On the Effects of Incorporating Memory in GC-AIS for the Set Cover Problem

Learning is an important part of the immune system by which the immune system maintains a memory of the infections it has encountered to protect against future attacks. In this paper we incorporate the mechanism of maintaining a memory in the recently proposed GC-AIS algorithm. GC-AIS has shown good performance on the static set cover problem (SCP) in recent work [13] and we are interested in investigating the merits of GC-AIS in a dynamic setting. We compare the affect of GC-AIS with and without a memory approach on the dynamic SCP instances, which are created with varying degrees of modifications to instances from [2]. Three types of modifications are proposed in the paper by adding, removing or editing the subsets from the original problem instances. It is shown that for the case of adding subsets to the original instance using our memory approach is always beneficial while for the case of removing subsets using our memory approach almost always results in worse performance than when not utilising memory. Finally in the cases with editing subsets it is shown that for lower levels of modification using our memory approach gives better results while when the level of modification is higher our memory based approach is worse than using no memory.authorsversio

The set covering problem revisited: an empirical study of the value of dual information

This paper investigates the role of dual information on the performances of heuristics designed for solving the set covering problem. After solving the linear programming relaxation of the problem, the dual information is used to obtain the two main approaches proposed here: (i) The size of the original problem is reduced and then the resulting model is solved with exact methods. We demonstrate the effectiveness of this approach on a rich set of benchmark instances compiled from the literature. We conclude that set covering problems of various characteristics and sizes may reliably be solved to near optimality without resorting to custom solution methods. (ii) The dual information is embedded into an existing heuristic. This approach is demonstrated on a well-known local search based heuristic that was reported to obtain successful results on the set covering problem. Our results demonstrate that the use of dual information significantly improves the efficacy of the heuristic in terms of both solution time and accuracy

Cocircuits of vector matroids

In this thesis, I present a set covering problem (SCP) formulation of the matroid cogirth problem, finding the cardinality of the smallest cocircuit of a matroid. Addressing the matroid cogirth problem can lead to significantly enhancing the design process of sensor networks. The solution to the matroid cogirth problem provides the degree of redundancy of the corresponding sensor network, and allows for the evaluation of the quality of the network. I provide an introduction to matroids, and their relation to the degree of redundancy problem. I also discuss existing methods developed to solve the matroid cogirth problem and the SCP. Computational results are provided to validate a branch-and-cut algorithm that addresses the SCP formulation

Un algorithme tabou stochastique pour le problème de recouvrement d'ensemble à coûts unitaires

RÉSUMÉ Le problème de recouvrement d’ensemble avec coûts unitaires (USCP) est un problème NP-difficile. Ce problème possède plusieurs applications importantes comme le problème d’affectation des équipages. Le but de notre travail est de résoudre de manière efficace le problème USCP. Pour atteindre cet objectif, nous avons commencé par développer un algorithme tabou qui s’inspire du meilleur algorithme conçu pour résoudre ce problème. L’un des points faibles de ce dernier algorithme est l’absence d’une technique permettant un réglage efficace des paramètres. Notre principal objectif était de trouver une manière efficace de régler les paramètres. Durant notre travail, nous avons exploré plusieurs approches. La première approche consistait à trouver des formules générales pour nos listes taboues. Nous n’avons pas réussi à trouver des formules simples, mais les résultats des tests réalisés avec nos deux formules compliquées sont meilleurs que ceux obtenus par le meilleur algorithme de la littérature. La deuxième approche consistait à adapter l’algorithme tabou réactif à notre problème USCP. Les tests réalisés avec cette approche ont montré que l’algorithme ne produit pas des cycles avec les jeux de grande taille, donc incapable de régler dynamiquement les longueurs des listes taboues. Notre troisième idée consistait à combiner le recuit simulé avec l’algorithme tabou. Nos tests ont révélé que l’algorithme obtient des résultats médiocres lorsque la température n’est pas suffisamment basse. Grâce aux résultats obtenus avec la troisième approche, nous avons développé notre algorithme tabou stochastique STS. Notre algorithme STS nous a permis de régler plus facilement les longueurs des listes taboues. Les résultats de STS sont meilleurs que ceux obtenus par RWLS -- le meilleur algorithme de la littérature publié récemment. Notre algorithme obtient 6 nouveaux records et atteint tous les meilleurs résultats sur le reste des jeux de données. Pour rendre nos algorithmes plus rapides, nous avons développé une implémentation efficace. Notre implémentation est fondée sur deux caractéristiques clés. La première est l’utilisation d’algorithmes de bas niveau incrémentaux. Le deuxième point fort de notre implémentation est l’utilisation des files de priorité qui rendent la sélection d’un mouvement plus rapide. Les tests effectués montrent l’efficacité de nos files de priorités sur la majorité des jeux de données traités dans notre travail.----------ABSTRACT The unicost set covering problem (USCP) is an NP-hard problem. This problem has many important real-life applications such as the crew scheduling problem. In this work, we aim to effectively solve the USCP. To achieve this goal, we first developed a tabu search algorithm inspired by the best algorithm designed to solve the USCP. One of the weaknesses of the latter algorithm is the absence of an effective technique for setting the parameters. Our main objective was to find an effective way to adjust the tabu lists parameters. During our work, we explored several approaches. The first approach was to find general formulas for our tabu lists. We have not managed to find simple formulas, but the results of the tests performed with our two complicated formulas are better than those obtained by the best performing algorithms in the literature. The second approach was to adapt the reactive tabu algorithm to our problem. Tests performed with this approach have shown that the algorithm does not produce cycles when applied to big instances, so it is unable to dynamically adjust the length of the tabu lists. Our third idea was to combine a simulated annealing algorithm with the tabu algorithm. Our tests revealed that the algorithm performs poorly when the temperature is not low enough. Thanks to the results obtained with the third approach, we have developed our stochastic tabu algorithm STS. Our STS algorithm allowed us to easily adjust the lengths of the tabu lists. STS results are better than those obtained by RWLS - the best algorithm in the literature which was recently published. Our algorithm obtains 6 new records and achieves the best results on all the remaining instances. To make our algorithm faster, we have developed an efficient implementation. Our implementation is based on two features. The first is the use of incremental low level algorithms. The second feature of our implementation is the use of priority queues that make the selection of the movements faster. The tests show the effectiveness of using the priority queues on the majority of the instances used in this work