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Weighted k-cardinality trees
(1992)

We consider the k -CARD TREE problem, i.e., the problem of finding in a given undirected graph G a subtree with k edges, having minimum weight. Applications of this problem arise in oil-field leasing and facility layout. While the general problem is shown to be strongly NP hard, it can be solved in polynomial time if G is itself a tree. We give an integer programming formulation of k-CARD TREE, and an efficient exact separation routine for a set of generalized subtour elimination constraints. The polyhedral structure of the convex huLl of the integer solutions is studied.

Max ordering (MO) optimization is introduced as tool for modelling production
planning with unknown lot sizes and in scenario modelling. In MO optimization a feasible solution set \(X\) and, for each \(x\in X, Q\) individual objective functions \(f_1(x),\dots,f_Q(x)\) are given. The max ordering objective
\(g(x):=max\) {\(f_1(x),\dots,f_Q(x)\)} is then minimized over all \(x\in X\).
The paper discusses complexity results and describes exact and approximative
algorithms for the case where \(X\) is the solution set of combinatorial
optimization problems and network flow problems, respectively.

We investigate two versions of multiple objective minimum spanning tree
problems defined on a network with vectorial weights. First, we want to minimize the maximum of Q linear objective functions taken over the set of all spanning trees (max linear spanning tree problem ML-ST). Secondly, we look for efficient spanning trees (multi criteria spanning tree problem MC-ST). Problem ML-ST is shown to be NP-complete. An exact algorithm which is based on ranking is presented. The procedure can also be used as an approximation scheme. For solving the bicriterion MC-ST, which in the worst case may have an exponential number of efficient trees, a two-phase procedure is presented. Based on the computation of extremal efficient spanning trees we use neighbourhood search to determine a sequence of solutions with the property that the distance
between two consecutive solutions is less than a given accuracy.

Es wird anhand von Beispielen, an denen der Autor in der Vergangenheit gearbeitet hat, gezeigt, wie man Modelle der exakten Naturwissenschaften auf wirtschaftliche Probleme
anwenden kann. Insbesondere wird diskutiert, wo Grenzen dieser Übertragbarkeit liegen. Die Arbeit ist eine Zusammenfassung eines Vortrags, der im SS 1992 im Rahmen des Studium Generale an der Universität Kaiserslautern gehalten wurde.

In this paper we consider the problem of finding in a given graph a minimal weight subtree of connected subgraph, which has a given number of edges. These NP-hard combinatorial optimization problems have various applications in the oil industry, in facility layout and graph partitioning. We will present different heuristic approaches based on spanning tree and shortest path methods and on an exact algorithm solving the problem in polynomial time if the underlying graph is a tree. Both the edge- and node weighted case are investigated and extensive numerical results on the behaviour of the heuristics compared to optimal solutions are presented. The best heuristic yielded results within an error margin of less than one percent from optimality for most cases. In a large percentage of tests even optimal solutions have been found.

Finding "good" cycles in graphs is a problem of great interest in graph theory as well as in locational analysis. We show that the center and median problems are NP hard in general graphs. This result holds both for the variable cardinality case (i.e. all cycles of the graph are considered) and the fixed cardinality case (i.e. only cycles with a given cardinality p are feasible). Hence it is of interest to investigate special cases where the problem is solvable in polynomial time. In grid graphs, the variable cardinality case is, for instance, trivially solvable if the shape of the cycle can be chosen freely. If the shape is fixed to be a rectangle one can analyse rectangles in grid graphs with, in sequence, fixed dimension, fixed cardinality, and variable cardinality. In all cases a com plete characterization of the optimal cycles and closed form expressions of the optimal objective values are given, yielding polynomial time algorithms for all cases of center rectangle problems. Finally, it is shown that center cycles can be chosen as rectangles for small cardinalities such that the center cycle problem in grid graphs is in these cases completely solved.

Location problems with Q (in general conflicting) criteria are considered. After reviewing previous results of the authors dealing with lexicographic and Pareto location the main focus of the paper is on max-ordering locations. In these location problems the worst of the single objectives is minimized. After discussing some general results (including reductions to single criterion problems and the relation to lexicographic and Pareto locations) three solution techniques are introduced and exemplified using one location problem class, each: The direct approach, the decision space approach and the objective space approach. In the resulting solution algorithms emphasis is on the representation of the underlying geometric idea without fully exploring the computational complexity issue. A further specialization of max-ordering locations is obtained by introducing lexicographic max-ordering locations, which can be found efficiently. The paper is concluded by some ideas about future research topics related to max-ordering location problems.

The Weber problem for a given finite set of existing facilities {cal E}x = {Ex_1,Ex_2, ... ,Ex_M} subset R^2 with positive weights w_m (m = 1, ... ,M) is to find a new facility X* in R^2 such that sum_{m=1}^{M} w_{m}d(X,Ex_m) is minimized for some distance function d. In this paper we consider distances defined by polyhedral gauges. A variation of this problem is obtained if barriers are introduced which are convex polygonal subsets of the plane where neither location of new facilities nor traveling is allowed. Such barriers like lakes, military regions, national parks or mountains are frequently encountered in practice.From a mathematical point of view barrier problems are difficult, since the prensence of barriers destroys the convexity of the objective function. Nevertheless, this paper establishes a descretization result: One of the grid points in the grid defined by the existing facilities and the fuundamental directions of the gauge distances can be proved to be an optimal location. Thus the barrier problem can be solved with a polynomial algorithm.

The problem of finding an optimal location X* minimizing the maximum Euclidean distance to existing facilities is well solved by e.g. the Elzinga-Hearn algorithm. In practical situations X* will however often not be feasible. We therefore suggest in this note a polynomial algorithm which will find an optimal location X^F in a feasible subset F of the plane R^2

There are several good reasons to introduce classification schemes for optimization problems including, for instance, the ability for concise problem statement opposed to verbal, often ambiguous, descriptions or simple data encoding and information retrieval in bibliographical information systems or software libraries. In some branches like scheduling and queuing theory classification is therefore a widely accepted and appreciated tool. The aim of this paper is to propose a 5-position classification which can be used to cover all location problems. We will provide a list of currentliy available symbols and indicate its usefulness in a - necessarily non-comprehensive - list of classical location problems. The classification scheme is in use since 1992 and has since proved to be useful in research, software development, classroom, and for overview articles.