Combinatoire et Optimisation UE 4M068 Résumé …michel.pocchiola/pdf4M068/Resume_4M... · M....

Combinatoire et Optimisation

UE 4M068

Résumé Cours et TD

◇ année 2017 ◇

Michel Pocchiola ([email protected])

Copyright © 2017 Michel PocchiolaAll Rights Reserved

Résumé

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M. Pocchiola Comb. & Optimisation - UE 4M068 - Résumé 2017

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Table des matières

1 Mots de Łukasiewicz . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5

2 Graphes, graphes planaires, lemme des croisements . . . . . . . . . . . . . 7

3 Arbres binaires de recherche et randomisation . . . . . . . . . . . . . . . . 9

4 Treillis des faces d’un polytope . . . . . . . . . . . . . . . . . . . . . . . . . 11

5 Forêts, arbres et arbres couvrants . . . . . . . . . . . . . . . . . . . . . . . . 13

6 Matroides . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15

7 Théorème de dualité pour la programmation linéaire . . . . . . . . . . . . 17

8 Epsilon-nets, cuttings et recherche simpliciale . . . . . . . . . . . . . . . . . 19

9 A suivre . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21

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1 Mots de Łukasiewicz

ê cours ê

Monoides libres - mots de Łukasiewicz sur un alphabet à deux lettres - nombres de Cata-lan - algèbre des séries formelles - algèbres universelles - exemples : monoides, groupes,anneaux, treillis - algèbres universelles libres - arborescences et arbres ordonnés - arbresbinaires et mots de Łukasiewicz

ê travaux dirigés ê

preuves bijectives - génération aléatoire d’un arbre binaire - longueur de cheminementinterne d’un arbre binaire aléatoire - hauteur d’un arbre binaire aléatoire - rotations dansles arbres binaires - associaèdre - vecteur de Loday d’un arbre binaire - énumération desarborescences - séquences de Davenport-Schinzel - code de Gray -

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The Philosophy of Mathematics and Logic in the 1920s and 1930s in Polandby R. Murawski

volume 48 of Science Networks Historical Studies. Birkhäuser, 2014.

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

Łukasiewicz dealt with philosophy (especially at the early stage of his scien-tific research) and above all, with mathematical logic. Although he receivededucation in philosophy, he had an excellent intuitive understanding of ma-thematics. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

Łukasiewicz’s achievements in mathematical logic allow us to treat him as oneof the most outstanding representatives of this field in the twentieth century.In particular, he might have been one of the most eminent creators of pro-positional calculi. His achievements include : (1) the elaboration of a speciallogical notation (called parenthesis-free symbolism, Łukasiewicz symbolismor Polish notation) that was excellent to conduct investigations on logicalcalculi in the Warsaw School of Logic ; (2) the creation of many-valued lo-gics ; (3) research—based on many- valued logics—on modal connectives andthe construction of the so-called systems of Ł-modal logic ; (4) the elabora-tion of a series of axiomatic systems for classical logic calculus (in particular,axiomatic implication-negation system for propositional calculus) ; (5) inves-tigations into the metalogical properties of various systems of propositionalcalculus. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . [5, page 61]

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Références

[1] P. Flajolet and A. M. Odlyzko. The average height of binary trees and other simple trees. J. Comput.

Syst. Sci., 25(2) :171–213, 1982. pdf.

[2] P. Flajolet and R. Sedgewick. Analytic Combinatorics. Cambridge, 2009.

[3] N. Jacobson. Basic Algebra I. Freeman, 1974.

[4] N. Jacobson. Basic Algebra II. Freeman, 1980.

[5] R. Murawski. The Philosophy of Mathematics and Logic in the 1920s and 1930s in Poland, volume 48of Science Networks Historical Studies. Birkhäuser, 2014.

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2 Graphes, graphes planaires, lemme des croisements

ê cours ê

Graphes et dessin de graphes - graphes isomorphes - graphes linéaires, graphes cycliques,graphes complets, graphes bipartis complets - sous-graphes, chemins et cycles d’un graphe- graphe connexe, composantes connexes d’un graphe - graphe planaire et graphe plan :relation d’Euler, lemme des croisements - arbre - graphe orienté -


score d’un graphe - segments bisecteurs - tournois et tounois libres - la méthode proba-biliste -

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Crossing-free subgraphs by M. Ajtai, V. Chvátal, M. Newborn, and E. Szemerédi,in Ann. Discrete Math., 12 :9–12, 1982.

Abstract. If m ≥ 4 then every planar drawing of a graph with n vertices and m edgescontains more than m3/100n2 edge-crossings and fewer than 1013n crossing-free sub-graphs. The first result settles a conjecture of Erdős and Guy and the second resultsettles a conjecture of Newborn and Moser.

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Références

[1] M. Aigner and G. M. Ziegler. Proofs from THE BOOK. Springer-Verlag, Berlin, 1998.

[2] N. Alon and J. Spencer. The Probabilistic Method. John Wiley & Sons, New York, NY, 1992.

