Randomized Contractions for Multiobjective Minimum Cuts

We show that Karger\u27s randomized contraction method (SODA 93) can be adapted to multiobjective global minimum cut problems with a constant number of edge or node budget constraints to give efficient algorithms. For global minimum cuts with a single edge-budget constraint, our extension of the randomized contraction method has running time tilde{O}(n^3) in an n-node graph improving upon the best-known randomized algorithm with running time tilde{O}(n^4) due to Armon and Zwick (Algorithmica 2006). Our analysis also gives a new upper bound of O(n^3) for the number of optimal solutions for a single edge-budget min cut problem. For the case of (k-1) edge-budget constraints, the extension of our algorithm saves a logarithmic factor from the best-known randomized running time of O(n^{2k} log^3 n). A main feature of our algorithms is to adaptively choose, at each step, the appropriate cost function used in the random selection of edges to be contracted. For the global min cut problem with a constant number of node budgets, we give a randomized algorithm with running time tilde{O}(n^2), improving the current best determinisitic running time of O(n^3) due to Goemans and Soto (SIAM Journal on Discrete Mathematics 2013). Our method also shows that the total number of distinct optimal solutions is bounded by O(n^2) as in the case of global min-cuts. Our algorithm extends to the node-budget constrained global min cut problem excluding a given sink with the same running time and bound on number of optimal solutions, again improving upon the best-known running time by a factor of O(n). For node-budget constrained problems, our improvements arise from incorporating the idea of merging any infeasible super-nodes that arise during the random contraction process. In contrast to cuts excluding a sink, we note that the node-cardinality constrained min-cut problem containing a given source is strongly NP-hard using a reduction from graph bisection

Ki-covers I: Complexity and polytopes

AbstractA Ki in a graph is a complete subgraph of size i. A Ki-cover of a graph G(V, E is a set C of Ki − 1's of G such that every Ki in G contains at least one Ki − 1 in C. Thus a K2-cover is a vertex cover. The problem of determining whether a graph has a Ki-cover (i ⩾ 2) of cardinality ⩽k is shown to be NP-complete for graphs in general. For chordal graphs with fixed maximum clique size, the problem is polynomial; however, it is NP-complete for arbitrary chordal graphs when i ⩾ 3. The NP-completeness results motivate the examination of some facets of the corresponding polytope. In particular we show that various induced subgraphs of G define facets of the Ki-cover polytope. Further results of this type are also produced for the K3-cover polytope. We conclude by describing polynomial algorithms for solving the separation problem for some classes of facets of the Ki-cover polytope

Distance Transformation for Network Design Problems

International audienceWe propose a new generic way to construct extended formulations for a large class of network design problems with given connectivity requirements. The approach is based on a graph transformation that maps any graph into a layered graph according to a given distance function. The original connectivity requirements are in turn transformed into equivalent connectivity requirements in the layered graph. The mapping is extended to the graphs induced by fractional vectors through an extended linear integer programming formulation. While graphs induced by binary vectors are mapped to isomorphic layered graphs, those induced by fractional vectors are mapped to a set of graphs having worse connectivity properties. Hence, the connectivity requirements in the layered graph may cut off fractional vectors that were feasible for the problem formulated in the original graph. Experiments over instances of the Steiner Forest and Hop-constrained Survivable Network Design problems show that significant gap reductions over the state-of-the art formulations can be obtained

The k-edge connected subgraph problem: Valid inequalities and Branch-and-Cut

International audienceIn this paper we consider the k-edge connected subgraph problem from a polyhedral point of view. We introduce further classes of valid inequalities for the associated polytope, and describe sufficient conditions for these inequalities to be facet defining. We also devise separation routines for these inequalities, and discuss some reduction operations that can be used in a preprocessing phase for the separation. Using these results, we develop a Branch-and-Cut algorithm and present some computational results

