Abstract
In the TSP with neighborhoods problem we are given a set of n regions (neighborhoods) in the plane, and seek to find a minimum length TSP tour that goes through all the regions. We give two approximation algorithms for the case when the regions are allowed to intersect: We give the first O(1)-factor approximation algorithm for intersecting convex fat objects of comparable diameters where we are allowed to hit each object only at a finite set of specified points. The proof follows from two packing lemmas that are of independent interest. For the problem in its most general form (but without the specified points restriction) we give a simple O(logn)-approximation algorithm.
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Preview
Unable to display preview. Download preview PDF.
References
Arkin, E.M., Hassin, R.: Approximation algorithms for the geometric covering salesman problem. Discrete Applied Mathematics 55(3), 197–218 (1994)
Arora, S.: Nearly linear time approximation schemes for euclidean TSP and other geometric problems. J. ACM 45(5), 1–30 (1998)
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. of Algorithms 57, 22–36 (2005)
Dumitrescu, A., Mitchell, J.S.B.: Approximation algorithms for TSP with neighborhoods in the plane. J. Algorithms 48(1), 135–159 (2003)
Elbassioni, K., Fishkin, A.V., Mustafa, N., Sitters, R.: Approximation algorithms for Euclidean group TSP. In: Caires, L., Italiano, G.F., Monteiro, L., Palamidessi, C., Yung, M. (eds.) ICALP 2005. LNCS, vol. 3580, pp. 1115–1126. Springer, Heidelberg (2005)
Garg, N., Konjevod, G., Ravi, R.: A polylogarithmic approximation algorithm for the group Steiner tree problem. J. Algorithms 37(1), 66–84 (2000)
Gudmundsson, J., Levcopoulos, C.: A fast approximation algorithm for TSP with neighborhoods. Nordic J. Computing 6(4), 469–488 (1999)
Halperin, E., Krauthgamer, R.: Polylogarithmic inapproximability. In: Proc. 35th Annual ACM Symposium on Theory of Computing, pp. 585–594 (2003)
Mata, C.S., Mitchell, J.S.B.: Approximation algorithms for geometric tour and network design problems (extended abstract). In: Proc. 11th Annual ACM Symposium on Computational Geometry, pp. 360–369 (1995)
Mitchel, J.S.B.: Handbook of computational geometry. In: Geometric shortest paths and network optimization, pp. 633–701. Elsevier, North-Holland, Amsterdam (2000)
Mitchell, J.S.B.: Guillotine subdivions approximate polygonal subdivisons: A simple polynomial-time approximation scheme for geometric TSP, k-MST and related problems. SIAM J. Computing 28(4), 1298–1309 (1999)
Mitchell, J.S.B.: A PTAS for TSP with neighborhoods among fat regions in the plane. In: Proc. 18th Annual ACM-SIAM Symposium on Discrete Algorithms (to appear, 2007)
Reich, G., Widmayer, P.: Beyond Steiner’s problem: a VLSI oriented generalization. In: Proc. 15th. Int. Workshop on Graph-theoretic Concepts in Computer Science, pp. 196–210. Springer, Heidelberg (1990)
Safra, S., Schwartz, O.: On the complexity of approximating TSP with Neighborhoods and related problems. In: Di Battista, G., Zwick, U. (eds.) ESA 2003. LNCS, vol. 2832, pp. 446–458. Springer, Heidelberg (2003)
Slavik, P.: The errand scheduling problem, Tech. report, SUNY, Buffalo, USA (1997)
van der Stappen, A.F.: Motion planning amidst fat obstacles, Ph.d. dissertation, Utrecht University, Utrecht, the Netherlands (1994)
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2006 Springer-Verlag Berlin Heidelberg
About this paper
Cite this paper
Elbassioni, K., Fishkin, A.V., Sitters, R. (2006). On Approximating the TSP with Intersecting Neighborhoods. In: Asano, T. (eds) Algorithms and Computation. ISAAC 2006. Lecture Notes in Computer Science, vol 4288. Springer, Berlin, Heidelberg. https://doi.org/10.1007/11940128_23
Download citation
DOI: https://doi.org/10.1007/11940128_23
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-540-49694-6
Online ISBN: 978-3-540-49696-0
eBook Packages: Computer ScienceComputer Science (R0)