New algorithms for k-center and extensions

The problem of interest is covering a given point set with homothetic copies of several convex containers C1, ..., Ck, 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 Ci 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 Ci.

[1]  David Avis,et al.  Diameter partitioning , 1986, Discret. Comput. Geom..

[2]  Peter Gritzmann,et al.  Modeling and Optimization of Correction Measures for Human Extremities , 2008 .

[3]  E. Helly.  Über Mengen konvexer Körper mit gemeinschaftlichen Punkte. , 1923 .

[4]  Micha Sharir,et al.  The 2-Center Problem with Obstacles , 2002, J. Algorithms.

[5]  Kurt Leichtweiss.  Zwei Extremalprobleme der Minkowski-Geometrie , 1955 .

[6]  Teofilo F. GONZALEZ,et al.  Clustering to Minimize the Maximum Intercluster Distance , 1985, Theor. Comput. Sci..

[7]  Kim-Chuan Toh,et al.  SDPT3 -- A Matlab Software Package for Semidefinite Programming , 1996 .

[8]  Nimrod Megiddo.  On the Complexity of Some Geometric Problems in Unbounded Dimension , 1990, J. Symb. Comput..

[9]  Piotr Indyk,et al.  Approximate clustering via core-sets , 2002, STOC '02.

[10]  Cecilia M. Procopiuc,et al.  Applications of Clustering Problems , 1997 .

[11]  John Hershberger.  A Faster Algorithm for the Two-Center Decision Problem , 1993, Inf. Process. Lett..

[12]  Heinrich W. E. Jung.  Ueber die kleinste Kugel, die eine räumliche Figur einschliesst. , 1901 .

[13]  Micha Sharir.  A Near-Linear Algorithm for the Planar 2-Center Problem , 1997, Discret. Comput. Geom..

[14]  John A. Hartigan,et al.  Clustering Algorithms , 1975 .

[15]  Michael R. Anderberg,et al.  Cluster Analysis for Applications , 1973 .

[16]  Sergey Bereg,et al.  Rectilinear 2-center problems , 1999, CCCG.

[17]  Kim-Chuan Toh,et al.  Solving semidefinite-quadratic-linear programs using SDPT3 , 2003, Math. Program..

[18]  Peter Gritzmann,et al.  On the Complexity of some Basic Problems in Computational Convexity: II. Volume and mixed volumes , 1994, Universität Trier, Mathematik/Informatik, Forschungsbericht.

[19]  Alan T. Murray,et al.  Solving the continuous space p-centre problem: planning application issues , 2006 .

[20]  Pankaj K. Agarwal,et al.  Exact and Approximation Algortihms for Clustering , 1997 .

[21]  Micha Sharir,et al.  Efficient algorithms for geometric optimization , 1998, CSUR.

[22]  Michael Hoffmann,et al.  A simple linear algorithm for computing rectilinear 3-centers , 2005, Comput. Geom..

[23]  Joseph S. B. Mitchell,et al.  Approximate minimum enclosing balls in high dimensions using core-sets , 2003, ACM J. Exp. Algorithmics.

[24]  Anil K. Jain,et al.  Algorithms for Clustering Data , 1988 .

[25]  Peter Gritzmann,et al.  Inner and outerj-radii of convex bodies in finite-dimensional normed spaces , 1992, Discret. Comput. Geom..

[26]  Jerzy W. Jaromczyk,et al.  An efficient algorithm for the Euclidean two-center problem , 1994, SCG '94.

[27]  V. Boltyanski,et al.  Excursions into Combinatorial Geometry , 1996 .

[28]  David Eppstein,et al.  Faster construction of planar two-centers , 1997, SODA '97.

[29]  F. Bohnenblust,et al.  Convex Regions and Projections in Minkowski Spaces , 1938 .

[30]  Peter Gritzmann,et al.  On the complexity of some basic problems in computational convexity: I. Containment problems , 1994, Discret. Math..

[31]  E. T. S. Informática.  On 2-SAT and Renamable Horn , 2000 .

[32]  Sergey Bereg,et al.  Covering a set of points by two axis-parallel boxes , 1997, CCCG.

[33]  Clustering and reconstructing large data sets , 2004 .

[34]  O. Mangasarian,et al.  Pattern Recognition Via Linear Programming: Theory and Application to Medical Diagnosis , 1989 .

[35]  Timothy M. Chan.  More planar two-center algorithms , 1999, Comput. Geom..

[36]  Jos F. Sturm,et al.  A Matlab toolbox for optimization over symmetric cones , 1999 .