Conference: Combinatorial Optimization - Aussois, 2003]]>

This paper presents new algorithms for the maximum flow problem, the Hitchcock transportation problem, and the general minimum-cost flow problem. Upper bounds on the numbers of steps in these algorithms are derived, and are shown to compale favorably with upper bounds on the numbers of steps required by earlier algorithms. First, the paper states the maximum flow problem, gives ...

Conference: Combinatorial Optimization - Aussois, pp. 31-33, 2001]]>The viewpoint of the subject of matroids, and related areas of lattice theory, has always been, in one way or another, abstraction of algebraic dependence or, equivalently, abstraction of the incidence relations in geometric representations of algebra. Often one of the main derived facts is that all bases have the same cardinality. (See Van der Waerden, Section 33.)

Conference: Combinatorial Optimization - Aussois, pp. 11-26, 2001]]>A main purpose of this work is to give a good algorithm for a certain well-described class of integer linear programming problems, called matching problems (or the matching problem). Methods developed for simple matching [2]

Conference: Combinatorial Optimization - Aussois, pp. 27-30, 2001]]>We discuss fast exponential time solutions for NP-complete problems. We survey known results and approaches, we provide pointers to the literature, and we discuss several open problems in this area. The list of discussed NP-complete problems includes the travelling salesman problem, scheduling under precedence constraints, satisfiability, knapsack, graph coloring, independent sets in graphs, bandwidth of a graph, and ...

Conference: Combinatorial Optimization - Aussois, pp. 185-208, 2001]]>Blind and Mani (2) proved that the entire combinatorial structure (the vertex-facet incidences) of a simple convex polytope is de- termined by its abstract graph. Their proof is not constructive. Kalai (15) found a short, elegant, and algorithmic proof of that result. However, his algorithm has always exponential running time. We show that the prob- lem to reconstruct the ...

Conference: Combinatorial Optimization - Aussois, pp. 105-118, 2001]]>This article deals with a new generalization of the well-known “Travelling Salesman Problem” (TSP) in which cities correspond to customers providing or requiring known amounts of a product, and the vehicle has a given capacity and is located in a special city called depot. Each customer and the depot must be visited exactly once by the vehicle serving the ...

Conference: Combinatorial Optimization - Aussois, pp. 89-104, 2001]]>Recently, Fischetti, Lodi and Toth [15] surveyed exact methods for the Asymmetric Traveling Salesman Problem (ATSP) and computationally compared branch-and-bound and branch-and-cut codes. The results of this comparison proved that branch-and-cut is the most effective method to solve hard ATSP instances. In the present paper the branch-and-cut algorithms by Fischetti and Toth [...

Conference: Combinatorial Optimization - Aussois, pp. 64-77, 2001]]>We discuss the role of mixed-integer value functions in the theoretical analysis of stochastic integer programs. It is shown how the interaction of value function properties with basic results from probability theory leads to structural statements in stochastic integer programming.

Conference: Combinatorial Optimization - Aussois, pp. 171-184, 2001]]>We present a finitely convergent cutting plane algorithm for 0-1 mixed integer programming. The algorithm is a hybrid between a strong cutting plane and a Gomory-type algorithm that generates violated facet-defining inequalities of a relaxation of the simplex tableau and uses them as cuts for the original problem. We show that the cuts can be computed in ...

Conference: Combinatorial Optimization - Aussois, pp. 158-170, 2001]]>In recent years the branch-and-cut method, a synthesis of the classical branch-and-bound and cutting plane methods, has proven to be a highly successful approach to solving large-scale integer programs to optimality. This is especially true for mixed 0-1 and pure 0-1 problems. However, other approaches to integer programming are possible. One alternative is ...

Conference: Combinatorial Optimization - Aussois, pp. 119-133, 2001]]>Solving the well known relaxations for large scale combinatorial optimization problems directly is out of reach. We use Lagrangian relaxations and solve it with the bundle method. The cutting plane model at each iteration which approximates the original problem can be kept moderately small and we can solve it very quickly. We report successful numerical results for approximating maximum cut.

Conference: Combinatorial Optimization - Aussois, pp. 78-88, 2001]]>We propose a new procedure of facet composition for the Symmetric Traveling Salesman Polytope(STSP). Applying this procedure to the well-known comb inequalities, we obtain completely or partially known classes of inequalities like clique-tree, star, hyperstar, ladder inequalities for STSP. This provides a proof that a large subset of hyperstar inequalities which are until now only known to ...

Conference: Combinatorial Optimization - Aussois, pp. 134-146, 2001]]>In this paper we relate the consecutive ones problem to the betweenness problem by pointing out connections between their associated polytopes. We will prove some results about the facet structure of the betweenness polytope and show how facets of this polytope can be used to generate facets of the consecutive ones polytope. Furthermore, the relations with the consecutive ones polytopes ...

Conference: Combinatorial Optimization - Aussois, pp. 147-157, 2001]]>An algorithmic characterization of a particular combinatorial optimization problem means that there is an algorithm that works exact if and only if applied to the combinatorial optimization problem under investigation. According to Jack Edmonds, the Greedy algorithm leads to an algorithmic characterization of matroids. We deal here with the algorithmic characterization of the intersection of two matroids. To this end ...

Conference: Combinatorial Optimization - Aussois, pp. 48-63, 2001]]>A connected matching M in a graph G is a matching such that every pair of edges of M is joined by an edge of G. Plummer, Stiebitz and Toft introduced connected matchings in connection with their study of the famous Hadwiger Conjecture. In this paper, I prove that the connected matching problem is NP-complete for 0-1-weighted ...

Conference: Combinatorial Optimization - Aussois, pp. 34-38, 2001]]>Odd cycles are well known to induce facet-defining inequalities for the stable set polytope. In graph coloring odd cycles represent the class of 3-critical graphs. We study Hajós’ construction to obtain a large class of n-critical graphs (n > 3), which properly generalize both cliques and odd cycles, and which again turn out to be facet-inducing...

Conference: Combinatorial Optimization - Aussois, pp. 39-47, 2001]]>Conference: Combinatorial Optimization - Aussois, pp. 1-10, 2001]]>

We consider a generalization of the Prize Collecting Steiner Tree Problem on a graph with special redundancy requirements on a subset of the customer nodes suitable to model a real world problem occurring in the extension of fiber optic communication networks. We strengthen an existing connection-based mixed integer programming formulation involving exponentially many variables using a recent result with ...

Conference: Combinatorial Optimization - Aussois]]>