<p>This Master’s Thesis deals with a special problem that may be of importance when planning a frequency hopping mobile communication network. In normal cases the Frequency Assignment Problem is solved, in order to plan the use of frequencies in a network. The special case discussed in this thesis occurs when the network operator requires that the frequencies must be arranged into groups. In this case the Frequency Assignment Problem must be solved with respect to the groups, i.e. a Group assignment Problem. </p><p>The thesis constitutes the final part of the Master of Science in Communication and Transport Systems Engineering education, at Linköping University, Campus Norrköping. The Group Arrangement Problem was presented by ComOpt, a company that has specialized in solving the Frequency Assignment Problem for network operators. </p><p>This thesis does not deal with solutions for the Frequency Assignment Problem, with respect to the groups. The main issue in the thesis is to construct a computer based algorithm that solves the Group Arrangement Problem, i.e. creating the groups. The goal is to construct an algorithm that creates groups which imply a better solution for the Frequency Assignment Problem than manually created groups. </p><p>Two algorithms are presented and tested on two cases. Their respective results for both cases are compared with the results from a manual grouping. The two computer based algorithms creates better groups than the manual grouping strategy, according to an artificial quality measure. As of spring 2003 a variant of one of the presented algorithms was implemented in ComOpt’s product for solving the Frequency Assignment Problem.</p>

This Master’s Thesis deals with a special problem that may be of importance when planning a frequency hopping mobile communication network. In normal cases the Frequency Assignment Problem is solved, in order to plan the use of frequencies in a network. The special case discussed in this thesis occurs when the network operator requires that the frequencies must be arranged into groups. In this case the Frequency Assignment Problem must be solved with respect to the groups, i.e. a Group assignment Problem. The thesis constitutes the final part of the Master of Science in Communication and Transport Systems Engineering education, at Linköping University, Campus Norrköping. The Group Arrangement Problem was presented by ComOpt, a company that has specialized in solving the Frequency Assignment Problem for network operators. This thesis does not deal with solutions for the Frequency Assignment Problem, with respect to the groups. The main issue in the thesis is to construct a computer based algorithm that solves the Group Arrangement Problem, i.e. creating the groups. The goal is to construct an algorithm that creates groups which imply a better solution for the Frequency Assignment Problem than manually created groups. Two algorithms are presented and tested on two cases. Their respective results for both cases are compared with the results from a manual grouping. The two computer based algorithms creates better groups than the manual grouping strategy, according to an artificial quality measure. As of spring 2003 a variant of one of the presented algorithms was implemented in ComOpt’s product for solving the Frequency Assignment Problem.

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.

Les applications réelles liées aux problèmes de satisfaction partielle de contraintes (PCSP : Partial Constraints Satisfaction Problem) sont de plus en plus nombreuses, ce qui justifie l’intérêt croissant des chercheurs pour cette classe de problèmes. La résolution d’un PCSP revient à affecter des valeurs à toutes ses variables tout en maximisant (ou minimisant) une fonction objectif prédéfinie. Ces problèmes sont NP-difficiles, par conséquent il n’existe aucune approche aussi bien exacte qu’heuristique efficace sur les grandes instances. Pour résoudre efficacement les instances difficiles, une multitude de solutions sont proposées, allant de l’hybridation à l’apprentissage en passant par la décomposition. Dans notre travail, nous nous intéressons à cette dernière proposition, qui consiste à fractionner le problème PCSP en plusieurs sous-problèmes PCSP de tailles raisonnables, puis proposer des algorithmes de résolution pour les problèmes décomposés. Cette approche a pour but de bénéficier de la structure du problème afin d’accélérer sa résolution tout en garantissant des solutions optimales ou sous-optimales. Deux grand axes sont explorés : les approches basées sur la décomposition et celles guidées par la décomposition. Les approches basées sur la décomposition consistent à résoudre séparément les parties difficiles du problème décomposé, puis combiner les solutions partielles obtenues en vue d’atteindre une solution globale du problème d’origine. Les approches guidées par la décomposition consistent à développer des métaheuristiques qui tiennent compte de la structure du problème décomposé. Les algorithmes proposés sont testés et validés sur des instances réelles des problèmes PSCP, comme le problème d’affectation de fréquences et le problème de coloration de graphes / The wide range of potential applications concerned by the resolution of Partial Constraints Satisfaction Problems (PCSP) justifies the growing interest of scientists in this class of problems. Solving a PCSP means searching for values to assign to the decision variables in order to maximize (or minimize) a predefined objective function. These problems are NP-hard, so there isn’t an exact approach nor an efficient heuristic able to provide the optimal solution for large instances. In order to solve effectively the difficult instances, numerous approaches based on hybridization, learning or decomposition are proposed. In the present work, we focus on the latter proposal, which consists in splitting the PCSP into several smaller size PCSPs and we propose some methods to solve the decomposed problem. Two wide axes are explored : the resolution based on the decomposition and the one guided by decomposition. The former solves separately the difficult parts of the decomposed problem (cuts or clusters) and then combines partial solutions obtained in order to achieve a global solution for the original problem. The latter aims at benefiting from the structure of the problem to be decomposed in order to accelerate its resolution while ensuring optimal or near optimal solutions. All the proposed algorithms are tested and validated on the well-known benchmarks of PCSP problems such as Frequency Assignment Problem (FAP) and graph coloring problem