# Approximation Schemes for the Generalized Traveling Salesman Problem

• M. Yu. Khachai
• E. D. Neznakhina
Article

## Abstract

The Generalized Traveling Salesman Problem (GTSP) is defined by a weighted graph G = (V,E,w) and a partition of its vertex set into k disjoint clusters V = V1 ∪... ∪ Vk. It is required to find a minimum-weight cycle that contains exactly one vertex of each cluster. We consider a geometric setting of the problem (we call it the EGTSP-k-GC), in which the vertices of the graph are points in the plane, the weight function corresponds to the Euclidean distances between the points, and the partition into clusters is specified implicitly by means of a regular integer grid with step 1. In this setting, a cluster is a subset of vertices lying in the same cell of the grid; the arising ambiguity is resolved arbitrarily. Even in this special setting, the GTSP remains intractable, generalizing in a natural way the classical planar Euclidean TSP. Recently, a $$(1.5 + 8\sqrt 2 + \varepsilon )$$ -approximation algorithm with complexity depending polynomially both on the number of vertices n and on the number of clusters k has been constructed for this problem. We propose three approximation schemes for this problem. For each fixed k, all the schemes are polynomial and the complexity of the first two is linear in the number of nodes. Furthermore, the first two schemes remain polynomial for k = O(log n), whereas the third scheme is polynomial for k = n − O(log n).

## Keywords

generalized traveling salesman problem NP-hard problem polynomial-time approximation scheme.

## References

1. 1.
A. N. Sesekin, A. A. Chentsov, and A. G. Chentsov, Routing Problems (Lan’, St. Petersburg, 2011) [in Russian].
2. 2.
E. M. Arkin and R. Hassin, “Approximation algorithms for the geometric covering salesman problem,” Discrete Appl. Math. 55 (3), 197–218 (1994).
3. 3.
S. Arora, “Polynomial time approximation schemes for Euclidean traveling salesman and other geometric problems,” J. ACM 45 (5), 753–782 (1998).
4. 4.
B. Bhattacharya, A. ´Custi´c, A. Rafiey, A. Rafiey, and V. Sokol, “Approximation algorithms for generalized MST and TSP in grid clusters,” in Combinatorial Optimization and Applications: Proceedings of the 9th International Conference, Houston, TX, USA, 2015 (Springer, Cham, 2015), pp. 110–125.
5. 5.
B. Bontoux, C. Artigues, and D. Feillet, “A memetic algorithm with a large neighborhood crossover operator for the generalized traveling salesman problem,” Comput. Oper. Res. 37 (11), 1844–1852 (2010).
6. 6.
A. Chentsov, M. Khachay, and D. Khachay, “Linear time algorithm for Precedence Constrained Asymmetric Generalized Traveling Salesman Problem,” IFAC-PapersOnLine 49 (12), 651–655 (2016).
7. 7.
N. Christofides, Worst-Case Analysis of a New Heuristic for the Traveling Salesman Problem, Technical Report No. AD-A025 602 (Carnegie Mellon Univ. Pittsburgh, 1976).Google Scholar
8. 8.
M. Dror and J. Orlin, “Combinatorial optimization with explicit delineation of the ground set by a collection of subsets,” SIAM J. Discrete Math. 21 (4), 1019–1034 (2008).
9. 9.
A. Dumitrescu and J. S. B. Mitchell, “Approximation algorithms for TSP with neighborhoods in the plane,” in Proceedings of the 12th Annual ACM–SIAM Symposium on Discrete Algorithms, Washington, DC, USA, 2001 (SIAM, Philadelphia, 2001), pp. 38–46.Google Scholar
10. 10.
A. Dumitrescu and C. D. Tóth, “The traveling salesman problem for lines, balls, and planes,” ACM Trans. Algorithms 12 (3), article 43 (2016).Google Scholar
11. 11.
C. Feremans, A. Grigoriev, and R. Sitters, “The geometric generalized minimum spanning tree problem with grid clustering,” 4OR 4 (4), 319–329 (2006).
12. 12.
G. Gutin and D. Karapetyan, “A memetic algorithm for the generalized traveling salesman problem,” Nat. Comput. 9 (1), 47–60 (2010).
13. 13.
M. Held and R. M. Karp, “A dynamic programming approach to sequencing problems,” J. Soc. Indust. Appl. Math. 10 (1), 196–210 (1962).
14. 14.
A. Henry-Labordere, “The record balancing problem: A dynamic programming solution of a generalized traveling salesman problem,” RAIRO Oper. Res. B2, 43–49 (1969).
15. 15.
K. Jun-man and Z. Yi, “Application of an improved ant colony optimization on generalized traveling salesman problem,” Energy Procedia 17A, 319–325 (2012).
16. 16.
G. Laporte, H. Mercure, and Y. Nobert, “Generalized travelling salesman problem through n sets of nodes: The asymmetrical case,” Discrete Appl. Math. 18 (2), 185–197 (1987).
17. 17.
C. S. Mata and J. S. B. Mitchell, “Approximation algorithms for geometric tour and network design problems,” in Proceedings of the 11th Annual Symposium on Computational Geometry, Vancouver, Canada, 1995 (ACM, New York, 1995), pp. 360–369.Google Scholar
18. 18.
J. S. B. Mitchell, “A PTAS for TSP with neighborhoods among fat regions in the plane,” in Proceedings of the 18th Annual ACM–SIAM Symposium on Discrete Algorithms, New Orleans, LA, USA, 2007 (SIAM, Philadelphia, 2007), pp. 11–18.Google Scholar
19. 19.
J. Saksena, “Mathematical model for scheduling clients through welfare agencies,” CORS J. 8, 185–200 (1970).
20. 20.
D. Williamson and D. Shmoys, The Design of Approximation Algorithms, (Cambridge Univ. Press, Cambridge, 2010).