Abstract
Christofides’ algorithm is a well known approximation algorithm for the metric travelling salesman problem. As a first step towards obtaining an average case analysis of Christofides’ algorithm, we provide a probabilistic analysis for the stochastic version of the algorithm for the Euclidean traveling salesman problem, where the input consists of n randomly chosen points in [0,1]d. Our main result provides bounds for the length of the computed tour that hold almost surely. We also provide an experimental evaluation of Christofides’s algorithm.
This project is partially funded by the DFG grant BL 511/7-1. The work was performed while the second author was affiliated with Max Planck Institute for Informatics in Saarbrücken.
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
Arora, S.: Polynomial time approximation schemes for euclidean traveling salesman and other geometric problems. J. ACM 45(5), 753–782 (1998)
Avis, D., Davis, B., Steele, J.M.: Probabilistic analysis of a greedy heuristic for euclidean matching. Probability in the Engineering and Informational Sciences 2(02), 143–156 (1988)
Baltz, A., Dubhashi, D.P., Srivastav, A., Tansini, L., Werth, S.: Probabilistic analysis for a multiple depot vehicle routing problem. Random Struct. Algorithms 30(1-2), 206–225 (2007)
Beardwood, J., Halton, J.H., Hammersley, J.M.: The shortest path through many points. Proc. Cambridge Philos. Soc. 55, 299–327 (1959)
Bertsimas, D.J., van Ryzin, G.: An asymptotic determination of the minimum spanning tree and minimum matching constants in geometrical probability. Operations Research Letters 9(4), 223–231 (1990)
Christofides, N.: Worst-case analysis of a new heuristic for the traveling salesman problem. Report 388, Graduate School of Industrial Administration, Carnegie Mellon University (1976)
Deineko, V., Tiskin, A.: Fast minimum-weight double-tree shortcutting for metric tsp: Is the best one good enough? J. Exp. Algorithmics 14, 4.6–4.16 (2010)
Frieze, A.M., Yukich, J.E.: Probabilistic analysis of the traveling salesman problem. In: Gutin, G., Punnen, A. (eds.) The Traveling Salesman Problem and Its Variations, pp. 257–308. Kluwer Academic Publisher (2002)
Goemans, M.X., Bertsimas, D.J.: Probabilistic analysis of the held and karp lower bound for the euclidean traveling salesman problem. Mathematics of Operations Research 16(1), 72–89 (1991)
Gutin, G., Punnen, A.P. (eds.): The traveling salesman problem and its variations. Combinatorial Optimization, vol. 12. Kluwer Academic Publishers (2002)
Johnson, D.S., McGeoch, L.A., Rothberg, E.E.: Asymptotic experimental analysis for the held-karp traveling salesman bound. In: SODA 1996, pp. 341–350 (1996)
Karp, R.M.: Probabilistic analysis of partitioning algorithms for the traveling-salesman problem in the plane. Math. of Operat. Research 2(3), 209–224 (1977)
Mitchell, J.: Guillotine subdivisions approximate polygonal subdivisions: A simple polynomial-time approximation scheme for geometric TSP, k-MST, and related problems. SIAM J. Comput. 28(4), 1298–1309 (1999)
Rhee, W.T.: On the travelling salesperson problem in many dimensions. Random Struct. Algorithms 3(3), 227–234 (1992)
Rhee, W.T.: A matching problem and subadditive euclidean functionals. Ann. Appl. Probab. 3(3), 794–801 (1993)
Steele, J.M.: Subadditive Euclidean functionals and nonlinear growth in geometric probability. Ann. Probab. 9(3), 365–376 (1981)
Steele, J.M.: Growth rates of Euclidean minimal spanning trees with power weighted edges. Ann. Probab. 16, 1767–1787 (1988)
Steele, J.M.: Probability Theory and Combinatorial Optimization. CBMS-NSF Regional Conference Series in Applied Mathematics, vol. 69. SIAM (1997)
Yukich, J.E.: Probability Theory of Classical Euclidean Optimization Problems. Lecture Notes in Mathematics, vol. 1675. Springer (1998)
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2012 Springer-Verlag Berlin Heidelberg
About this paper
Cite this paper
Bläser, M., Panagiotou, K., Rao, B.V.R. (2012). A Probabilistic Analysis of Christofides’ Algorithm. In: Fomin, F.V., Kaski, P. (eds) Algorithm Theory – SWAT 2012. SWAT 2012. Lecture Notes in Computer Science, vol 7357. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-31155-0_20
Download citation
DOI: https://doi.org/10.1007/978-3-642-31155-0_20
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-642-31154-3
Online ISBN: 978-3-642-31155-0
eBook Packages: Computer ScienceComputer Science (R0)