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International Conference on Combinatorial Optimization and Applications

COCOA 2008: Combinatorial Optimization and Applications pp 64–78Cite as

New Algorithms for k-Center and Extensions

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Part of the book series: Lecture Notes in Computer Science ((LNTCS,volume 5165))

Abstract

The problem of interest is covering a given point set with homothetic copies of several convex containers C 1, ..., C k , while the objective is to minimize the maximum over the dilatation factors. Such k-containment problems arise in various applications, e.g. in facility location, shape fitting, data classification or clustering. So far most attention has been paid to the special case of the Euclidean k-center problem, where all containers C i are Euclidean unit balls. New developments based on so-called core-sets enable not only better theoretical bounds in the running time of approximation algorithms but also improvements in practically solvable input sizes. Here, we present some new geometric inequalities and a Mixed-Integer-Convex-Programming formulation. Both are used in a very effective branch-and-bound routine which not only improves on best known running times in the Euclidean case but also handles general and even different containers among the C i .

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References

  1. Agarwal, P.K., Procopiuc, C.M.: Exact and approximation algorithms for clustering. In: Proc. 9th ACM-SIAM Symp. Discrete Alg., pp. 658–667 (1998)

    Google Scholar 

  2. Agarwal, P.K., Sharir, M.: Efficient algorithms for geometric optimization. ACM Comput. Surv. 30(4), 412–458 (1998)

    Article  Google Scholar 

  3. Anderberg, M.R.: Cluster analysis for applications. Probability and mathematical statistics. Academic Press, London (1973)

    MATH  Google Scholar 

  4. Avis, D.: Diameter partitioning. Discrete Comput. Geom. 1, 265–276 (1986)

    Article  MATH  MathSciNet  Google Scholar 

  5. Bespamyatnikh, S., Kirkpatrick, D.: Rectilinear 2-center problems. In: Proc. 11th Canad. Conf. Comp. Geom., pp. 68–71 (1999)

    Google Scholar 

  6. Bespamyatnikh, S., Segal, M.: Covering a set of points by two axis-parallel boxes. Inf. Process. Lett. 75(3), 95–100 (2000)

    Article  MathSciNet  Google Scholar 

  7. Bohnenblust, H.F.: Convex regions and projections in Minkowski spaces. Ann. Math. 39, 301–308 (1938)

    Article  MathSciNet  Google Scholar 

  8. Boltyanski, V., Martini, H., Soltan, P.S.: Excursions into Combinatorial Geometry. Springer, Heidelberg (1997)

    MATH  Google Scholar 

  9. Brandenberg, R., Gerken, T., Gritzmann, P., Roth, L.: Modeling and optimization of correction measures for human extremities. In: Jäger, W., Krebs, H.-J. (eds.) Mathematics – Key Technology for the Future. Joint Projects between Universities and Industry 2004-2007, pp. 131–148. Springer, Heidelberg (2008)

    Google Scholar 

  10. Brandenberg, R., Roth, L.: Optimal containment under homothetics, a practical approach (submitted, 2007)

    Google Scholar 

  11. Bădoiu, M., Har-Peled, S., Indyk, P.: Approximate clustering via core-sets. In: Proc. 34th Annu. ACM Symp. Theor. Comput., pp. 250–257 (2002)

    Google Scholar 

  12. Bunschoten, R.: A fully vectorized function that computes the Euclidean distance matrix between two sets of vectors (1999), http://www.mathworks.com/matlabcentral/fileexchange/loadFile.do?objectId=71

  13. Chan, T.M.: More planar two-center algorithms. Comp. Geom. Theor. Appl. 13(3), 189–198 (1999)

    MATH  Google Scholar 

  14. Eppstein, D.: Faster construction of planar two-centers. In: Proc. 8th ACM-SIAM Symp. Discrete Alg., pp. 131–138 (1997)

    Google Scholar 

  15. Gonzalez, T.F.: Clustering to minimize the maximum intercluster distance. Theor. Comput. Sci. 38, 293–306 (1985)

    Article  MATH  Google Scholar 

  16. Gritzmann, P., Klee, V.: Inner and outer j-radii of convex bodies in finite-dimensional normed spaces. Discrete Comput. Geom. 7, 255–280 (1992)

    Article  MATH  MathSciNet  Google Scholar 

  17. Gritzmann, P., Klee, V.: On the complexity of some basic problems in computational convexity I: Containment problems. Discrete Math. 136, 129–174 (1994)