[3] M. Heyvaert and F. T. Bruss. La méthode probabiliste. Gazette des mathématiciens, 124 :15–29,Apr. 2010.

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3 Arbres binaires de recherche et randomisation

ê cours ê

Arbres binaires de recherche - sous-ensemble canonique associé à un nœud - chemin derecherche d’une clé : partition induite - insertion d’une clé - arbre binaire de recherchealéatoire - hauteur d’un arbre binaire de recherche aléatoire - queue de distribution :borne de Chernoff pour les variables aléatoires harmoniques - notation O(f(n)) -


dynamisation des arbres binaires de recherche aléatoires - arbres de segments -

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Randomized Search Treesby R. Seidel and C.R. Aragon, in Algorithmica 16, pages 464–497, 1996 (FOCS’89).

Abstract. We present a randomized strategy for maintaining balance in dynamicallychanging search trees that has optimal expected behavior. In particular, in the expec-ted case a search or an update takes logarithmic time, with the update requiring fewerthan two rotations. Moreover, the update time remains logarithmic, even if the cost ofa rotation is taken to be proportional to the size of the rotated subtree. Finger searchesand splits and joins can be performed in optimal expected time also. We show that theseresults continue to hold even if very little true randomness is available, i.e., if only alogarithmic number of truely random bits are available. Our approach generalizes natu-rally to weighted trees, where the expected time bounds for accesses and updates againmatch the worst-case time bounds of the best deterministic methods. We also discussways of implementing our randomized strategy so that no explicit balance informationis maintained. Our balancing strategy and our algorithms are exceedingly simple andshould be fast in practice.

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Références

[1] K. Mulmuley. Computational Geometry : An Introduction Through Randomized Algorithms. PrenticeHall, Englewood Cliffs, NJ, 1993.

[2] R. Seidel and C. Aragon. Randomized search trees. Algorithmica, 16(4-5) :464–497, 1996.

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4 Treillis des faces d’un polytope

ê cours ê

Terminologie des treillis - cônes, cônes finiment engendrés, cônes polyèdriques - le lan-gage de la géométrie affine - convexes, convexes finiment engendrés, convexes poly-èdriques, polytopes - Lemme de Farkas et procèdure d’élimination de Fourier-Motzkin- autres formulations du Lemme de Farkas - partie linéaire et cône de récession - treillisdes faces d’un polytope - treillis et étoiles - treillis et polarité - graphe d’un polytope -simplexes, cocubes, permutaèdres, polytopes cycliques - théorème de la borne supérieurepour les polytopes


procédure d’élimination de Gauss - résolution d’un système d’équations linéaires par laméthode du pivot de Gauss - une alternative à la procédure d’élimination de Fourier-Motzkin (criss-cross method) - le treillis des faces d’un polytope simple est déterminépar son graphe (Bland-Mani theorem) - relation d’Euler-Poincaré - nombre de cellulesd’un arrangement d’hyperplans - théorème de la zone pour les arrangements d’hyperplans- polytopes des ordres (order polytope) - polytope associé à un réseau de tri -

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The maximal number of faces of a convex polytopeby P. McMullen, in Mathematika, 17 :179–184, 1970

Abstract. In this paper we give a proof of the long-standing Upper-bound Conjecturefor convex polytopes, which states that, for 1 ≤ j < d < v, the maximum possible numberof j-faces of a d-polytope with v vertices is achieved by a cyclic polytope C(v, d).

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Références

[1] M. Berger. Géométrie : Convexes et Polytopes, Polyèdres Réguliers, Aires et Volumes, volume 3.Cedic/Fernand Nathan, 1978.

[2] B. Grünbaum. Convex Polytopes, volume 221 of Graduate Texts in Mathematics. Springer-Verlag,2nd edition, 2003.

[3] P. McMullen. The maximal number of faces of a convex polytope. Mathematika, 17 :179–184, 1970.

[4] K. Mulmuley. Computational Geometry : An Introduction Through Randomized Algorithms. Prentice-Hall, 1994.

[5] G. M. Ziegler. Lectures on Polytopes, volume 152 of Graduate Texts in Mathematics. Springer-Verlag,Heidelberg, 1994.

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5 Forêts, arbres et arbres couvrants

ê cours ê

Matrice d’adjacence, matrice d’incidence orientée et matrice laplacienne d’un graphe -matrices totalement unimodulaires et formule de Binet-Cauchy - nombre d’arbres sur unensemble à n éléments : la formule de Cayley - nombre d’arbres couvrants d’un graphesimple : le théorème de Kirchhoff - arbres couvrants optimaux -


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Références

[1] R. A. Brualdi and H. J. Ryser. Combinatorial Matrix Theory. Cambridge University Press, 1991.

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6 Matroides

ê cours ê

Axiomatique des indépendants - matroides vectoriels, graphiques, transversals et de cou-plage - axiomatique des rangs - matroide dual - bases optimales et algorithme glouton -théorème de Rado-Hall -


Polytopes des indépendants d’un matroide -

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H. Whitney. On the abstract properties of linear dependence.by H. Whitney in American Journal of Mathematics, 57(3) :509–533, 1935.

