Bi-rank-width
Rank-width over GF (4)
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Preliminaries
Layout
Let f : 2V → N be a symmetric function.
A layout of f is a pair (T , L ) where T is a tree of degree ≤ 3 and
L : V → LT is a bijection of the set of vertices of G onto the set of leaves
of T .
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Preliminaries
Layout
9
1
8
2
7
3
4
T
5
6
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Preliminaries
Layout
Let f : 2V → N be a symmetric function.
A layout of f is a pair (T , L ) where T is a tree of degree ≤ 3 and
L : V → LT is a bijection of the set of vertices of G onto the set of leaves
of T .
Any edge e of T induces a bipartition (Xe , Ye ) of the set LT , thus a
bipartition (X , V − X ) of V :wd (e) = f (X ).
wd (T , L ) = max{wd (e ) | e edge of T }.
The rank-width of f , rwd (f ), is the minimum width over all layouts of f .
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Preliminaries
Rank-width of undirected graphs
For X ⊆ VG , we let cutrkG (X ) = rk (AG [X , VG − X ]) where rk is the matrix
rank function over GF (2).
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Preliminaries
Rank-width of undirected graphs
1
9
2
8
3
7
6
rk(AG [A, B]) = 2
4
B
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1
0
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0
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0
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A
G
AG [A, B] =
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Preliminaries
Rank-width of undirected graphs
For X ⊆ VG , we let cutrkG (X ) = rk (AG [X , VG − X ]) where rk is the matrix
rank function over GF (2).
Rank-width of G
The rank-width of G, denoted by rwd (G), is the rank-width of cutrkG .
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Preliminaries
Clique-width and rank-width
Clique-width
⊕ = disjoint union, ηi ,j = edge-cretation, ρi →j = relabeling.
for directed graphs, we use αi ,j instead of ηi ,j .
Fk = {⊕, ηi ,j , ρi →j | i , j ∈ [k ], i 6= j } and Ck = {i | i ∈ [k ]}
cwd (G) = min{k | G = val (t ) ∧ t ∈ T (Fk , Ck )}
Clique-width and Rank-width (Oum and Seymour)
rwd (G) ≤ cwd (G) ≤ 2rwd (G)+1 − 1.
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Rank-width of directed graphs
Bi-rank-width
Plan
1
Preliminaries
2
Rank-width of directed graphs
Bi-rank-width
Rank-width over GF (4)
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Rank-width of directed graphs
Bi-rank-width
Bi-rank-width
Let G = (VG , EG ) be a directed graph.
AG the adjacency matrix of G over GF (2) such that AG [x , y ] = 1 if and only
if (x , y ) ∈ EG .
For X ⊆ VG , we let A+
G [X , V − X ] = AG [X , V − X ] and
T
A−
G [X , V − X ] = AG [V − X , X ] .
(bi )
−
For X ⊆ VG we let cutrkG (X ) = rk (A+
G [X , V − X ]) + rk (AG [X , V − X ]).
(bi )
The function cutrkG
is symmetric and submodular.
(bi )
The bi-rank-width of G, denoted by brwd (G), is the rank-width of cutrkG .
−
→
brwd ( G ) = 2 · rwd (G).
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Rank-width of directed graphs
Bi-rank-width
Vectorial colorings
Let k ∈ N, B = {0, 1}. A Bk -coloring of a graph G is a mapping γ : VG → Bk .
x ∈ VG has color i (among others) iff γ(x )[i ] (the i-th component of γ(x )) is
1.
Graph Products
M , M ′ be (k × ℓ)-matrices, N and P be respectively (k × m) and
(ℓ × m)-matrices.
G, Bk -colored and H , Bℓ -colored. We let K = G ⊗M ,M ′ ,N ,P H where
VK = VG ∪ VH ,
EK = EG ∪ EH ∪ {(x , y ) | x ∈ VG , y ∈ VH ∧ γG (x ) · M · γH (y )T = 1}
∪ {(y , x ) | x ∈ VG , y ∈ VH ∧ γG (x ) · M ′ · γH (y )T = 1},
γK (x ) =
(
γG (x ) · N
γH (x ) · P
if x ∈ VG ,
if x ∈ VH .
