A note on compact and compact circular edge-colorings of graphs

Graphs and Algorithm

Interval Edge-Colorings of Graphs

A proper edge-coloring of a graph G by positive integers is called an interval edge-coloring if the colors assigned to the edges incident to any vertex in G are consecutive (i.e., those colors form an interval of integers). The notion of interval edge-colorings was first introduced by Asratian and Kamalian in 1987, motivated by the problem of finding compact school timetables. In 1992, Hansen described another scenario using interval edge-colorings to schedule parent-teacher conferences so that every person\u27s conferences occur in consecutive slots. A solution exists if and only if the bipartite graph with vertices for parents and teachers, and edges for the required meetings, has an interval edge-coloring. A well-known result of Vizing states that for any simple graph G, χ0(G) ≤ ∆(G)+1, where χ0(G) and ∆(G) denote the edge-chromatic number and maximum degree of G, respectively. A graph G is called class 1 if χ0(G) = ∆(G), and class 2 if χ0(G) = ∆(G) + 1. One can see that any graph admitting an interval edge-coloring must be of class 1, and thus every graph of class 2 does not have such a coloring. Finding an interval edge-coloring of a given graph is hard. In fact, it has been shown that determining whether a bipartite graph has an interval edge-coloring is NP-complete. In this thesis, we survey known results on interval edge-colorings of graphs, with a focus on the progress of (a, b)-biregular bipartite graphs. Discussion of related topics and future work is included at the end. We also give a new proof of Theorem 3.15 on the existence of proper path factors of (3, 4)-biregular graphs. Finally, we obtain a new result, Theorem 3.18, which states that if a proper path factor of any (3, 4)-biregular graph has no path of length 8, then it contains paths of length 6 only. The new result we obtained and the method we developed in the proof of Theorem 3.15 might be helpful in attacking the open problems mentioned in the Future Work section of Chapter 5

Metaheuristic approach for solving scheduling and financial derivative problems

This thesis was submitted for the award of Doctor of Philosophy and was awarded by Brunel University LondonThe objective of this thesis is to implement metaheuristic approaches to solve di erent types of combinatorial problems. The thesis is focused on neighborhood heuristic optimisation techniques such as Variable Neighborhood Search (VNS) and Ant Colony Optimisation (ACO) algorithms. The thesis will focus on two diverse combinatorial problems. A job shop scheduling problem, and a nancial derivative matching problem. The rst is a NP-hard 2-stage assembly problem, where we will be focussing on the rst stage. It consists of sequencing a set of jobs having multiple components to be processed. Each job has to be worked on independently on a speci c machine. We consider these jobs to form a vector of tasks. Our objective is to schedule jobs on the particular machines in order to minimise the completion time before the second stage starts. In this thesis, we have constructed a new hybrid metaheuristic approach to solve this unique job shop scheduling problem. The second problem has arisen in the nancial sector, where the nancial regulators collects transaction data across regulated assets classes. Our focus is to identify any unhedged derivative, Contract for Di erence (CFD), with its corresponding underlying asset that has been reported to the corresponding component authorities. The underlying asset and CFD transaction contain di erent variables, like volume and price. Therefore, we are looking for a combination of underlying asset variables that may hedge the equivalent CFD variables. Our aim is to identify unhedged or unmatched CFD's with their corresponding underlying asset. This problem closely relates to the goal programming problem with variable parameters. We have developed two new local search methods and embedded the newly constructed local search methods with basic VNS, to attain a new modi ed variant of the VNS algorithm. We then used these newly constructed VNS variants to solve this nancial matching problem. In tackling the Vector Job Scheduling problem, we developed a new hybrid optimisation heuristic algorithm by combining VNS and ACO. We then compared the results of this hybrid algorithm with VNS and ACO on their own. We found that the hybrid algorithm performance is better than the other two independent heuristic algorithms. In tackling the nancial derivative problem, our objective is to match the CFD trades with their corresponding underlying equity trades. Our goal is to identify the mismatched CFD trades while optimising the search process. We have developed two new local search techniques and we have implemented a VNS algorithm with the newly developed local search techniques to attain better solutions

