the rank width of directed graphskante/articles/kantejga07.pdf · oum deﬁnes the notion of...

university-logo

The Rank Width of Directed graphs

Mamadou Moustapha KANTÉ

LaBRI, Université Bordeaux 1, CNRS.

November 08 2007JGA 2007 (Paris)

(LaBRI, Universite Bordeaux 1, CNRS) Directed Rank-width 1 / 20

university-logo

Introduction

Introduction

Clique-width based under algebraic expressions, is defined for directedgraphs as well as undirected graphs.

Many algorithms that output a tree-decomposition of width at most k if thetree-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 theclique-width of the graph is at most k .

Oum defines the notion of rank-width of an undirected graph and give anO(n3) time algorithm that outputs a layout of witdth k of G if rank-width ofG at most k .

We extend the notion of rank-width to directed graphs

=⇒ Polynomial time approximation algorithm for clique-width of directedgraphs.


university-logo

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.


university-logo

Introduction

Plan

1 Preliminaries

2 Rank-width of directed graphsBi-rank-widthRank-width over GF(4)


university-logo

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 andL : V → LT is a bijection of the set of vertices of G onto the set of leavesof T .


university-logo

Preliminaries

Layout

1

56

7

8

9

T

2

3

4


university-logo

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 andL : V → LT is a bijection of the set of vertices of G onto the set of leavesof T .

Any edge e of T induces a bipartition (Xe,Ye) of the set LT , thus abipartition (X ,V −X) of V :wd(e) = f (X).

wd(T ,L) = max{wd(e) | e edge ofT}.

The rank-width of f , rwd(f ), is the minimum width over all layouts of f .


university-logo

Preliminaries

Rank-width of undirected graphs

For X ⊆ VG, we let cutrkG(X) = rk(AG[X ,VG −X ]) where rk is the matrixrank function over GF(2).


university-logo

Preliminaries

Rank-width of undirected graphs9

8

7

6

5

4

3

2

1

G

A

B

1

2

3

4

5 6 8 97

1 1

1 1

1 1

0 0 0

0 0

000 0 0

00

0

0

AG[A, B] =

rk(AG[A, B]) = 2


university-logo

Preliminaries

Rank-width of undirected graphs

For X ⊆ VG, we let cutrkG(X) = rk(AG[X ,VG −X ]) where rk is the matrixrank function over GF(2).

Rank-width of GThe rank-width of G, denoted by rwd(G), is the rank-width of cutrkG.


university-logo

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.


university-logo

Rank-width of directed graphs Bi-rank-width

Plan

1 Preliminaries



university-logo


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 onlyif (x,y) ∈ EG.

For X ⊆ VG, we let A+G [X ,V −X ] = AG[X ,V −X ] andA−G [X ,V −X ] = AG[V −X ,X ]

T .

For X ⊆ VG we let cutrk(bi)G (X) = rk(A

+G [X ,V −X ])+ rk(A

−G [X ,V −X ]).

The function cutrk (bi)G is symmetric and submodular.

The bi-rank-width of G, denoted by brwd(G), is the rank-width of cutrk (bi)G .

brwd(−→G ) = 2 · rwd(G).


university-logo


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)) is1.

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 if x ∈ VG,γH(x) ·P if x ∈ VH .


university-logo


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 valueof the term in the corresponding algebra.

The minimum k such that G is the value of a term t in T (Rn,Cn) is denotedby mrk(G).


university-logo


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

RemarkTo prove (ii) we encode directed graphs by undirected ones and use a characterizationof rank-width of undirected graphs by operations based on bilinear transformations(Courcelle and Kanté (2007).


university-logo


Recognizing bi-rank-width

For G we let B(G) be the simple undirected bipartite graph associatedwith 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 alayout of rank-width k of B(G) such that (x,1) and (x,4) are children ofthe same node.

Oum and Hliněný have proved that if such decomposition exists, it can beconstructed in O(n3)-time.


university-logo


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 deletingthe leaves (x,1) and (x,4).

Otherwise confirms that the bi-rank-width is at least k +1.

RemarkWe can transform this algorithm into one that constructs a clique-width expression ofwidth at most 22k+1 −1 if cwd(G) ≤ k.


university-logo

Rank-width of directed graphs Rank-width over GF (4)

Plan

1 Preliminaries



university-logo


Matrices over GF(4)

We can use matrices over GF(4) to represent directed graphs.

FG[i, j] =

0 iff (i, j) /∈ EG ∧ (j, i) /∈ EGa iff (i, j) ∈ EG ∧ (j, i) /∈ EGa2 iff (j, i) ∈ EG ∧ (i, j) /∈ EG1 iff (i, j) ∈ EG ∧ (j, i) ∈ EG

For X ⊆ VG, we let rk(4)G (X) = rk(FG[X ,X ]). It is clear that rk

(4)G is symmetric

and sub-modular.

The GF(4)-rank-width is the rank-width of rk (4)G .


university-logo


GF(4)-rank-width

Proposition

rwd (4)(G) ≤ brwd(G).

rwd (4)(G) ≤ cwd(G) ≤ 4rwd(4)(G)+1 −1.


university-logo


GF(4)-rank-width

Proposition

rwd (4)(G) ≤ brwd(G).

rwd (4)(G) ≤ cwd(G) ≤ 4rwd(4)(G)+1 −1.

Proof

FG[X ,X ] = a ·A+G [X ,X ]+ a

2 ·A−G [X ,X ].

We use the algebraic characterization of GF(4)-rank-width.


university-logo


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 anF

k -coloring of < VG,EG >.


university-logo



Graph Product

M (k × ℓ)-matrix over GF(4) and N and P (k ×m) and (ℓ×m)-matricesover 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 if x ∈ VG,κH(x) ·P if x ∈ VH .


university-logo



Graph Product

M (k × ℓ)-matrix over GF(4) and N and P (k ×m) and (ℓ×m)-matricesover 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 if x ∈ VG,κH(x) ·P if x ∈ VH .

TheoremGF(4)-rank-width at most k if and only if G is the value of a term written withsymbols ⊗M,N,P and constants u ∈ Fk


university-logo


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 polynomialalgorithm to recongnize graphs of bi-rank-width ≤ k for fixed k . Thisimplies a polynomial time approximation algorithm for clique-width.

Algebraic operations are also proposed even if they don’t characterizebi-rank-width, they allow to solve in Courcelle-manner MSO problems.

GF(4)-rank-width similar to rank-width is characterized by algebraicoperations, but we don’t have recognition algorithms. It is a challenge togive one.

Another challenge is to define operation like local complementation thatdoes’t increase GF(4)-rank-width and use it to characterizeGF(4)-rank-width at most k by forbidden graphs.


IntroductionPreliminariesRank-width of directed graphsBi-rank-widthRank-width over GF(4)

the rank width of directed graphskante/articles/kantejga07.pdf · oum deﬁnes the notion of...

Documents