In this paper we describe the convex hull of all solutions of the integer bounded knapsack problem in the special case when the weights of the items are divisible. The corresponding inequalities are defined via an inductive scheme that can also be used in a more general setting.

We investigate dominance relations between basic semidefinite relaxations and classes of cuts. We show that simple semidefinite relaxations are tighter than corresponding linear relaxations even in case of linear cost functions. Numerical results are presented illustrating the quality of these relaxations.

This book was typeset with TEX using L ATEX and many further formatting packages. The pictures were prepared using pstricks, xfig, gnuplot and gmt. All numerals in this text are recycled. Für meine Eltern Preface Avoid reality at all costs — fortune(6) As the inclined reader will find out soon enough, this thesis is not about deeply involved mathematics as a mean in itself, but about how to apply mathematics to solve real-world problems. We will show how to shape, forge, and yield our tool of choice to rapidly answer questions of concern to people outside the world of mathematics. But there is more to it. Our tool of choice is software. This is not unusual, since it has become standard practice in science to use software as part of experiments and sometimes even for proofs. But in order to call an experiment scientific it must be reproducible. Is this the case?

In order to gain insight into the quality of semidefinite relaxations of constrained quadratic 0/1 programming problems we study the quadratic knapsack problem. We investigate several basic semidefinite relaxations of this problem and compare their strength in theory and in practice. Various possibilities to improve these basic relaxations by cutting planes are discussed. The cutting planes either arise from quadratic representations of linear inequalities or from linear inequalities in the quadratic model. In particular, a large family of combinatorial cuts is introduced for the linear formulation of the knapsack problem in quadratic space. Computational results on a small class of practical problems illustrate the quality of these relaxations and cutting planes.

Let a set N of items, a capacity F 2 IN and weights a i 2 IN, i 2 N be given. The 0/1 knapsack polytope is the convex hull of all 0/1 vectors that satisfy the inequality X i2N a i x i F: In this paper we present a linear description of the 0/1 knapsack polytope for the special case where a i 2 f; g for all items i 2 N and 1 ! b are two natural numbers. The inequalities needed for this description involve elements of the Hilbert basis of a certain cone. The principle of generating inequalities based on elements of a Hilbert basis suggests further extensions. Keywords: complete description, facets, Hilbert basis, knapsack polytope, knapsack problem, separation 1 Introduction and Notation Let a set N of items, a capacity F 2 IN and weights a i 2 IN, i 2 N be given. The problem considered in this paper is the special case of the 0/1 knapsack problem, P i2N a i x i F , x i 2 f0; 1g; i 2 N where ! are given natural numbers, N = N 1 [N 2 is the set of items and N 1 contains all ...

. Recent advances in optical fiber systems and transmission equipments play a primary role in today's telecommunications networks. The standardized SDH (Synchronous Digital Hierarchy) technology made the rings the most survivable architecture for designing high speed networks. Designing such networks involves not only the configuration of rings at the logical level but also the mapping of such configurations on the physical level, i.e. fiber paths. This paper deals with the problem of finding optimal clusters of nodes to be used as the basis for designing logical rings. In the simplest form, this problem is equivalent to k-equipartition, and can be modeled as a quadratic pseudo-boolean problem. We consider both linear and semidefinite relaxations and present numerical results on real data from France Telecom networks with up 900 nodes, and also on randomly generated problems. Key words. Graph Partitioning, Semidefinite Programming. 1. Introduction In the early eighties, a new broadb...

We use column generation and a specialized branching technique for solving constrained clustering problems. We also develop and implement an innovative combinatorial method for solving the pricing subproblems. Computational experiments comparing the resulting branch-and-price method to competing methodologies in the literature are presented and suggest that our technique yields a significant improvement on the hard instances of this problem. 1 Introduction Given a graph G(V; E) where V is the vertex set and E is the edge set, and edge weights w e ; e 2 E, we consider clustering problems that involve partitioning G into connected subgraphs or clusters such that the sum of the edge weights in every cluster is maximized (or, equivalently, the total weight on edges between clusters in minimized). Several graph partitioning type problems have been studied in the literature. Uncapacitated versions of this problem model a common problem in qualitative data analysis of partitioning a number...

The topic of this paper are integer programming models in which a subset of 0/1-variables encode a partitioning of a set of objects into disjoint subsets. Such models can be surprisingly hard to solve by branch-and-cut algorithms if the order of the subsets of the partition is irrelevant. This kind of symmetry unnecessarily blows up the branchand-cut tree. We present a general tool, called orbitopal fixing, for enhancing the capabilities of branch-and-cut algorithms in solving such symmetric integer programming models. We devise a linear time algorithm that, applied at each node of the branch-and-cut tree, removes redundant parts of the tree produced by the above mentioned symmetry. The method relies on certain polyhedra, called orbitopes, which have been investigated in [11]. It does, however, not add inequalities to the model, and thus, it does not increase the difficulty of solving the linear programming relaxations. We demonstrate the computational power of orbitopal fixing at the example of a graph partitioning problem motivated from frequency planning in mobile telecommunication networks.

This paper investigates relations among combinatorial optimization problems. To establish such relations we introduce a transformation technique ---aggregation--- that allows to relax an integer program by means of another integer program. We prove that various families of prominent inequalities for the acyclic subdigraph problem, the multiple knapsack problem, the max cut, graph, and the clique partitioning problem, the set covering problem, and the set packing problem can be derived and separated in polynomial time in this way. Our technique is algorithmic. It has been implemented and used in a set partitioning code.

This paper investigates a technique of building up discrete relaxations of combinatorial optimization problems. To establish such a relaxation we introduce a transformation technique - aggregation - that allows one to relax an integer program by means of another integer program. We show that knapsack and set packing relaxations give rise to combinatorial cutting planes in a simple and straightforward way. The constructions are algorithmic.