It is known to be an NP-complete problem to color a graph with a given number of colors. We present some approximation algorithms which come close to the desired number of colors. We also develop an algorithm that colors k-colorable graphs with ~O(n^a(k)) colors, where a(2)=0, a(3)=3/14 and a(k)=1 - 6/(k+4+3(1-2/k)/(1-a(k-2))) for k >= 4, as presented in [20]. This formula has been generalized for new possible base algorithms. / Das Problem, einen Graphen mit einer gegebenen Anzahl Farben zu färben, ist als NP-vollständig bekannt. Hier werden einige Algorithmen vorgestellt, die für dieses Problem eine gute Approximation liefern. Des Weiteren wird ein allgemeines Färbungsverfahren hergeleitet, das für k-färbbare Graphen den bisher besten existierenden Algorithmus darstellt. Es können k-färbbare Graphen mit ~O(n^a(k)) Farben gefärbt werden, wobei a(2)=0, a(3)=3/14 und a(k) = 1 - 6/(k+4+3(1-2/k)/(1-a(k-2))) für k >= 4 gilt [20]. Diese Formel wurde für neue Basisalgorithmen verallgemeinert.

The frequency assignment (FAP) consists in assigning the frequency on the radio links of a network which satisfiesthe electromagnetic interference among the links. Given the limited spectrum resources for each application, the fre-quency resources are often insufficient to deploy a wireless network without interference. In this case, the network isover-contrained and the problem is infeasible. Our objective is to identify an area with heavy interference.The work presented here concerns the detection for one of these areas with an algorithmic approach based onmodeling the problem by CSP. The problem of frequency assignment can be modeled as a constraint satisfactionproblem (CSP) which is represented by a triple: a set of variables (radio links), a set of constraints (electromagneticinterference) and a set of available frequencies.The interfered area in CSP can be considered a subset of irreducible feasible subset (IIS). An IIS is a infeasiblesubproblem with irreducible size, that is to say that all subsets of an IIS are feasible. The identification of an IIS ina CSP refers to two general interests. First, locating an IIS can easily prove the infeasibility of the problem. Becausethe size of IIS is assumed to be smaller compared to the entire problem, its infeasibility is relatively easier to prove.Second, we can locate the reason of infeasibility, in this case, the decision maker can provide the solutions to relax theconstraints inside IIS, which perhaps leads to a feasible solution to the problem.This work proposes algorithms to identify an IIS in the over-constrained CSP. These algorithms have tested on the well known benchmarks of the FAP and of the problem of graph k-coloring. The results show a significant improve-ment on instances of FAP compared to known methods.