Introduction. Let C1, C2, . . . , Cn be the columns of a matrix M . Any subset of these columns is either linearlyindependent or linearly dependent ; the subsets thus fall into two classes. These classes are not arbitrary ; forinstance, the two following theorems must hold :

(a) Any subset of an independent is an independent.(b) If Np and Np+1 are independent sets of p and p + 1 columns respectively, then Np together with some

column of Np+1 forms an independent set of p + 1 columns.

There are other theorems not deducible from these ; for in §16 we give an example of a system satisfying thesetwo theorems but not representing any matrix. Further theorems seem, however, to be quite difficult to find. Letus call a system obeying (a) and (b) a “matroid.” The present paper is devoted to the study of the elementaryproperties of matroids. The fundamental question of completely characterizing systems which represent matricesis left unsolved. In place of the columns of a matrix we may equally well consider points or vectors in a Euclideanspace, or polynomails, etc.

This paper has a close connection with a paper of the author on linear graphs ; 1 we say a subgraph of a graph isindependent if it contains no circuit. Although graphs are, abstractly, a very small subclass of the class of matrids,(see the appendix), many of the simpler theorems on graphs, especially on non-separable and dual graphs, applyalso to matroids. For these reason, we carry over various terms in the theory of graphs to the present theory.Remarkably enough, for matroids representing matrices, dual matroids have a simple geometrical interpretationquite different from that in the case of graphs (see §13).

The contents of the paper are as follows : In Part I . . .

1. “Non-separable and planar graphs”. Transactions of the American Mathematical Society, vol. 34(1932), pp. 339–362. We refer to this paper as G.

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Références

[1] D. C. Kozen. The Design and Analysis of Algorithms. Springer-Verlag, 1992.

[2] H. Whitney. On the abstract properties of linear dependence. American Journal of Mathematics,57(3) :509–533, 1935.

[3] R. J. Wilson. Introduction to Graph Theory. Longman, 1996. “Graph Theory has recently emerged asa subject in its own right, as well as being an important mathematical tool in such diverse subjects asoperational research, chemistry, sociology and genetics. Robin Wilson’s book has been widely used asa text for undergraduate courses in mathematics, computer science and economics, and as a readableintroduction to the subject for non-mathematicians. The opening chapters provide a basic foundationcourse, containing such topics as trees, algorithms, Eulerien and Hamiltonian graphs, planar graphsand colouring, with special reference to the four-colour theorem. Following, there are two chapterson directed graphs and transversal theory, relating these areas to such subjects as Markov chains andnetwork flows. Finally, there is a chapter on matroid theory, which is used to consolidate some of thematerial from earlier chapters. For this new edition, the text has been completely revised, and thereis a full range of exercices of varying difficulty. There is new material on algorithms, tree-searches,and graph-theoritical puzzles. Full solutions are provided for many of the exercices”.

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7 Théorème de dualité pour la programmation linéaire

ê cours ê

Programmes linéaires : faisable/infaisable, borné/non-borné, solution admissible, solu-tion optimale, optimum - programmes linéaires équivalents, variables d’écarts - théorèmede dualité pour la programmation linéaire - conditions des écarts complémentaires - pro-grammation linéaire (cf [3, chap.1]) -

Hypergraphe H, ensemble transversal ou transversal, cœfficient de transversalité (trans-versal number) τ(H), couplage (packing, matching), cœfficient de couplage (matchingnumber) ν(H), transversal fractionnaire, couplage fractionnaire, cœfficients fraction-naires de transversalité et de couplage τ∗(H), ν∗(H).

Polytope des vecteurs caractéristiques des ensembles d’arêtes des arbres couvrants d’ungraphe simple connexe.


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Références

[1] R. Cottle, E. Johnson, and R. Wets. Georges B. Dantzig (1914–2005). Notices Amer. Math. Society,54(3) :344–362, march 2007.

[2] D. Gale. Linear programming and the simplex method. Notices Amer. Math. Society, 54(3) :364–369,march 2007.

[3] J. Matoušek and B. Gärtner. Understanding and using linear programming. Universitext. springer,2007.

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8 Epsilon-nets, cuttings et recherche simpliciale

ê cours ê

Hypergraphes - dimension de Vapnik-Chervonenkis ou cœfficient de trace - epsilon-nets- théorème des epsilon-nets - arbres de recherche simpliciale - cuttings - partitions sim-pliciales - arrangements de droites : triangulation canonique, zone d’une droite, dualitépoint-droite, etc. -


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Références

[1] D. Haussler and E. Welzl. Epsilon-nets and simplex range queries. Discrete Comput. Geom., 2 :127–151, 1987.

[2] J. Matoušek. Geometric range searching. ACM Comput. Surv., 26 :421–461, 1994.

[3] J. Matoušek. Lectures on Discrete Geometry. Number 212 in Graduate texts in Mathematics.Springer-Verlag, 2002.

[4] K. Mulmuley. Computational Geometry : An Introduction Through Randomized Algorithms. Prentice-Hall, 1994.

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9 A suivre

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