Complexité et cassage de symétrie pour le problème de la déficience d’un graphe

RÉSUMÉ : Une coloration d’arête d’un graphe G=(V,E) est une fonction c qui assigne un entier c(e) (appelé une couleur) dans {0, 1, 2,...} à chaque arête e dans E de sorte que des couleurs différentes soient assignées à des arêtes adjacentes. Une coloration d’arête est compacte si les couleurs des arêtes incidentes à chaque sommet forment un ensemble d’entiers consécutifs. Le problème appelé déficience consiste à déterminer le nombre minimum d’arêtes pendantes à rajouter au graphe pour que le graphe résultant ait une coloration d’arête compacte. Parmi les variations de ce problème, on compte le problème de la coloration d’arête compacte linéaire (k−LCCP) où il est possible d’utiliser uniquement les k couleurs dans {0, 1, ... , k−1}, et le problème de la coloration d’arête compacte cyclique (k−CCCP) où additionnellement la couleur 0 est considérée consécutive à la couleur k−1. Nous proposons une réduction polynomiale du k−LCCP (optionnellement avec des couleurs imposées ou interdites sur certaines arêtes) au k−CCCP lorsque k≥12, et au 12-CCCP lorsque k<12. Nous proposons et comparons également la performance de 3 modélisations en Programmation en Nombres Entiers et un modèle en Programmation par Contraintes pour le problème de la déficience, et déterminons le dernier comme étant le plus approprié pour ce problème. En raison des symétries, une instance du problème de déficience peut avoir de nombreuses solutions optimales équivalentes. Nous présentons une approche pour générer un petit ensemble de contraintes, appelées GAMBLLE, destinée à casser la symétrie, qui peuvent être incorporées au modèle en programmation par contrainte. Les contraintes GAMBLLE sont inspirées des contraintes de Lex-Leader, basées sur les automorphismes de graphe, et agissent sur des familles de variables permutables. Nous analysons leur impact sur la réduction du nombre de solutions optimales, ainsi que le gain de temps obtenu lors de la résolution d’une modélisation en programmation par contrainte.----------ABSTRACT : An edge-coloring of a graph G=(V,E) is a function c that assigns an integer c(e) (called color) in {0, 1, 2,...} to every edge e in E so that adjacent edges are assigned different colors. An edge-coloring is compact if the colors of the edges incident to every vertex form a set of consecutive integers. The deficiency problem is to determine the minimum number of pendant edges that must be added to a graph such that the resulting graph admits a compact edge-coloring. Variations of this problem include the linear compact k-edge-coloring problem (k−LCCP) where there are only the k colors of {0, 1, ... , k−1} available, and the cyclic compact k-edge-coloring problem (k−CCCP) where additionally color 0 is considered consecutive to color k−1. We demonstrate a polynomial reduction of the k−LCCP (with optionally additional imposed or forbidden colors on some edges) to the k−CCCP when k≥12, and to the 12−CCCP when k<12. We also propose and compare the performance of three integer programming models and one constraint programming model for the deficiency problem, and determine the latter to be the best suited to model this problem. Because of symmetries, an instance of the deficiency problem can have many equivalent optimal solutions. We present a way to generate a small set of symmetry breaking constraints, called GAMBLLE constraints, that can be added to a constraint programming model. The GAMBLLE constraints are inspired by the Lex-Leader ones, based on automorphisms of graphs, and act on families of permutable variables. We analyze their impact on the reduction of the number of optimal solutions as well as on the speed-up of the constraint programming model

Programmation linéaire en nombres entiers pour l'ordonnancement cyclique sous contraintes de ressources

Un problème d'ordonnancement cyclique consiste à ordonner dans le temps l'exécution répétitive d'un ensemble d'opérations liées par des contraintes de précédence, en utilisant un nombre limité de ressources. Ces problèmes ont des applications immédiates dans les systèmes de production ou en informatique parallèle. Particulièrement, ils permettent de modéliser l'ensemble des contraintes de précédence et de ressource à prendre en compte pour l'ordonnancement d'instructions dans les processeurs de type VLIW (Very Long Instruction Word). Dans ce cas, une opération représente une instance d'une instruction dans un programme. L'ordonnancement d'instructions de boucles internes est connu sous le nom de pipeline logiciel. Le pipeline logiciel désigne une méthode efficace pour l'optimisation de boucles qui permet la réalisation en parallèle des opérations des différentes itérations de la boucle. Dans cette thèse, nous nous intéressons principalement au problème d'ordonnancement périodique qui est un cas particulier de l'ordonnancement cyclique et qui est également la base du pipeline logiciel. Le terme ordonnancement modulo désigne un ordonnancement périodique tel que l'allocation de ressources pour une opération donnée n'est pas modifiée d'une itération sur l'autre. Pour résoudre le problème, nous nous intéressons aux formulations de programmation linéaire en nombres entiers, et notamment à la résolution du problème par des techniques de séparation, évaluation, génération de colonnes, relaxation lagrangienne et des méthodes hybrides. En particulier, nous proposons des nouvelles formulations basées sur des variables binaires représentant l'exécution d'ensembles d'instructions en parallèle. Enfin, les méthodes développées ont été validées sur des jeux d'instances industrielles pour des processeurs de type VLIW.The resource-constrained modulo scheduling problem (RCMSP) is a general periodic cyclic scheduling problem, abstracted from the problem solved by compilers when optimizing inner loops at instruction level for very long instruction word parallel processors. Since solving the instruction scheduling problem at compilation phase in less time critical than for real time scheduling, integer linear programming (ILP) is a relevant technique for the RCMSP. In this work, we are interested in the methods based on the integer linear programming for the RCMSP. At first, we present a study of the two classic integer linear formulation for the RCMSP. A theoretical evidence of the equivalence between the classic formulations is shown in terms of linear programming (LP) relaxation. Secondly, based on the ILP formulations for the RCMSP, stronger formulations for the RCMSP derived from Dantzig-Wolfe decomposition are presented. In these formulations, the number of variables can be huge, for this reason, we proposed a column generation scheme to solve their LP relaxations. We propose also the heuristics methods based on the Lagragian relaxation and decomposed software pipelining. The heuristic methods search the transformation of the classic integer linear programming for the RCMSP for the performance improvement in the time for the search of solutions. All formulations are compared experimentally on problem instances generated from real data issued from the STMicroelectronics ST200 VLIW processor family

Coloration de graphes et attribution d'activités dans des quarts de travail

Revue de littérature -- Organisation de la thèse -- Lower bounds and a tabu search algorithm for the minimum deficiency problem -- On a reduction of the interval coloring problem to a series of bandwidth coloring problems -- About equivalent interval colorings of weighted graphs -- Une approche de programmation en nombres entiers pour la résolution d'un problème d'horaire -- Discussion générale et conclusion