Oriented Coloring:

Jean-François CulusUniversité Toulouse 2 Grimm culus@univ-tlse2.fr

Marc DemangeEssec Sid, Paris demange@essec.fr

Complexity and Approximation

SOFSEM 2006

Presentation• 1. Introduction

What is an oriented coloring ?

•

•

• Notations: G=(V,E) graph G=(V,A) oriented graph

2. Complexity How difficult is it ?

3. Approximation How to solve it ?

Introduction: Oriented Coloration & Coloration

Coloration as vertices partition

Homomorphism

Oriented Homomorphism

Oriented Coloring asvertices partition

1. IntroductionHomomorphism

• Let G=(V,E) and K=(V’,E’) be graphs.• An homomorphism from G to K is an application

f: V V’ such that {x;y} E {f(x);f(y)} E’

x y a

z t c b

f(x)=f(t)=af(y)=bf(z)=c

G K

1. Introduction

Coloration and Homomorphism

• G admits a k-coloration if and only if

(G)or

it exists a k-graph K and an homomorphism from G to K.

there exists an homomorphism from G to Kk

K3

G

=minimum k such that G admits a k-coloring

Coloration as Vertices partition into independent sets

1. IntroductionOriented homomorphism

• Let G=(V,A) and K=(V’,A’) be oriented graphs.

• An oriented homomorphism from G to K is an application f: V V’ such that: (x;y) A (f(x);f(y)) A’

x y z a b

t u v c

f(x)=f(t)=f(v)=af(y)=f(u)=bf(z)=c

1. IntroductionOriented Coloring as Oriented Homomorphism

• Digraph G admits an oriented k-coloring iff

o(G)=

x y z

u v

there exists an oriented k-graph K and an oriented homomorphism from G to K.

the minimum k such that G admits an k-oriented coloring.

Call K-coloring

G K

1. IntroductionOriented Coloring as vertex partition

• An k-oriented coloring of digraph G=(V,A) is a k-partition of V into independent sets such that x,x’Vi; y,y’Vj; (x,y)A (y’,y) A

x x’

y y’

Unidirection property

Oriented coloring: Example

x y z

A B Non locality of the oriented coloring

Note: Digraphs are antisymmetric

X Y

2. Complexity: Plan

Oriented k-coloring Homomorphism

NP-complete caseNP-complete case

Polynomial Case: Oriented Tree

Extention ?

Extention ? Another polynomial case!

Def

2. Complexity: Homomorphism

• G=(V,A) digraph admits an oriented k-coloring iff there exists K an oriented k-graph such that G K

• Theorem [Bang-Jensen et al., 90]

T-coloring is NP-complete iff• Smaller tournament T :

Hom

T has 2 circuits

3-Oriented Coloring is Polynomial4-Oriented Coloring is NP-Complete

[Klostermeyer & al., 04]… even for connected graphH4

2. ComplexityPolynomial case

• Easy on oriented trees

Tree Oriented

o(G) ≤3 polynomial algorithm

Bipartite oriented graph

Circuit-free oriented graph

NP-complete !!

Sketch of proof: BipartiteReduction from 3-Sat

• L admits a H4-coloring

• •

T R

FB

H4

For each litteral xi

For each clause Cj

Cj= z1 z2 z3

L

2. Complexity: NP-Complete

• Theorem: k-Col is NP-Complete for k≥4 even if G is a Connected oriented graph

Planar Bounded degree

even if G is circuit free

even if G is a bipartite

Complexity: Bipartite and Planar?

For each litteral xi

• Reduction from Planar 3-Sat.For each clause

2. Complexity: Polynomial case

• k-colo is polynomial for complete multipartite oriented graphs.

G1 G2

x y

z

u t v

G1 is a cograph: [Golumbic, 80](G1) could be obtain in polynomial time.

o(G)= (G1) + (G2) +…+ (Gp)

3. Approximation: Plan

• Introduction: What is it?

Negative result !

Inapproximability

Analysis of the Greedy Algorithm

Positive Result Minimum Oriented Coloring (MOC)

3. ApproximationWhat is an approximation ?

• Min Oriented Coloring (MOC) Minimization problem• Let G be a n-digraph

Optimum: o(G); Worst: n; Algorithm A(G)

0 o(G) A(G) n

• Classical ratio :• Differential ratio:

r(n) = o(G) / A(G) ≤ 1

r(n) = (n-A(G)) / (n - o(G)) ≤ 1

Min |G|=n

3. ApproximationReduction from Max Independent Set (MIS)

• Theorem: There exists a reduction from MIS to MOC transforming any differential ratio r(n) for the MOC into a r(3n) ratio for MIS.

• Corollary: If PNP, then Min Oriented Coloring is not approximable within a constant differential ratio.

If PZPP, then Min Oriented Coloring is not approximable within a differential ratio of O(nε-1), ε>0.

For undirected graphs, all coloring problems are approximable within a constant differential ratio [Demange & al., Hassin & Lahav, Duh & Fürer]

3. ApproximationThe greedy algorithm (Ideas)

G

S1

S2

S1 independent set

S3

S2 independent set

S3 independent set

Si

Theorem [Jonhson,74] Greedy algorithm guarantee a ratio of O(log(n)/n) for Min Coloring Problem.

3. ApproximationGreedy Algorithm (Problem)

x y z

t u v

w a

Contradict Unidirection property

3. ApproximationThe greedy Algorithm (Solution)

G

S1

+(S1)-(S1)

Min(|-(S1)|;|+(S1)|)

S2

Theorem: Greedy algorithm guarantee a differential ratio of O( log2(n)/ (n log k) )

In case k boudedO(log2(n)/n)

References:References:

Oriented coloring: Eric Sopena: Oriented Graph Coloring

Discrete Mathematics 1990

Homomorphism

•

Approximation:

Hell, Nesetril(04) Graphs and HomomorphismsBang Jensen, Hell,MacGillivray The complexity of Colouring by Semicomplete digraphs, J. of Discrete Mathematics; 1998 Bang Jensen, Hell: The effect of 2 cycles on the complexity of coulouring by directed graphs, Discrete Mathematics; 1990Klostermayer & MacGillivray: Homomorphisms and oriented colorings of equivalence classes of oriented graphs, Discrete Mathematics (2004)

Ausiello, Crescenzi, Gambozi, Kann & al. Complexity and Approximation; 2003Demange, Grisoni, Paschos: Approximation results for the minimum graph coloring problem

•

culus@univ-tlse2.fr

demange@essec.fr

Sketch of Proof for Bipartite digraphsReduction from 3-Sat

• H4 -Coloring with H4: T R

FB

yT

xF

xT

xR

xB

yF

yR

yB

Complexity: For each litteral xi

One must be colored by T and the other by F

T R

FBDigraph Gi admits a H4-coloring

Gi

H4

Complexity:For each Clause Cj: z1 z2 z3

T R

FB

FT or F ?

R

T or B

B or F

F or R

B or T

R or T

R or F

F or B or T

B

T

RF

T

B

F or R

T

If one of the litteral is True, then digraph Gj admits a H4-coloringGj

H4

Clause Cj satisfies iff oriented Graph Gj admits a H4-coloring