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Rank-width of directed graphs
Bi-rank-width
Vectorial colorings
Let k ∈ N, B = {0, 1}. A Bk -coloring of a graph G is a mapping γ : VG → Bk .
x ∈ VG has color i (among others) iff γ(x )[i ] (the i-th component of γ(x )) is
1.
Graph Products
M , M ′ be (k × ℓ)-matrices, N and P be respectively (k × m) and
(ℓ × m)-matrices.
G, Bk -colored and H , Bℓ -colored. We let K = G ⊗M ,M ′ ,N ,P H where
VK = VG ∪ VH ,
EK = EG ∪ EH ∪ {(x , y ) | x ∈ VG , y ∈ VH ∧ γG (x ) · M · γH (y )T = 1}
∪ {(y , x ) | x ∈ VG , y ∈ VH ∧ γG (x ) · M ′ · γH (y )T = 1},
γK (x ) =
(
γG (x ) · N
γH (x ) · P
if x ∈ VG ,
if x ∈ VH .
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Rank-width of directed graphs
Bi-rank-width
Vectorial colorings
Let k ∈ N, B = {0, 1}. A Bk -coloring of a graph G is a mapping γ : VG → Bk .
x ∈ VG has color i (among others) iff γ(x )[i ] (the i-th component of γ(x )) is
1.
Graph Products
M , M ′ be (k × ℓ)-matrices, N and P be respectively (k × m) and
(ℓ × m)-matrices.
G, Bk -colored and H , Bℓ -colored. We let K = G ⊗M ,M ′ ,N ,P H where
VK = VG ∪ VH ,
EK = EG ∪ EH ∪ {(x , y ) | x ∈ VG , y ∈ VH ∧ γG (x ) · M · γH (y )T = 1}
∪ {(y , x ) | x ∈ VG , y ∈ VH ∧ γG (x ) · M ′ · γH (y )T = 1},
γK (x ) =
(
γG (x ) · N
γH (x ) · P
if x ∈ VG ,
if x ∈ VH .
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Rank-width of directed graphs
Bi-rank-width
Some definitions ...
u = the graph with a single vertex colored by u ∈ Bn and Cn = {u | u ∈ Bn }.
Rn = {⊗M ,M ′ ,N ,P } where M , M ′ , N , P are matrices of size at most n.
val (t ) = the graph defined by a term t ∈ T (Rn , Cn ). This graph is the value
of the term in the corresponding algebra.
The minimum k such that G is the value of a term t in T (Rn , Cn ) is denoted
by mrk (G).
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Rank-width of directed graphs
Bi-rank-width
Properties
Theorem 1
(i) brwd (G) ≤ 2 · cwd (G).
(ii) 12 brwd (G) ≤ mrk (G) ≤ brwd (G).
(iii) mrk (G) ≤ cwd (G) ≤ 2mrk (G)+1 − 1
Remark
To prove (ii) we encode directed graphs by undirected ones and use a characterization
of rank-width of undirected graphs by operations based on bilinear transformations
(Courcelle and Kanté (2007).
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Rank-width of directed graphs
Bi-rank-width
Recognizing bi-rank-width
For G we let B (G) be the simple undirected bipartite graph associated
with G where:
VB(G) = VG × {1, 4},
EB(G) = {{(v , 1), (w , 4)} | (v , w ) ∈ EG }.
One can prove easily that any layout of bi-rank-width k of G is also a
layout of rank-width k of B (G) such that (x , 1) and (x , 4) are children of
the same node.
Oum and Hliněný have proved that if such decomposition exists, it can be
constructed in O (n3 )-time.
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Rank-width of directed graphs
Bi-rank-width
Recognizing bi-rank-width
Algorithm for fixed k
Construct in O (n2 )-time the graph B (G).
Apply the algorithm of Oum and Hliněný.
If it outputs a layout of B (G), transforms it into a layout of G by deleting
the leaves (x , 1) and (x , 4).
Otherwise confirms that the bi-rank-width is at least k + 1.
Remark
We can transform this algorithm into one that constructs a clique-width expression of
width at most 22k +1 − 1 if cwd (G) ≤ k .
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Rank-width of directed graphs
Rank-width over GF (4)
Plan
1
Preliminaries
2
Rank-width of directed graphs
Bi-rank-width
Rank-width over GF (4)
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Directed Rank-width
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Rank-width of directed graphs
Rank-width over GF (4)
Matrices over GF (4)
We can use matrices over GF (4) to represent directed graphs.
