Document

# Parameterized Approximation Algorithms for TSP

## File

LIPIcs.ISAAC.2022.50.pdf
• Filesize: 0.7 MB
• 16 pages

## Acknowledgements

The authors thank the reviewers for their valuable comments and constructive suggestions.

## Cite As

Jianqi Zhou, Peihua Li, and Jiong Guo. Parameterized Approximation Algorithms for TSP. In 33rd International Symposium on Algorithms and Computation (ISAAC 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 248, pp. 50:1-50:16, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2022)
https://doi.org/10.4230/LIPIcs.ISAAC.2022.50

## Abstract

We study the Traveling Salesman problem (TSP), where given a complete undirected graph G = (V,E) with n vertices and an edge cost function c:E↦R_{⩾0}, the goal is to find a minimum-cost cycle visiting every vertex exactly once. It is well-known that unless P = NP, TSP cannot be approximated in polynomial time within a factor of ρ(n) for any computable function ρ, while the metric case of TSP, that the edge cost function satisfies the △-inequality, admits a polynomial-time 1.5-approximation. We investigate TSP on general graphs from the perspective of parameterized approximability. A parameterized ρ-approximation algorithm returns a ρ-approximation solution in f(k)⋅|I|^O(1) time, where f is a computable function and k is a parameter of the input I. We introduce two parameters, which measure the distance of a given TSP-instance from the metric case, and achieve the following two results: - A 3-approximation algorithm for TSP in O((3k₁)! 8^k₁⋅ n²+n³) time, where k₁ is the number of triangles in which the edge costs violate the △-inequality. - A 3-approximation algorithm for TSP in O(n^O(k₂)) time and a (6k₂+9)-approximation algorithm for TSP in O(k₂^O(k₂)⋅n³) time, where k₂ is the minimum number of vertices, whose removal results in a metric graph. To our best knowledge, the above algorithms are the first non-trivial parameterized approximation algorithms for TSP on general graphs.

## Subject Classification

##### ACM Subject Classification
• Theory of computation → Graph algorithms analysis
• Theory of computation → Approximation algorithms analysis
##### Keywords
• FPT-approximation algorithms
• the Traveling Salesman problem
• the triangle inequality
• fixed-parameter tractability
• metric graphs

