Oriented Coloring: Complexity and Approximation Jean-François Culus Université Toulouse 2 Grimm culus@univ-tlse2.fr Marc Demange Essec Sid, Paris demange@essec.fr SOFSEM 2006

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 ?

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

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

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

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

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

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

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

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

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

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

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

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

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

2. Complexity: Polynomial case k-colo is polynomial for complete multipartite oriented graphs. G1 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.

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

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) 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

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.

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

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

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

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; 1998 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

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

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

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

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