1. Oriented Coloring: Complexity and Approximation
Jean-François Culus Université Toulouse 2 Grimm Marc Demange Essec Sid, Paris
SOFSEM 2006

2. Presentation Notations: 1. Introduction What is an oriented coloring ?
G=(V,E) graph G=(V,A) oriented graph
2. Complexity How difficult is it ?
3. Approximation How to solve it ?

3. Introduction: Oriented Coloration & Coloration
Coloration as vertices partition
Homomorphism
Oriented Coloring as vertices partition
Oriented Homomorphism

4. 1. Introduction Homomorphism
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)=a
f(y)=b
f(z)=c K G

5. 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.
=minimum k such that G admits a k-coloring
there exists an homomorphism from G to Kk
Coloration as Vertices partition into independent sets K3 G

6. 1. Introduction Oriented 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)=a f(y)=f(u)=b f(z)=c

7. 1. Introduction Oriented 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

8. 1. Introduction Oriented 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

9. Oriented coloring: Example
x y z
Non locality of the oriented coloring
A B
Note: Digraphs are antisymmetric
X Y

10. 2. Complexity: Plan Oriented k-coloring Homomorphism NP-complete case
Extention ?
Extention ?
Polynomial Case: Oriented Tree
Another polynomial case!

11. 2. Complexity: Homomorphism
Def
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 Polynomial 4-Oriented Coloring is NP-Complete
[Klostermeyer & al., 04] … even for connected graph H4

12. 2. Complexity Polynomial case
Easy on oriented trees
o(G) ≤3 polynomial algorithm
Tree
Oriented
Circuit-free oriented graph
Bipartite oriented graph
NP-complete !!

13. Sketch of proof: Bipartite Reduction from 3-Sat
L admits a H4-coloring
For each litteral xi
For each clause Cj Cj= z1  z2  z3 L

14. 2. Complexity: NP-Complete
Theorem: k-Col is NP-Complete for k≥ even if G is
a Connected oriented graph
even if G is a bipartite
Planar
Bounded degree
even if G is circuit free

15. Complexity: Bipartite and Planar?
Reduction from Planar 3-Sat.
For each clause
For each litteral xi

16. 2. Complexity: Polynomial case
k-colo is polynomial for complete multipartite oriented graphs G G2 x y z u t v
o(G)= (G1) + (G2) +…+ (Gp)
G1 is a cograph: [Golumbic, 80] (G1) could be obtain in polynomial time.

17. 3. Approximation: Plan Introduction: What is it? Negative result !
Inapproximability
Analysis of the Greedy Algorithm
Positive Result
Minimum Oriented Coloring (MOC)

18. 3. Approximation What is an approximation ?
Min Oriented Coloring (MOC) Minimization problem
Let G be a n-digraph Optimum: o(G); Worst: n; Algorithm A(G) o(G) A(G) n
Classical ratio :
Differential ratio:
Min|G|=n
r(n) = o(G) / A(G) ≤ 1
r(n) = (n-A(G)) / (n - o(G)) ≤ 1

19. 3. Approximation Reduction 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 PNP, then Min Oriented Coloring is not approximable within a constant differential ratio.
For undirected graphs, all coloring problems are approximable within a constant differential ratio [Demange & al., Hassin & Lahav, Duh & Fürer]
If PZPP, then Min Oriented Coloring is not approximable within a differential ratio of O(nε-1), ε>0.

20. 3. Approximation The greedy algorithm (Ideas)
S1 independent set S1
S2 independent set
S3 independent set S2 S3
Theorem [Jonhson,74] Greedy algorithm guarantee a ratio of O(log(n)/n) for Min Coloring Problem. Si G

21. 3. Approximation Greedy Algorithm (Problem)
x y z t u v w a
Contradict Unidirection property

22. 3. Approximation The greedy Algorithm (Solution)
Min(|-(S1)|;|+(S1)|)
Theorem: Greedy algorithm guarantee a differential ratio of O( log2(n)/ (n log k) )
-(S1)
+(S1) S1
In case k bouded O(log2(n)/n) G S2

23. Oriented coloring: Eric Sopena: Oriented Graph Coloring Discrete Mathematics 1990
Homomorphism
Approximation:
References:
Hell, Nesetril(04) Graphs and Homomorphisms Bang Jensen, Hell,MacGillivray The complexity of Colouring by Semicomplete digraphs, J. of Discrete Mathematics; Bang Jensen, Hell: The effect of 2 cycles on the complexity of coulouring by directed graphs, Discrete Mathematics; 1990 Klostermayer & MacGillivray: Homomorphisms and oriented colorings of equivalence classes of oriented graphs, Discrete Mathematics (2004)
Ausiello, Crescenzi, Gambozi, Kann & al Complexity and Approximation; 2003 Demange, Grisoni, Paschos: Approximation results for the minimum graph coloring problem

24. culus@univ-tlse2.fr demange@essec.fr
Thank You !
Questions ?

25. Sketch of Proof for Bipartite digraphs Reduction from 3-Sat
H4 -Coloring with H4: T R F B xF xT xR xB yF yR yB yT

26. Complexity: For each litteral xiB
Digraph Gi admits a H4-coloring H4
One must be colored by T and the other by F Gi

27. Complexity: For each Clause Cj: z1  z2  z3
T or F ? F T R F B T
B or T B R H4
T or B
F or R
R or T T T
Clause Cj satisfies iff oriented Graph Gj admits a H4-coloring
B or F R
R or F B F
F or B or T
If one of the litteral is True, then digraph Gj admits a H4-coloring
F or R Gj