Curve Reconstruction, the Traveling Salesman Problem, and Menger's Theorem on Length

We give necessary and sufficient regularity conditions under which the curve reconstruction problem is solved by a traveling salesman tour or path, respectively. For the proof we have to generalize a theorem of Menger [12], [13] on arc lengt

Fast Hierarchical Clustering and Other Applications of Dynamic Closest Pairs

We develop data structures for dynamic closest pair problems with arbitrary distance functions, that do not necessarily come from any geometric structure on the objects. Based on a technique previously used by the author for Euclidean closest pairs, we show how to insert and delete objects from an n-object set, maintaining the closest pair, in O(n log^2 n) time per update and O(n) space. With quadratic space, we can instead use a quadtree-like structure to achieve an optimal time bound, O(n) per update. We apply these data structures to hierarchical clustering, greedy matching, and TSP heuristics, and discuss other potential applications in machine learning, Groebner bases, and local improvement algorithms for partition and placement problems. Experiments show our new methods to be faster in practice than previously used heuristics.Comment: 20 pages, 9 figures. A preliminary version of this paper appeared at the 9th ACM-SIAM Symp. on Discrete Algorithms, San Francisco, 1998, pp. 619-628. For source code and experimental results, see http://www.ics.uci.edu/~eppstein/projects/pairs

SCIL - Symbolic Constraints in Integer Linear Programming

We describe SCIL. SCIL introduces symbolic constraints into branch-and-cut-and-price algorithms for integer linear programs. Symbolic constraints are known from constraint programming and contribute significantly to the expressive power, ease of use, and efficiency of constraint programs

Spanning trees short or small

We study the problem of finding small trees. Classical network design problems are considered with the additional constraint that only a specified number $k$ of nodes are required to be connected in the solution. A prototypical example is the $k$MST problem in which we require a tree of minimum weight spanning at least $k$ nodes in an edge-weighted graph. We show that the $k$MST problem is NP-hard even for points in the Euclidean plane. We provide approximation algorithms with performance ratio $2\sqrt{k}$ for the general edge-weighted case and $O(k^{1/4})$ for the case of points in the plane. Polynomial-time exact solutions are also presented for the class of decomposable graphs which includes trees, series-parallel graphs, and bounded bandwidth graphs, and for points on the boundary of a convex region in the Euclidean plane. We also investigate the problem of finding short trees, and more generally, that of finding networks with minimum diameter. A simple technique is used to provide a polynomial-time solution for finding $k$-trees of minimum diameter. We identify easy and hard problems arising in finding short networks using a framework due to T. C. Hu.Comment: 27 page

ZASTOSOWANIE SZTUCZNEJ INTELIGENCJI W ZAUTOMATYZOWANYCH CENTRACH LOGISTYCZNYCH

The paper deals with the problem of works transport organization in logistic center with the use of artificial intelligence algorithms. The presented approach is based on non-changeable path during travel along a given loop. The ordered set of containers requesting transport service was determined by fuzzy logic, while the sequence of containers in a loop was optimized by genetic algorithms. A solution for semi-autonomous transport vehicles wherein the control system informs the driver about optimal route was presented. The obtained solution was verified by a computer simulation.Artykuł dotyczy problematyki sterowania transportem wewnątrzzakładowym w zautomatyzowanych centrach logistycznych z zastosowaniem metod sztucznej inteligencji. Zaprezentowane podejście zakłada predykcję niezmiennej trasy  przejazdu środka transportu. Kolejność zbioru regałów wymagających obsługi transportowej jest determinowana przez logikę rozmytą, natomiast do optymalizacja trasy przejazdu wykorzystano algorytmy genetyczne. Zaprezentowano koncepcję środka transportu, w którym system sterowania informuje kierowcę dokąd ma jechać. Uzyskane rozwiązanie zostało zweryfikowane z wykorzystaniem metod symulacji komputerowej