## Metrics

• Access Statistics
• Total Accesses (updated on a weekly basis)
0

## References

1. Thomas Andreae. On the traveling salesman problem restricted to inputs satisfying a relaxed triangle inequality. Networks, 38(2):59-67, 2001.
2. Thomas Andreae and Hans-Jürgen Bandelt. Performance guarantees for approximation algorithms depending on parametrized triangle inequalities. SIAM Journal on Discrete Mathematics, 8(1):1-16, 1995.
3. Sanjeev Arora. Polynomial time approximation schemes for Euclidean traveling salesman and other geometric problems. Journal of the ACM, 45(5):753-782, 1998.
4. Richard Bellman. Dynamic programming treatment of the travelling salesman problem. Journal of the ACM, 9(1):61-63, 1962.
5. Michael A. Bender and Chandra Chekuri. Performance guarantees for the TSP with a parameterized triangle inequality. Information Processing Letters, 73(1-2):17-21, 2000.
6. Hans-Joachim Böckenhauer, Juraj Hromkovič, Ralf Klasing, Sebastian Seibert, and Walter Unger. Towards the notion of stability of approximation for hard optimization tasks and the traveling salesman problem. Theoretical Computer Science, 285(1):3-24, 2002.
7. Hans-Joachim Bockenhauer, Juraj Hromkovic, Joachim Kneis, and Joachim Kupke. The parameterized approximability of TSP with deadlines. Theory of Computing Systems, 41(3):431-444, 2007.
8. Edouard Bonnet, Bruno Escoffier, Eun Jung Kim, and Vangelis Th. Paschos. On subexponential and FPT-time inapproximability. Algorithmica, 71(3):541-565, 2015.
9. Parinya Chalermsook, Marek Cygan, Guy Kortsarz, Bundit Laekhanukit, Pasin Manurangsi, Danupon Nanongkai, and Luca Trevisan. From Gap-ETH to FPT-inapproximability: Clique, dominating set, and more. In Proceedings of the IEEE 58th Annual Symposium on Foundations of Computer Science, pages 743-754. IEEE, 2017.
10. Yijia Chen and Bingkai Lin. The constant inapproximability of the parameterized dominating set problem. In Proceedings of the IEEE 57th Annual Symposium on Foundations of Computer Science, pages 505-514. IEEE, 2016.
11. Nicos Christofides. Worst-case analysis of a new heuristic for the travelling salesman problem. Technical Report 388, Carnegie-Mellon University, 1976.
12. Vladimir G. Deǐneko, Michael Hoffmann, Yoshio Okamoto, and Gerhard J. Woeginger. The traveling salesman problem with few inner points. Operations Research Letters, 34(1):106-110, 2006.
13. Erik D. Demaine and MohammadTaghi Hajiaghayi. Equivalence of local treewidth and linear local treewidth and its algorithmic applications. In Proceedings of the 15th Annual ACM-SIAM Symposium on Discrete Algorithms, pages 840-849. SIAM, 2004.
14. Rodney G. Downey and Michael R. Fellows. Fundamentals of Parameterized Complexity. Springer, 2013.
15. Andreas Emil Feldmann, C.S. Karthik, Euiwoong Lee, and Pasin Manurangsi. A survey on approximation in parameterized complexity: Hardness and algorithms. Algorithms, 13(6):146, 2020.
16. Michael R. Garey and David S. Johnson. Computers and intractability: A guide to the theory of NP-completeness. Bulletin (New Series) of the AMS, 3(2):898-904, 1980.
17. Lee-Ad Gottlieb. A light metric spanner. In Proceedings of the IEEE 56th Annual Symposium on Foundations of Computer Science, pages 759-772. IEEE, 2015.
18. Martin Grohe. Local tree-width, excluded minors, and approximation algorithms. Combinatorica, 23:613-632, 2003.
19. Martin Grohe, Ken-ichi Kawarabayashi, and Bruce Reed. A simple algorithm for the graph minor decomposition - logic meets structural graph theory-. In Proceedings of the 24th Annual ACM-SIAM Symposium on Discrete Algorithms, pages 414-431. SIAM, 2013.
20. Jiong Guo, Falk Hüffner, and Rolf Niedermeier. A structural view on parameterizing problems: Distance from triviality. In Proceedings of the 1st International Workshop on Parameterized and Exact Computation, pages 162-173. Springer, 2004.
21. Michael Held and Richard M. Karp. A dynamic programming approach to sequencing problems. Journal of the SIAM, 10(1):196-210, 1962.
22. Anna R. Karlin, Nathan Klein, and Shayan Oveis Gharan. A (slightly) improved approximation algorithm for metric TSP. In Proceedings of the 53rd Annual ACM SIGACT Symposium on Theory of Computing, pages 32-45. ACM, 2021.
23. Richard M. Karp. Reducibility among combinatorial problems. In Proceedings of a symposium on the Complexity of Computer Computations, pages 85-103. Plenum Press, 1972.
24. Ioannis Katsikarelis, Michael Lampis, and Vangelis Th. Paschos. Structural parameters, tight bounds, and approximation for (k,r)-center. Discrete Applied Mathematics, 264:90-117, 2019.
25. Michael Khachay and Katherine Neznakhina. Approximability of the minimum-weight k-size cycle cover problem. Journal of Global Optimization, 66(1):65-82, 2016.
26. Christian Knauer and Andreas Spillner. A fixed-parameter algorithm for the minimum weight triangulation problem based on small graph separators. In Proceedings of the 32nd International Workshop on Graph-Theoretic Concepts in Computer Science, pages 49-57. Springer, 2006.
27. Bingkai Lin. Constant approximating k-clique is W[1]-hard. In Proceedings of the 53rd Annual ACM SIGACT Symposium on Theory of Computing, pages 1749-1756. ACM, 2021.
28. Dániel Marx. Parameterized complexity and approximation algorithms. The Computer Journal, 51(1):60-78, 2008.
29. Usha Mohan, Sivaramakrishnan Ramani, and Sounaka Mishra. Constant factor approximation algorithm for TSP satisfying a biased triangle inequality. Theoretical Computer Science, 657:111-126, 2017.
30. Tobias Mömke. An improved approximation algorithm for the traveling salesman problem with relaxed triangle inequality. Information Processing Letters, 115(11):866-871, 2015.
31. Rolf Niedermeier. Invitation to Fixed-Parameter Algorithms. Oxford University Press, 2006.
32. Christos H. Papadimitriou. The Euclidean travelling salesman problem is NP-complete. Theoretical Computer Science, 4(3):237-244, 1977.
33. Karthik C. S., Bundit Laekhanukit, and Pasin Manurangsi. On the parameterized complexity of approximating dominating set. In Proceedings of the 50th Annual ACM SIGACT Symposium on Theory of Computing, pages 1283-1296. ACM, 2018.
34. Sartaj Sahni and Teofilo Gonzalez. P-complete approximation problems. Journal of the ACM, 23(3):555-565, 1976.
35. Anatolii Ivanovich Serdyukov. On some extremal walks in graphs. Upravliaemie systemy, 17:76-79, 1978.
X

Feedback for Dagstuhl Publishing