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#### Schlagworte

- Grid Graphs (1)
- Minkowski space (1)
- Optimierung (1)
- Standorttheorie (1)
- Verkehsplanung (1)
- activity-based model (1)
- center and median problems (1)
- center hyperplane (1)
- centrally symmetric polytope (1)
- common transversal (1)

#### Fachbereich / Organisatorische Einheit

- Fachbereich Mathematik (15)
- Fraunhofer (ITWM) (1)

In this paper we deal with locating a line in the plane. If d is a distance measure our objective is to find a straight line l which minimizes f(l) of g(l) (see the paper for the definition of these functions). We show that for all distance measures d derived from norms, one of the lines minimizing f(l) contains at least two of the existing facilities. For the center objective we always get an optimal line which is at maximum distance from at least three of the existing facilities. If all weights are equal, there is an optimal line which is parallel to one facet of the convex hull of the existing facilities.

In line location problems the objective is to find a straight line which minimizes the sum of distances, or the maximum distance, respectively to a given set of existing facilities in the plane. These problems have well solved. In this paper we deal with restricted line location problems, i.e. we have given a set in the plane where the line is not allowed to pass through. With the help of a geometric duality we solve such problems for the vertical distance and then extend these results to block norms and some of them even to arbitrary norms. For all norms we give a finite candidate set for the optimal line.

In this survey we deal with the location of hyperplanes in n-dimensional normed spaces, i.e., we present all known results and a unifying approach to the so-called median hyperplane problem in Minkowski spaces. We describe how to find a hyperplane H minimizing the weighted sum f(H) of distances to a given, finite set of demand points. In robust statistics and operations research such an optimal hyperplane is called a median hyperplane.After summarizing the known results for the Euclidean and rectangular situation, we show that for all distance measures d derived from norms one of the hyperplanes minimizing f(H) is the affine hull of n of the demand points and, moreover, that each median hyperplane is a halving one (in a sense defined below) with respect to the geiven point set. Also an independence of norm result for finding optimal hyperplanes with fixed slope will be given. Furthermore we discuss how these geometric criteria can be used for algorithmical approaches to median hyperplanes, with an extra discussion for the case of polyhedral norms. And finally a characterizatio of all smooth norms by a sharpened incidence criterion for median hyperplanes is mentioned.

In this paper we consider the problem of optimizing a piecewise-linear objective function over a non-convex domain. In particular we do not allow the solution to lie in the interior of a prespecified region R. We discuss the geometrical properties of this problems and present algorithms based on combinatorial arguments. In addition we show how we can construct quite complicated shaped sets R while maintaining the combinatorial properties.

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

In this paper we deal with the location of hyperplanes in n-dimensional normed spaces. If d is a distance measure, our objective is to find a hyperplane H which minimizes f(H) = sum_{m=1}^{M} w_{m}d(x_m,H), where w_m ge 0 are non-negative weights, x_m in R^n, m=1, ... ,M demand points and d(x_m,H)=min_{z in H} d(x_m,z) is the distance from x_m to the hyperplane H. In robust statistics and operations research such an optimal hyperplane is called a median hyperplane. We show that for all distance measures d derived from norms, one of the hyperplanes minimizing f(H) is the affine hull of n of the demand points and, moreover, that each median hyperplane is (ina certain sense) a halving one with respect to the given point set.

The anchored hyperplane location problem is to locate a hyperplane passing through some given points P IR^n and minimizing either the sum of weighted distances (median problem), or the maximum weighted distance (center problem) to some other points Q IR^n . If the distances are measured by a norm, it will be shown that in the median case there exists an optimal hyperplane that passes through at least n - k affinely independent points of Q, if k is the maximum number of affinely independent points of P. In the center case, there exists an optimal hyperplane which isatmaximum distance to at least n - k + 1 affinely independent points of Q. Furthermore, if the norm is a smooth norm, all optimal hyperplanes satisfy these criteria. These new results generalize known results about unrestricted hyperplane location problems.

We consider the problem of locating a line or a line segment in three- dimensional space, such that the sum of distances from the linear facility to a given set of points is minimized. An example is planning the drilling of a mine shaft, with access to ore deposits through horizontal tunnels connecting the deposits and the shaft. Various models of the problem are developed and analyzed, and effcient solution methods are given.

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.