Oriented Coloring:

1. Oriented Coloring: Complexity and Approximation Jean-François CulusUniversité Toulouse 2 Grimm culus@univ-tlse2.frMarc DemangeEssec Sid, Paris demange@essec.fr SOFSEM 2006

2. 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 ?

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

4. 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’ xyaz t c b f(x)=f(t)=a f(y)=b f(z)=c K G

5. 1. IntroductionColoration and Homomorphism • G admits a k-coloration if and only if • (G)or it exists a k-graph K and an homomorphismfrom 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. 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

7. 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

8. 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

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

10. 2. Complexity: Plan Oriented k-coloring Homomorphism NP-complete case 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 Polynomial4-Oriented Coloring is NP-Complete [Klostermeyer & al., 04]… even for connected graph H4

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

13. T R B F Sketch of proof: BipartiteReduction from 3-Sat H4 • L admits a H4-coloring For each litteral xi For each clause CjCj= z1 z2  z3 L

14. 2. Complexity: NP-Complete • Theorem: k-Col is NP-Complete for k≥4 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. G1 G2x 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. ApproximationWhat is an approximation ? • Min Oriented Coloring (MOC) Minimization problem • Let G be a n-digraphOptimum: o(G); Worst: n; Algorithm A(G)0 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. 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 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. ApproximationThe 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. ApproximationGreedy Algorithm (Problem) x y z t u v w a Contradict Unidirection property

22. 3. ApproximationThe 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 boudedO(log2(n)/n) G S2

23. 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

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

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

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

27. T R B F Complexity:For each Clause Cj: z1 z2  z3 T or F ? F 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