A min-max relation for K3-covers in graphs noncontractible to K5e

AbstractIn Euler and Mahjoub (1991) it is proved that the triangle-free subgraph polytope of a graph noncontractible to K5e is completely described by the trivial inequalities and the so-called triangle and odd wheel inequalities. In this paper we show that the system denned by those inequalities is TDI for a subclass of that class of graphs. As a consequence we obtain the following min-max relation: If G is a graph noncontractible to K5e, then the minimum number of edges covering all the triangles of G equals the maximum profit of a partition of the edge set of G into edges, triangles and odd wheels. Here the profit of an edge is 0, the profit of a triangle is 1 and the profit of a 2k + 1-wheel (k ϵ N) is equal to k + 1

Étude de structures combinatoires issues de la physique statistique et d'autres domaines

Université : Université scientifique et médicale de GrenobleÉtude de certains problèmes d'optimisation combinatoire. Le premier concerne un problème de régulation de trafic pour lequel on donne une formulation mathématique et on propose une méthode permettant de le résoudre. Le deuxième problème traité est un des problèmes de la physique statistique qui relève de la combinatoire et de l'optimisation, celui du fondamental d'un verre de spins (modèle d'Ising). Enfin on étudie, deux autres problèmes d'optimisation combinatoire: l'absorbant et le Ki-recouvrement de poids minimu

I. Résolution d'un problème de régulation de trafic<br />II. Polytope des absorbants d'un graphe à seuil

Universités : Université scientifique et médicale de Grenoble et Institut national polytechnique de GrenobleThe first part deals with a network of roads on which traffic from several origins going toward several destinations is engaged. All characteristics of the network, as well as the traffic data, are presumed constant in time. Given that the drivers on the network affect each other as specified in the Wardrop principle, how can the real capacity of the network arcs be lowered, and can the total time spent by all the users of the network be minimized. In the second part, the absorbers of a class of graphs defined by C. Benzahen and P. L. Hammer are studiedDans la première partie on étudie un réseau routier sur lequel circule un trafic entre plusieurs origines et plusieurs destinations. Toutes les caractéristiques du réseau ainsi que les données de trafic sont supposées constantes dans le temps. Sachant que les usagers de ce réseau s'affectent suivant le principe de Wardrop, comment alors en abaissant éventuellement les capacités réelles des arcs du réseau, peut-on minimiser le temps total passé par tous les usagers de ce réseau. Dans la deuxième partie, on étudie les absorbants d'une classe de graphes définie par Claude Benzaken et P. L. Hammer

Conception de réseaux 2-connexes avec contraintes de bornes

CLERMONT FD-BCIU Sci.et Tech. (630142101) / SudocSudocFranceF

On the linear relaxation of the 2-node connected subgraph polytope

In this paper, we study the linear relaxation P(G) of the 2-node connected subgraph polytope of a graph G. We introduce an ordering on the fractional extreme points of P(G) and we give a characterization of the minimal extreme points with respect to that ordering. This yields a polynomial method to separate a minimal extreme point of P(G) from the 2-node connected subgraph polytope. It also provides a new class of facet defining inequalities for this polytope. © 1999 Elsevier Science B.V. All rights reserved.SCOPUS: ar.jinfo:eu-repo/semantics/publishe

Sécurisation et dimensionnement de réseaux multicouches (modèles et polyèdres)

Les réseaux de télécommunications évoluent vers des modèles qui consistent en un certain nombre de routeurs IP interconnectés par un réseau optique intelligent. Cette nouvelle infrastructure multicouche nécessite un haut niveau de fiabilité, de telle manière que les services du réseau puissent être rétablis en cas de panne. Dans cette thèse, nous considérons différents problèmes de sécurisation et de dimensionnement liés à cette infrastructure multicouche. Nous donnons des formulations en terme de programmes linéaires mixtes, et nous discutons des polytopes associés. Nous décrivons plusieurs classes d'inégalités valides et étudions les conditions pour qu'elles définissent des facettes. Nous discutons de procédures de séparation pour ces inégalités et induisons des opérations de réduction. Nous développons des algorithmes de coupes et branchements et de coupes génération de colonnes et branchements et présentons une étude expérimentale sur des instances aléatoires et réellesCLERMONT FD-BCIU Sci.et Tech. (630142101) / SudocSudocFranceF