Connecting a set of circles with minimum sum of radii

Abstract We consider the problem of assigning radii to a given set of points in the plane, such that the resulting set of disks is connected, and the sum of radii is minimized. We prove that the problem is NP-hard in planar weighted graphs if there are upper bounds on the radii and sketch a similar proof for planar point sets. For the case when there are no upper bounds on the radii, the complexity is open; we give a polynomial-time approximation scheme. We also give constant-factor approximation guarantees for solutions with a bounded number of disks; these results are supported by lower bounds, which are shown to be tight in some of the cases. Finally, we show that the problem is polynomially solvable if a connectivity tree is given, and we conclude with some experimental results.

[1]  Rina Panigrahy,et al.  Clustering to minimize the sum of cluster diameters , 2001, STOC '01.

[2]  Ion I. Mandoiu,et al.  Selecting Forwarding Neighbors in Wireless Ad Hoc Networks , 2001, DIALM '01.

[3]  Esther M. Arkin,et al.  Minimum-cost coverage of point sets by disks , 2006, SCG '06.

[4]  Andrea E. F. Clementi,et al.  On the Power Assignment Problem in Radio Networks , 2004, Mob. Networks Appl..

[5]  Joseph S. B. Mitchell,et al.  The minimum area spanning tree problem , 2006, EuroCG.

[6]  Matt Gibson,et al.  On clustering to minimize the sum of radii , 2008, SODA '08.

[7]  Xiang-Yang Li,et al.  Minimum-energy broadcast routing in static ad hoc wireless networks , 2001, Proceedings IEEE INFOCOM 2001. Conference on Computer Communications. Twentieth Annual Joint Conference of the IEEE Computer and Communications Society (Cat. No.01CH37213).

[8]  Hanan Shpungin,et al.  Fault-tolerant power assignment and backbone in wireless networks , 2006, Fourth Annual IEEE International Conference on Pervasive Computing and Communications Workshops (PERCOMW'06).

[9]  Ioannis Caragiannis,et al.  Energy-Efficient Wireless Network Design , 2005, Theory of Computing Systems.

[10]  Jorge Urrutia,et al.  Augmenting the connectivity of geometric graphs , 2008, Comput. Geom..

[11]  Vittorio Bilò,et al.  Geometric Clustering to Minimize the Sum of Cluster Sizes , 2005, ESA.

[12]  Joseph S. B. Mitchell,et al.  Guillotine Subdivisions Approximate Polygonal Subdivisions: A Simple Polynomial-Time Approximation Scheme for Geometric TSP, k-MST, and Related Problems , 1999, SIAM J. Comput..

[13]  David Peleg,et al.  Polynomial time approximation schemes for base station coverage with minimum total radii , 2005, Comput. Networks.

[14]  Peng-Jun Wan,et al.  Range Assignment for High Connectivity in Wireless Ad Hoc Networks , 2003, ADHOC-NOW.

[15]  David Avis,et al.  Reverse Search for Enumeration , 1996, Discret. Appl. Math..

[16]  Joseph S. B. Mitchell,et al.  Connecting a Set of Circles with Minimum Sum of Radii , 2011, WADS.

[17]  Bernhard Fuchs,et al.  On the hardness of range assignment problems , 2006, Networks.

[18]  David Lichtenstein,et al.  Planar Formulae and Their Uses , 1982, SIAM J. Comput..

[19]  Madhav V. Marathe,et al.  Algorithmic Aspects of Topology Control Problems for Ad Hoc Networks , 2005, Mob. Networks Appl..

[20]  Sándor P. Fekete,et al.  Approximation of Geometric Dispersion Problems , 1998, APPROX.

[21]  Xiang-Yang Li,et al.  Minimum-Energy Broadcasting in Static Ad Hoc Wireless Networks , 2002, Wirel. Networks.

[22]  Gregory Dudek,et al.  Hybrid Inference for Sensor Network Localization Using a Mobile Robot , 2007, AAAI.

[23]  Madhav V. Marathe,et al.  Approximation Algorithms for Clustering to Minimize the Sum of Diameters , 2000, Nord. J. Comput..