Approximate Clustering via Metric Partitioning

In this paper we consider two metric covering/clustering problems - \textit{Minimum Cost Covering Problem} (MCC) and $k$-clustering. In the MCC problem, we are given two point sets $X$ (clients) and $Y$ (servers), and a metric on $X \cup Y$. We would like to cover the clients by balls centered at the servers. The objective function to minimize is the sum of the $\alpha$-th power of the radii of the balls. Here $\alpha \geq 1$ is a parameter of the problem (but not of a problem instance). MCC is closely related to the $k$-clustering problem. The main difference between $k$-clustering and MCC is that in $k$-clustering one needs to select $k$ balls to cover the clients. For any \eps > 0, we describe quasi-polynomial time (1 + \eps) approximation algorithms for both of the problems. However, in case of $k$-clustering the algorithm uses (1 + \eps)k balls. Prior to our work, a $3^{\alpha}$ and a ${c}^{\alpha}$ approximation were achieved by polynomial-time algorithms for MCC and $k$-clustering, respectively, where $c > 1$ is an absolute constant. These two problems are thus interesting examples of metric covering/clustering problems that admit (1 + \eps)-approximation (using (1+\eps)k balls in case of $k$-clustering), if one is willing to settle for quasi-polynomial time. In contrast, for the variant of MCC where $\alpha$ is part of the input, we show under standard assumptions that no polynomial time algorithm can achieve an approximation factor better than $O(\log |X|)$ for $\alpha \geq \log |X|$.Comment: 19 page

Програмна реалізація методу k-середніх інтелектуальної інформаційно-управляючої системи виробництва комбікорму

The issue of classifying data when managing technological processes and compound feed production was substantiated. The program complex performs functions of integrated optimization and analysis of compound feed and premix rations. The programs involve new pattern of ration which first takes into account losses caused by unbalanced feeding (reduction of productivity, reproduction, health and breeding abilities). The programs allow comprehensive optimization of rations, determination of necessary feed additives and formulation of recipes of compound feeds, premixes, PVMA (protein-vitamin-mineral additives), which are very well combined with basic feed and considered when planning the feed expenditure. The issue of optimization of feed production plan is important for all agricultural enterprises with livestock industries, but it is extremely topical for livestock-based industries producing feeds, since it allows finding additional feed production reserves by means of improving the structure of sown areas and feed expenditure. Before creating livestock complexes, it is necessary to identify the sources and amounts of feed supply. Substantiation of the feed base and computing of the plan variants should be implemented using methods of mathematical modeling and ECM. The analysis of data classification was made and the choice of the k-means method for feed components classification was substantiated. The software for the k-means algorithm implementation was developed and various algorithm patterns, depending on initial conditions, were worked out.Обоснована задача классификации данных при управлении технологическими процессами и производством комбикорма. Проведен анализ классификации данных, обоснован выбор метода k-средних для классификации компонентов комбикорма. Разработано программное обеспечение для реализации алгоритма k-средних и отработаны разные варианты поведения алгоритма в зависимости от начальных условий.Обгрунтована задача класифікації даних при керуванні технологічними процесами та виробництвом комбікорму. Проведений аналіз класифікації даних, обґрунтований вибір методу k-середніх для класифікації компонентів комбікорму. Розроблено програмне забезпечення для реалізації алгоритму k-середніх та відпрацьовані різні варіанти поведінки алгоритму в залежності від початкових умов

Програмна реалізація методу k-середніх інтелектуальної інформаційно-управляючої системи виробництва комбікорму

The issue of classifying data when managing technological processes and compound feed production was substantiated. The program complex performs functions of integrated optimization and analysis of compound feed and premix rations. The programs involve new pattern of ration which first takes into account losses caused by unbalanced feeding (reduction of productivity, reproduction, health and breeding abilities). The programs allow comprehensive optimization of rations, determination of necessary feed additives and formulation of recipes of compound feeds, premixes, PVMA (protein-vitamin-mineral additives), which are very well combined with basic feed and considered when planning the feed expenditure. The issue of optimization of feed production plan is important for all agricultural enterprises with livestock industries, but it is extremely topical for livestock-based industries producing feeds, since it allows finding additional feed production reserves by means of improving the structure of sown areas and feed expenditure. Before creating livestock complexes, it is necessary to identify the sources and amounts of feed supply. Substantiation of the feed base and computing of the plan variants should be implemented using methods of mathematical modeling and ECM. The analysis of data classification was made and the choice of the k-means method for feed components classification was substantiated. The software for the k-means algorithm implementation was developed and various algorithm patterns, depending on initial conditions, were worked out.Обоснована задача классификации данных при управлении технологическими процессами и производством комбикорма. Проведен анализ классификации данных, обоснован выбор метода k-средних для классификации компонентов комбикорма. Разработано программное обеспечение для реализации алгоритма k-средних и отработаны разные варианты поведения алгоритма в зависимости от начальных условий.Обгрунтована задача класифікації даних при керуванні технологічними процесами та виробництвом комбікорму. Проведений аналіз класифікації даних, обґрунтований вибір методу k-середніх для класифікації компонентів комбікорму. Розроблено програмне забезпечення для реалізації алгоритму k-середніх та відпрацьовані різні варіанти поведінки алгоритму в залежності від початкових умов

Air Force Institute of Technology Research Report 2007

This report summarizes the research activities of the Air Force Institute of Technology’s Graduate School of Engineering and Management. It describes research interests and faculty expertise; lists student theses/dissertations; identifies research sponsors and contributions; and outlines the procedures for contacting the school. Included in the report are: faculty publications, conference presentations, consultations, and funded research projects. Research was conducted in the areas of Aeronautical and Astronautical Engineering, Electrical Engineering and Electro-Optics, Computer Engineering and Computer Science, Systems and Engineering Management, Operational Sciences, Mathematics, Statistics and Engineering Physics

Hinged Dissections Exist

We prove that any finite collection of polygons of equal area has a common hinged dissection. That is, for any such collection of polygons there exists a chain of polygons hinged at vertices that can be folded in the plane continuously without self-intersection to form any polygon in the collection. This result settles the open problem about the existence of hinged dissections between pairs of polygons that goes back implicitly to 1864 and has been studied extensively in the past ten years. Our result generalizes and indeed builds upon the result from 1814 that polygons have common dissections (without hinges). We also extend our common dissection result to edge-hinged dissections of solid 3D polyhedra that have a common (unhinged) dissection, as determined by Dehn's 1900 solution to Hilbert's Third Problem. Our proofs are constructive, giving explicit algorithms in all cases. For a constant number of planar polygons, both the number of pieces and running time required by our construction are pseudopolynomial. This bound is the best possible, even for unhinged dissections. Hinged dissections have possible applications to reconfigurable robotics, programmable matter, and nanomanufacturing.Comment: 22 pages, 14 figure