L’affectation de fr´equences (AFP) consiste `a attribuer des fr´equences radio aux liens de communications d’un r´eseauen respectant un spectre de fr´equences donn´e et des contraintes d’interf´erence ´electromagn´etique sur les liens. Vu lalimitation des ressources spectrales pour chaque application, les ressources en fr´equences sont souvent insuffisantespour d´eployer un r´eseau sans interf´erence. Dans ce cas, le r´eseau est surcontraint et le probl`eme est irr´ealisable.R´esoudre le probl`eme consiste alors `a identifier les zones surcontraintes pour en revoir la conception.Le travail que nous pr´esentons concerne la recherche d’une de ces zones surcontraintes avec une approche algo-rithmique bas´ee sur la mod´elisation du probl`eme par un CSP. Le probl`eme de l’affectation de fr´equences doit doncˆetre mod´elis´e comme un probl`eme de satisfaction de contraintes (CSP) qui est repr´esent´e par un tripl´e : un ensemblede variables (les liens radio), un ensemble de contraintes (les interf´erences ´electromagn´etiques), et un ensemble dedomaines (les fr´equences admises).Sous forme de CSP, une zone perturb´ee peut ˆetre consid´er´ee comme un sous-ensemble irr´ealisable irr´eductible duprobl`eme (IIS pour Irreductible Infeasible Subset). Un IIS est un sous probl`eme de taille minimale qui est irr´ealisable,c’est-`a-dire que tous les sous-ensembles d’un IIS sont r´ealisables. L’identification d’un IIS dans un CSP se rapporte `a deux r´esultats g´en´eraux int´eressants. Premi`erement, en localisant un IIS on peut plus facilement prouver l’irr´ealisabilit´ed’un probl`eme donn´e car l’irr´ealisabilit´e d’un IIS, qui est suppos´e ˆetre petit par rapport au probl`eme complet, est plusrapidement calculable que sur le probl`eme entier. Deuxi`emement, on peut localiser la raison de l’irr´ealisabilit´e; dansce cas, sur un probl`eme r´eel, le d´ecideur peut proposer des solutions pour relˆacher des contraintes de l’IIS, et peut-ˆetre aboutir `a une solution r´ealisable pour son probl`eme. La recherche d’IIS consiste donc `a r´esoudre un probl`emefondamental qui fait partie des outils de prise de d´ecision.Ce travail propose des algorithmes pour identifier un IIS dans un CSP incoh´erent. Ces algorithmes ont ´et´e test´essur des instances connues du probl`eme de l’affectation des fr´equences et du probl`eme de k-coloration de graphe. Lesr´esultats ont montr´es d’une grande am´elioration sur des instances du probl`eme de l’affectation des fr´equences parrapport aux m´ethodes connues. / The frequency assignment (FAP) consists in assigning the frequency on the radio links of a network which satisfiesthe electromagnetic interference among the links. Given the limited spectrum resources for each application, the fre-quency resources are often insufficient to deploy a wireless network without interference. In this case, the network isover-contrained and the problem is infeasible. Our objective is to identify an area with heavy interference.The work presented here concerns the detection for one of these areas with an algorithmic approach based onmodeling the problem by CSP. The problem of frequency assignment can be modeled as a constraint satisfactionproblem (CSP) which is represented by a triple: a set of variables (radio links), a set of constraints (electromagneticinterference) and a set of available frequencies.The interfered area in CSP can be considered a subset of irreducible feasible subset (IIS). An IIS is a infeasiblesubproblem with irreducible size, that is to say that all subsets of an IIS are feasible. The identification of an IIS ina CSP refers to two general interests. First, locating an IIS can easily prove the infeasibility of the problem. Becausethe size of IIS is assumed to be smaller compared to the entire problem, its infeasibility is relatively easier to prove.Second, we can locate the reason of infeasibility, in this case, the decision maker can provide the solutions to relax theconstraints inside IIS, which perhaps leads to a feasible solution to the problem.This work proposes algorithms to identify an IIS in the over-constrained CSP. These algorithms have tested on the well known benchmarks of the FAP and of the problem of graph k-coloring. The results show a significant improve-ment on instances of FAP compared to known methods.

It is known to be an NP-complete problem to color a graph with a given number of colors. We present some approximation algorithms which come close to the desired number of colors. We also develop an algorithm that colors k-colorable graphs with ~O(n^a(k)) colors, where a(2)=0, a(3)=3/14 and a(k)=1 - 6/(k+4+3(1-2/k)/(1-a(k-2))) for k >= 4, as presented in [20]. This formula has been generalized for new possible base algorithms. / Das Problem, einen Graphen mit einer gegebenen Anzahl Farben zu färben, ist als NP-vollständig bekannt. Hier werden einige Algorithmen vorgestellt, die für dieses Problem eine gute Approximation liefern. Des Weiteren wird ein allgemeines Färbungsverfahren hergeleitet, das für k-färbbare Graphen den bisher besten existierenden Algorithmus darstellt. Es können k-färbbare Graphen mit ~O(n^a(k)) Farben gefärbt werden, wobei a(2)=0, a(3)=3/14 und a(k) = 1 - 6/(k+4+3(1-2/k)/(1-a(k-2))) für k >= 4 gilt [20]. Diese Formel wurde für neue Basisalgorithmen verallgemeinert.