Read e-book online Algorithmes d’approximation PDF

By Vijay V. Vazirani

ISBN-10: 228700677X

ISBN-13: 9782287006777

ISBN-10: 2287310207

ISBN-13: 9782287310201

Le champ des algorithmes d'approximation est aujourd'hui l'un des domaines de recherche les plus actifs en informatique. Il allie los angeles profondeur de los angeles th?orie math?matique aux promesses d'applications pratiques d'un int?r?t consid?rable. los angeles plupart des probl?mes issus d'applications suitable de domaines aussi diff?rents que los angeles perception de circuits VLSI, los angeles perception et l. a. planification de r?seaux, l'ordonnancement, l. a. th?orie des jeux, los angeles biologie ou l. a. th?orie des nombres, sont des probl?mes NP-difficiles. Leur r?solution exacte demanderait des ressources informatiques inaccessibles et ne peut donc ?tre envisag?e. Pour faire face ? cette state of affairs, un grand nombre d'algorithmes proposant des options approch?es ? ces probl?mes ont ?t? d?velopp?s. Une quantit? consid?rable de r?sultats nouveaux a ?t? ?tablie lors de los angeles derni?re d?cennie et a r?volutionn? ce champ d'?tude. Le d?fi relev? par cet ouvrage est de pr?senter clairement les th?ories et m?thodologies sous-jacentes sans rien ?ter ? l. a. beaut? des r?sultats. Ce livre disclose ces questions algorithmiques complexes en proposant des d?monstrations simples et intuitives accompagn?es de nombreux exemples.

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Indication : D´emontrez qu’il suffit de « casser » tous les cycles de longueur 3. Utilisez la f -approximation pour la couverture par ensembles minimum. 15 (Hochbaum [133]) Consid´erons le probl`eme suivant. 18 (Couverture Maximum)10 Etant donn´e un univers U contenant n ´el´ements munis de poids positifs, une collection de sousensembles S1 , . . , Sl de U et un entier k, s´electionner k ensembles de fa¸con `a maximiser le poids total des ´el´ements couverts. 10 Maximum coverage, en anglais. 2. 16 En vous appuyant sur le probl`eme de la couverture par sommets, donnez des algorithmes d’approximation pour les variantes suivantes du surfacteur minimum (on note ici sR le mot miroir de s) : 1.

5 (N. 1 et le fait que le probl`eme de la couverture par sommets se r´esout en temps polynomial lorsque le graphe est biparti. 13. H est biparti et pour tout sommet v, degH (v) (1/2) degG (v). 6 (Wigderson [266]) Consid´erons le probl`eme suivant. 16 (Coloration de sommets)8 Etant orient´e G = (V, E), colorier avec un nombre minimal de couleurs ses sommets de sorte que les extr´emit´es de chaque arˆete re¸coivent des couleurs diff´erentes. 1. Donnez un algorithme glouton qui colorie G avec ∆ + 1 couleurs, o` u∆ est le degr´e maximum d’un sommet de G.

2 : ... 1/n 1/(n-1) 1+ε 1 Sur cette instance, l’algorithme glouton produit la couverture constitu´ee de n singletons car, `a chaque it´eration, c’est un singleton qui a le coˆ ut effectif minimal. Ainsi, la couverture produite a pour coˆ ut : 1 1 + + · · · + 1 = Hn . n n−1 Or, la couverture optimale coˆ ute 1 + ε. 9). Au chapitre 1, nous avons insist´e sur le fait que trouver un bon minorant de OPT est une premi`ere ´etape pour la conception d’algorithmes d’approximation d’un probl`eme de minimisation.

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