Oriented Coloring: Jean-François Culus Université Toulouse 2 Grimm [email protected] Marc...
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Transcript of Oriented Coloring: Jean-François Culus Université Toulouse 2 Grimm [email protected] Marc...
Oriented Coloring:
Jean-François CulusUniversité Toulouse 2 Grimm [email protected]
Marc DemangeEssec Sid, Paris [email protected]
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. 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
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