No CrossRef data available.
Article contents
Analysis of a near-metric TSP approximation algorithm∗
Published online by Cambridge University Press: 06 August 2013
Abstract
The traveling salesman problem (TSP) is one of the most fundamental optimization problems. We consider the β-metric traveling salesman problem (Δβ-TSP), i.e., the TSP restricted to graphs satisfying the β-triangle inequality c({v,w}) ≤ β(c({v,u}) + c({u,w})), for some cost function c and any three vertices u,v,w. The well-known path matching Christofides algorithm (PMCA) guarantees an approximation ratio of 3β2/2 and is the best known algorithm for the Δβ-TSP, for 1 ≤ β ≤ 2. We provide a complete analysis of the algorithm. First, we correct an error in the original implementation that may produce an invalid solution. Using a worst-case example, we then show that the algorithm cannot guarantee a better approximation ratio. The example can also be used for the PMCA variants for the Hamiltonian path problem with zero and one prespecified endpoints. For two prespecified endpoints, we cannot reuse the example, but we construct another worst-case example to show the optimality of the analysis also in this case.
- Type
- Research Article
- Information
- Copyright
- © EDP Sciences 2013
References
H.-C. An, R. Kleinberg and D.B. Shmoys, Improving Christofides’ algorithm for the s-t path TSP. In Proc. of the 44th symposium on Theory of Computing (STOC 2012) 875–886.
Andreae, T., On the Traveling Salesman Problem Restricted to Inputs Satisfying a Relaxed Triangle Inequality. Networks 38 (2001) 59–67. Google Scholar
Bender, M.A. and Chekuri, C., Performance guarantees for the TSP with a parameterized triangle inequality. Inf. Proc. Lett. 73 (2000) 17–21. Google Scholar
H.-J. Böckenhauer and J. Hromkovič, Stability of approximation algorithms or parameterization of the approximation ratio. In Proc. of the 9th International Symposium on Operations Research in Slovenia (SOR 2007) 23–28.
Böckenhauer, H.-J., Hromkovič, J., Klasing, R., Seibert, S. and Unger, W., Towards the notion of stability of approximation for hard optimization tasks and the traveling salesman problem. Theor. Comput. Sci. 285 (2002) 3–24. Google Scholar
H.-J. Böckenhauer, J. Hromkovič and S. Seibert, Stability of Approximation. In Handbook of Approximation Algorithms and Metaheuristics, edited by T.F. Gonzalez. Chapman & Hall, Boca Raton (2007).
N. Christofides, Worst-case analysis of a new heuristic for the travelling salesman problem. Technical Report 388. Carnegie Mellon University, Graduate School of Industrial Administration (1976).
T.H. Cormen, C.E. Leiserson, R.L. Rivest and C. Stein, Introduction to Algorithms. MIT Press, Cambridge (2009).
Forlizzi, L., Hromkovič, J., Proietti, G. and Seibert, S., On the Stability of Approximation for Hamiltonian Path Problems. Alg. Oper. Res. 1 (2006) 31–45. Google Scholar
E.G. Goodaire and M.M. Parmenter, Discrete Mathematics with Graph Theory. Prentice Hall, Upper Saddle River (2005).
Hoogeveen, J.A., Analysis of Christofides’ heuristic: Some paths are more difficult than cycles. Oper. Res. Lett. 10 (1991) 291–295. Google Scholar
J. Hromkovič, Algorithmics for Hard Problems. Introduction to Combinatorial Optimization, Randomization, Approximation, and Heuristics. Springer, Heidelberg (2004).
S. Krug, Analysis of Approximation Algorithms for the Traveling Salesman Problem in Near-Metric Graphs. Master’s thesis. ETH Zurich, Department of Computer Science (2011).
T. Mömke, Structural Properties of Hard Metric TSP Inputs. In Proc. 37th Int. Conf. on Current Trends in Theory and Practice of Computer Science (SOFSEM 2011) 394–405.