Preprint
8 search hits
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Convex hulls of polyominoes
(2007)
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Sascha Kurz
- In this article we prove a conjecture of Bezdek, Brass, and Harborth concerning the maximum volume of the convex hull of any facet-to-facet connected system of $n$ unit hypercubes in $mathbb{R}^d$. For $d=2$ we enumerate the extremal polyominoes and determine the set of possible areas of the convex hull for each $n$.
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Zwei auf einen Streich: Optimierte dynamische Einsatzplanung für Gelbe Engel und Lastenaufzüge
(2007)
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Jörg Rambau
Cornelius Schwarz
- Wir modellieren zwei verschiedene dynamische Einsatzplanungsprobleme: die dynamische Einsatzplanung Gelber Engel beim ADAC und die Steuerung von Lastenaufzügen in einem Versandlager der Herlitz PBS AG. Wir benutzen eine Reoptimierungspolitik, die die Steuerung des Systems mit Hilfe der Lösung von statischen Schnappschussproblemen durchführt. Für die auftretenden Schnappschussprobleme vergleichen wir zwei Modellierungsansätze (Flussmodell versus Tourenmodell), von denen nur einer echtzeittauglich ist. Das Verfahren zur dynamischen Einsatzplanung Gelber Engel ist beim ADAC in Betrieb.
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Integral point sets over finite fields
(2007)
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Sascha Kurz
- We consider point sets in the affine plane GF(q)^2 where each Euclidean distance of two points is an element of GF(q). These sets are called integral point sets and were originally defined in m-dimensional Euclidean spaces. We determine their maximal cardinality I(GF(q),2). For arbitrary commutative rings R instead of GF(q) or for further restrictions as no three points on a line or no four points on a circle we give partial results. Additionally we study the geometric structure of the examples with maximum cardinality.
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Inclusion-maximal integral point sets over finite fields
(2007)
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Michael Kiermaier
Sascha Kurz
- We consider integral point sets in affine planes over finite fields. Here an integral point set is a set of points in $GF(q)^2$ where the formally defined Euclidean distance of every pair of points is an element of $GF(q)$. From another point of view we consider point sets over $GF(q)^2$ with few and prescribed directions. So this is related to Redeis work. Another motivation comes from the field of ordinary integral point sets in Euclidean spaces. In this article we study the spectrum of integral point sets over $GF(q)^2$ which are maximal with respect to inclusion. We give some theoretical results, constructions, conjectures, and some numerical data.
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Lotsize optimization leading to a p-median problem with cardinalities
(2007)
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Constantin Gaul
Sascha Kurz
Jörg Rambau
- We consider the problem of approximating the branch and size dependent demand of a fashion discounter with many branches by a distributing process being based on the branch delivery restricted to integral multiples of lots from a small set of available lot-types. We propose a formalized model which arises from a practical cooperation with an industry partner. Besides an integer linear programming formulation and a primal heuristic for this problem we also consider a more abstract version which we relate to several other classical optimization problems like the p-median problem, the facility location problem or the matching problem.
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On the minimum diameter of plane integral point sets
(2007)
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Sascha Kurz
Alfred Wassermann
- Since ancient times mathematicians consider geometrical objects with integral side lengths. We consider plane integral point sets P, which are sets of n points in the plane with pairwise integral distances where not all the points are collinear. The largest occurring distance is called its diameter. Naturally the question about the minimum possible diameter d(2,n) of a plane integral point set consisting of n points arises. We give some new exact values and describe state-of-the-art algorithms to obtain them. It turns out that plane integral point sets with minimum diameter consist very likely of subsets with many collinear points. For this special kind of point sets we prove a lower bound for d(2,n) achieving the known upper bound n^{c_2loglog n} up to a constant in the exponent.
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There are integral heptagons, no three points on a line, no four on a circle
(2007)
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Tobias Kreisel
Sasch Kurz
- We give two configurations of seven points in the plane, no three points in a line, no four points on a circle with pairwise integral distances. This answers a famous question of Paul Erdös.
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Integral point sets over Z_n^m
(2007)
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Axel Kohnert
Sascha Kurz
- There are many papers studying properties of point sets in the Euclidean space or on integer grids, with pairwise integral or rational distances. In this article we consider the distances or coordinates of the point sets which instead of being integers are elements of Z_n, and study the properties of the resulting combinatorial structures.
