Abstract
We present the first nontrivial approximation algorithm for the bottleneck asymmetric traveling salesman problem. Given an asymmetric metric cost between n vertices, the problem is to find a Hamiltonian cycle that minimizes its bottleneck (or maximum-length edge) cost. We achieve an O(logn / loglogn) approximation performance guarantee by giving a novel algorithmic technique to shortcut Eulerian circuits while bounding the lengths of the shortcuts needed. This allows us to build on the recent result of Asadpour, Goemans, Mądry, Oveis Gharan, and Saberi to obtain this guarantee. Furthermore, we show how our technique yields stronger approximation bounds in some cases, such as the bounded orientable genus case studied by Oveis Gharan and Saberi.
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
Archer, A.: Inapproximability of the asymmetric facility location and k-median problems (2000), http://www2.research.att.com/~aarcher/Research/asym-hard.ps
Arya, V., Garg, N., Khandekar, R., Meyerson, A., Munagala, K., Pandit, V.: Local search heuristics for k-median and facility location problems. SIAM J. Comput. 33(3), 544–562 (2004)
Asadpour, A., Goemans, M.X., Mądry, A., Oveis Gharan, S., Saberi, A.: An O(logn/loglogn)-approximation algorithm for the asymmetric traveling salesman problem. In: SODA 2010: Proceedings of the 21st Annual ACM-SIAM Symposium on Discrete Algorithms, pp. 379–389 (2010)
Chekuri, C., Vondrák, J., Zenklusen, R.: Dependent randomized rounding for matroid polytopes and applications. CoRR abs/0909.4348 (2009), http://arxiv.org/abs/0909.4348
Christofides, N.: Worst-case analysis of a new heuristic for the travelling salesman problem. Tech. Rep. 388, Graduate School of Industrial Administration, CMU (1976)
Chuzhoy, J., Guha, S., Halperin, E., Khanna, S., Kortsarz, G., Krauthgamer, R., Naor, J.: Asymmetric k-center is log* n-hard to approximate. J. ACM 52(4), 538–551 (2005)
Fleischner, H.: The square of every two-connected graph is Hamiltonian. Journal of Combinatorial Theory, Series B 16(1), 29–34 (1974)
Frieze, A.M., Galbiati, G., Maffioli, F.: On the worst-case performance of some algorithms for the asymmetric traveling salesman problem. Networks 12, 23–39 (1982)
Grötschel, M., Lovász, L., Schrijver, A.: The ellipsoid method and its consequences in combinatorial optimization. Combinatorica 1(2), 169–197 (1981)
Hall, P.: On Representatives of Subsets. J. London Math. Soc. 10, 26–30 (1935)
Held, M., Karp, R.M.: The traveling-salesman problem and minimum spanning trees. Operations Research 18(6), 1138–1162 (1970)
Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. J. ACM 33(3), 533–550 (1986)
Hsu, W.L., Nemhauser, G.L.: Easy and hard bottleneck location problems. Discrete Applied Mathematics 1(3), 209–215 (1979)
Jain, K., Mahdian, M., Markakis, E., Saberi, A., Vazirani, V.V.: Greedy facility location algorithms analyzed using dual fitting with factor-revealing LP. J. ACM 50(6), 795–824 (2003)
Kleinberg, J., Tardos, É.: Algorithm Design. Addison-Wesley Longman Publishing Co., Inc., Boston (2005)
Kleinberg, J., Williamson, D.P.: Unpublished manuscript, pp. 124–126 (1998), http://legacy.orie.cornell.edu/~dpw/cornell.ps
Lau, H.T.: Finding EPS-Graphs. Monatshefte für Mathematik 92(1), 37–40 (1981)
Lawler, E.L., Lenstra, J.K., Rinnooy Kan, A.H.G., Shmoys, D.B. (eds.): The Traveling Salesman Problem: A Guided Tour of Combinatorial Optimization. Wiley, Chichester (1985)
Lin, J.H., Vitter, J.S.: ε-approximations with minimum packing constraint violation (extended abstract). In: STOC 1992: Proceedings of the 24th Annual ACM Symposium on Theory of Computing, pp. 771–782. ACM, New York (1992)
Oveis Gharan, S., Saberi, A.: The asymmetric traveling salesman problem on graphs with bounded genus. CoRR abs/0909.2849 (2009), http://arxiv.org/abs/0909.2849
Panigrahy, R., Vishwanathan, S.: An O(log* n) approximation algorithm for the asymmetric p-center problem. J. Algorithms 27(2), 259–268 (1998)
Papadimitriou, C.H., Vempala, S.: On the approximability of the traveling salesman problem. Combinatorica 26(1), 101–120 (2006)
Parker, R.G., Rardin, R.L.: Guaranteed performance heuristics for the bottleneck traveling salesman problem. Operations Research Letters 2(6), 269–272 (1984)
Schrijver, A.: Combinatorial Optimization: Polyhedra and Efficiency. Springer, Heidelberg (2003)
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2010 Springer-Verlag Berlin Heidelberg
About this paper
Cite this paper
An, HC., Kleinberg, R.D., Shmoys, D.B. (2010). Approximation Algorithms for the Bottleneck Asymmetric Traveling Salesman Problem. In: Serna, M., Shaltiel, R., Jansen, K., Rolim, J. (eds) Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques. RANDOM APPROX 2010 2010. Lecture Notes in Computer Science, vol 6302. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-15369-3_1
Download citation
DOI: https://doi.org/10.1007/978-3-642-15369-3_1
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-642-15368-6
Online ISBN: 978-3-642-15369-3
eBook Packages: Computer ScienceComputer Science (R0)