Abstract
We study a generalization of the well known bottleneck spanning tree problem called Best Case Connectivity with Uncertainty: Given a family of geometric regions, choose one point per region, such that the length of the longest edge in a spanning tree of a disc intersection graph is minimized. We show that this problem is NP-hard even for very simple scenarios such as line segments and squares. We also give exact and approximation algorithms for the case of line segments and unit discs respectively.
The authors are grateful for two Bellairs workshops supporting this research: the 8th and 9th McGill—INRIA Workshop on Computational Geometry in 2009 and 2010.
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References
Alt, H., Arkin, E., Brönnimann, H., Erickson, J., Fekete, S., Knauer, C., Lenchner, J., Mitchell, J., Whittlesey, K.: Minimum-cost coverage of point sets by disks. In: Proc. 22nd ACM Symp. Comp. Geom. (SoCG), pp. 449–458 (2006)
Arkin, E.M., Hassin, R.: Approximation algorithms for the geometric covering salesman problem. Disc. Appl. Mathematics 55(3), 197–218 (1994)
Chambers, E., Erickson, A., Fekete, S., Lenchner, J., Sember, J., Venkatesh, S., Stege, U., Stolpner, S., Weibel, C., Whitesides, S.: Connectivity graphs of uncertainty regions. arXiV:1009.3469 (2010)
Clementi, A.E.F., Penna, P., Silvestri, R.: On the power assignment problem in radio networks. Technical Report TR00-054, Electronic Colloquium on Computational Complexity (2000)
de Berg, M., Gudmundsson, J., Katz, M.J., Levcopoulos, C., Overmars, M.H., van der Stappen, A.F.: TSP with neighborhoods of varying size. J. Algorithms 57(1), 22–36 (2005)
Duchet, P., Hamidoune, Y.O., Vergnas, M.L., Meyniel, H.: Representing a planar graph by vertical lines joining different levels. Disc. Mathematics 46(3), 319–321 (1983)
Dumitrescu, A., Mitchell, J.S.B.: Approximation algorithms for TSP with neighborhoods in the plane. In: Proc. 12th ACM-SIAM Symp. on Disc. Algorithms (SODA), pp. 38–46 (2001)
Fuchs, B.: On the hardness of range assignment problems. In: Calamoneri, T., Finocchi, I., Italiano, G.F. (eds.) CIAC 2006. LNCS, vol. 3998, pp. 127–138. Springer, Heidelberg (2006)
Garey, M.R., Johnson, D.S.: Computers and Intractability: A Guide to the Theory of NP-Completeness. W.H. Freeman, New York (1979)
Gudmundsson, J., Levcopoulos, C.: A fast approximation algorithm for TSP with neighborhoods. Nord. J. Comput. 6(4), 469 (1999)
Lev-Tov, N., Peleg, D.: Exact algorithms and approximation schemes for base station placement problems. In: Penttonen, M., Schmidt, E.M. (eds.) SWAT 2002. LNCS, vol. 2368, pp. 90–99. Springer, Heidelberg (2002)
Lev-Tov, N., Peleg, D.: Polynomial time approximation schemes for base station coverage with minimum total radii. Computer Networks 47(4), 489–501 (2005)
Mitchell, J.S.B.: A PTAS for TSP with neighborhoods among fat regions in the plane. In: Proc. 18th ACM-SIAM Symp. on Disc. Algorithms (SODA), pp. 11–18 (2007)
Parker, G., Rardin, R.L.: Guaranteed performance heuristics for the bottleneck traveling salesman problem. Operations Research Letters 2(6), 269–272 (1984)
Rosenstiehl, P., Tarjan, R.E.: Rectilinear planar layouts and bipolar orientations of planar graphs. Disc. Comp. Geom. 1, 343–353 (1986)
Yang, Y., Lin, M., Xu, J., Xie, Y.: Minimum spanning tree with neighborhoods. In: Kao, M.-Y., Li, X.-Y. (eds.) AAIM 2007. LNCS, vol. 4508, pp. 306–316. Springer, Heidelberg (2007)
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Chambers, E. et al. (2010). Connectivity Graphs of Uncertainty Regions. In: Cheong, O., Chwa, KY., Park, K. (eds) Algorithms and Computation. ISAAC 2010. Lecture Notes in Computer Science, vol 6507. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-17514-5_37
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DOI: https://doi.org/10.1007/978-3-642-17514-5_37
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