    Article  MATH  MathSciNet  Google Scholar 

  18. Halperin, D., Sharir, M., Goldberg, K.: The 2-center problem with obstacles. J. Alg. 42(1), 109–134 (2002)

    Article  MATH  MathSciNet  Google Scholar 

  19. Hartigan, J.A.: Clustering algorithms. Wiley series in probability and mathematical statistics. John Wiley and Sons, New York (1975)

    MATH  Google Scholar 

  20. Helly, E.: Über Mengen konvexer Körper mit gemeinschaftlichen Punkten. Jahresbericht Deutsch. Math. Verein 32, 175–176 (1923)

    MATH  Google Scholar 

  21. Hershberger, J.: A faster algorithm for the two-center decision problem. Inf. Process. Lett. 47(1), 23–29 (1993)

    Article  MATH  MathSciNet  Google Scholar 

  22. Hoffmann, M.: A simple linear algorithm for computing rectilinear 3-centers. Comput. Geom. Theor. Appl. 31(3), 150–165 (2005)

    MATH  Google Scholar 

  23. Jain, A.K., Dubes, R.C.: Algorithms for clustering data. Prentice Hall, Englewood Cliffs (1988)

    MATH  Google Scholar 

  24. Jaromczyk, J.W., Kowaluk, M.: An efficient algorithm for the Euclidean two-center problem. In: Symp. Comp. Geom, pp. 303–311 (1994)

    Google Scholar 

  25. John, F.: Extremum problems with inequalities as subsidiary conditions. In: Courant Anniversary Volume, pp. 187–204. Interscience (1948)

    Google Scholar 

  26. Jung, H.W.E.: Über die kleinste Kugel, die eine räumliche Figur einschließt. J. Reine Angew. Math. 123, 241–257 (1901)

    MATH  Google Scholar 

  27. Kumar, P.: Clustering and reconstructing large data sets. PhD thesis, Department of Computer Science, Stony Brook University (2004)

    Google Scholar 

  28. Mangasarian, O.L., Setiono, R., Wolberg, W.H.: Pattern recognition via linear programming: theory and application to medical diagnosis. In: Coleman, T.F., Li, Y. (eds.) Large-Scale Numerical Optimization, pp. 22–31. SIAM, Philadelphia (1990); Computer Sciences TR 878 (1989)

    Google Scholar 

  29. Megiddo, N.: On the complexity of some geometric problems in unbounded dimension. J. Symb. Comput. 10(3/4), 327–334 (1990)

    Article  MATH  MathSciNet  Google Scholar 

  30. Pólik, I.: Addendum to the sedumi user guide version 1.1. Technical report, Advanced Optimization Laboratory, McMaster University (2005)

    Google Scholar 

  31. Procopiuc, C.M.: Clustering problems and their applications: A survey. Department of Computer Science, Duke University (1997)

    Google Scholar 

  32. Sharir, M.: A near-linear algorithm for the planar 2-center problem. In: Proc. Symp. Comp. Geom., pp. 106–112 (1996)

    Google Scholar 

  33. Sturm, J.F.: Using SeDuMi 1.02, a Matlab toolbox for optimization over symmetric cones. Optim. Method. Softw. 11-12, 625–653 (1999)

    Article  MathSciNet  Google Scholar 

  34. del Val, A.: On 2-SAT and renamable Horn. In: Proc. 17th Nat. Conf. on Artif. Intel. AAAI / MIT Press (2000)

    Google Scholar 

  35. Wei, H., Murray, A.T., Xiao, N.: Solving the continuous space p-centre problem: planning application issues. IMA J. Management Math. 17, 413–425 (2006)

    Article  MATH  MathSciNet  Google Scholar 

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Authors and Affiliations

  1. Zentrum Mathematik, Technische Universität München, Boltzmannstr. 3, D–85747, Garching b. München, Germany

    René Brandenberg & Lucia Roth

Authors
  1. René Brandenberg
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  2. Lucia Roth
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Editor information

Boting Yang Ding-Zhu Du Cao An Wang

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© 2008 Springer-Verlag Berlin Heidelberg

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Brandenberg, R., Roth, L. (2008). New Algorithms for k-Center and Extensions. In: Yang, B., Du, DZ., Wang, C.A. (eds) Combinatorial Optimization and Applications. COCOA 2008. Lecture Notes in Computer Science, vol 5165. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-540-85097-7_7

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  • DOI: https://doi.org/10.1007/978-3-540-85097-7_7

  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-540-85096-0

  • Online ISBN: 978-3-540-85097-7

  • eBook Packages: Computer ScienceComputer Science (R0)

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