FG [i , j ] =
(4 )

0



a

a2



1
iff (i , j ) ∈
/ EG
iff (i , j ) ∈ EG
iff (j , i ) ∈ EG
iff (i , j ) ∈ EG
∧
∧
∧
∧
(j , i ) ∈
/ EG
(j , i ) ∈
/ EG
(i , j ) ∈
/ EG
(j , i ) ∈ EG
(4 )
For X ⊆ VG , we let rkG (X ) = rk (FG [X , X ]). It is clear that rkG is symmetric
and sub-modular.
(4 )
The GF (4)-rank-width is the rank-width of rkG .
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Rank-width of directed graphs
Rank-width over GF (4)
GF (4)-rank-width
Proposition
rwd (4) (G) ≤ brwd (G).
rwd (4) (G) ≤ cwd (G) ≤ 4rwd
(4) (G)+1
− 1.
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Rank-width of directed graphs
Rank-width over GF (4)
GF (4)-rank-width
Proposition
rwd (4) (G) ≤ brwd (G).
rwd (4) (G) ≤ cwd (G) ≤ 4rwd
(4) (G)+1
− 1.
Proof
−
2
FG [X , X ] = a · A+
G [X , X ] + a · AG [X , X ].
We use the algebraic characterization of GF (4)-rank-width.
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Rank-width of directed graphs
Rank-width over GF (4)
Algebraic characterization
F = {0, 1, a, a2 }. An Fk -coloring of a graph G is a mapping κ : VG → Fk .
An Fk -colored graph is a triple G =< VG , EG , κG > where κG is an
Fk -coloring of < VG , EG >.
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Rank-width of directed graphs
Rank-width over GF (4)
Algebraic characterization
Graph Product
M (k × ℓ)-matrix over GF (4) and N and P (k × m) and (ℓ × m)-matrices
over GF (4).
G, Fk -colored and H , Fℓ -colored, we let K = G ⊗M ,N ,P H
VK = VG ∪ VH ,
EK = EG ∪ EH ∪ {(x , y ) | x ∈ VG , y ∈ VH , κG (x ) · M · κH (y )T ∈ {1, a}}
∪ {(y , x ) | x ∈ VG , y ∈ VH , κG (x ) · M · κH (y )T ∈ {1, a2 }},
κK (x ) =
(
κG (x ) · N
κH (x ) · P
if x ∈ VG ,
if x ∈ VH .
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Rank-width of directed graphs
Rank-width over GF (4)
Algebraic characterization
Graph Product
M (k × ℓ)-matrix over GF (4) and N and P (k × m) and (ℓ × m)-matrices
over GF (4).
G, Fk -colored and H , Fℓ -colored, we let K = G ⊗M ,N ,P H
VK = VG ∪ VH ,
EK = EG ∪ EH ∪ {(x , y ) | x ∈ VG , y ∈ VH , κG (x ) · M · κH (y )T ∈ {1, a}}
∪ {(y , x ) | x ∈ VG , y ∈ VH , κG (x ) · M · κH (y )T ∈ {1, a2 }},
κK (x ) =
(
κG (x ) · N
κH (x ) · P
if x ∈ VG ,
if x ∈ VH .
Theorem
GF (4)-rank-width at most k if and only if G is the value of a term written with
symbols ⊗M ,N ,P and constants u ∈ Fk
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Rank-width of directed graphs
Rank-width over GF (4)
Conclusion
2 notions of rank-width : Bi-rank-width and GF (4)-rank-width.
Bi-rank-width is based on GF (2)-matrices and we propose a polynomial
algorithm to recongnize graphs of bi-rank-width ≤ k for fixed k . This
implies a polynomial time approximation algorithm for clique-width.
Algebraic operations are also proposed even if they don’t characterize
bi-rank-width, they allow to solve in Courcelle-manner MSO problems.
GF (4)-rank-width similar to rank-width is characterized by algebraic
operations, but we don’t have recognition algorithms. It is a challenge to
give one.
Another challenge is to define operation like local complementation that
does’t increase GF (4)-rank-width and use it to characterize
GF (4)-rank-width at most k by forbidden graphs.
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