CIAC 2006: Algorithms and Complexity pp 223-235

# An Approximation Algorithm for a Bottleneck Traveling Salesman Problem

• Ming-Yang Kao
• Manan Sanghi
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 3998)

## Abstract

Consider a truck running along a road. It picks up a load L i at point β i and delivers it at α i , carrying at most one load at a time. The speed on the various parts of the road in one direction is given by f(x) and that in the other direction is given by g(x). Minimizing the total time spent to deliver loads L 1,...,L n is equivalent to solving the Traveling Salesman Problem (TSP) where the cities correspond to the loads L i with coordinates (α i , β i ) and the distance from L i to L j is given by $$\int^{\beta_j}_{\alpha_i} f(x)dx$$ if β j α i and by $$\int^{\alpha_i}_{\beta_j} g(x)dx$$ if β j < α i . This case of TSP is polynomially solvable with significant real-world applications.

Gilmore and Gomory obtained a polynomial time solution for this TSP [6]. However, the bottleneck version of the problem (BTSP) was left open. Recently, Vairaktarakis showed that BTSP with this distance metric is NP-complete [10].

We provide an approximation algorithm for this BTSP by exploiting the underlying geometry in a novel fashion. This also allows for an alternate analysis of Gilmore and Gomory’s polynomial time algorithm for the TSP. We achieve an approximation ratio of (2+γ) where $$\gamma \geq \frac{f(x)}{g(x)} \geq \frac{1}{\gamma} \; \forall x$$. Note that when f(x)=g(x), the approximation ratio is 3.

## Keywords

Nuclear Magnetic Resonance Approximation Algorithm Minimum Span Tree Travel Salesman Problem Travel Salesman Problem
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

## Preview

Unable to display preview. Download preview PDF.

## References

1. 1.
Bailey-Kellogg, C., Chainraj, S., Pandurangan, G.: A Random Graph Approach to NMR Sequential Assignment. In: Proceedings of the 8th Annual International Conference on Computational Molecular Biology, pp. 58–67 (2004)Google Scholar
2. 2.
Ball, M.O., Magazine, M.J.: Sequencing of Insertions in Printed Circuit Board Assembly. Operations Research 36, 192–201 (1988)
3. 3.
Cavanagh, J., Fairbrother, W.J., Palmer III, A.G., Skelton, N.J.: Protein NMR Spectroscopy: Principles and Practice. Academic Press, New York (1996)Google Scholar
4. 4.
Chen, Z.-Z., Jiang, T., Lin, G., Wen, J., Xu, D., Xu, J., Xu, Y.: Approximation Algorithms for NMR Spectral Peak Assignment. Theoretical Computer Science 299, 211–229 (2003)
5. 5.
Garey, M.R., Johnson, D.S.: Computers and Intractability: A Guide to the Theory of NP-Completeness. W. H. Freeman & Co., New York (1979)
6. 6.
Gilmore, P.C., Gomory, R.E.: Sequencing a One State-Variable Machine: A Solvable Case of the Traveling Salesman Problem. Operations Research 12, 655–679 (1964)
7. 7.
Gutin, G., Punnen, A.P.: The Traveling Salesman Problem and Its Variations. Kluwer Academic Publishers, Dordrecht (2002)
8. 8.
Hitchens, T.K., Lukin, J.A., Zhan, Y., McCallum, S.A., Rule, G.S.: MONTE: An Automated Monte Carlo Based Approach to Nuclear Magnetic Resonance Assignment of Proteins. Journal of Biomolecular NMR 25, 1–9 (2003)
9. 9.
Reddi, S.S., Ramamoorthy, C.V.: On the Flow-Shop Sequencing Problem with No Wait in Process. Operational Research Quarterly 23, 323–331 (1972)
10. 10.
Vairaktarakis, G.L.: On Gilmore-Gomory’s open question for the bottleneck TSP. Operations Research Letters 31, 483–491 (2003)
11. 11.
Vairaktarakis, G.L.: Simple Algorithms for Gilmore-Gomory’s Traveling Salesman and Related Problems. Journal of Scheduling 6, 499–520 (2003)
12. 12.
Vitek, O., Vitek, J., Craig, B., Bailey-Kellogg, C.: Model-Based Assignment and Inference of Protein Backbone Nuclear Magnetic Resonances. Statistical Applications in Genetics and Molecular Biology 3, 1–22 (2004)
13. 13.
Wan, X., Xu, D., Slupsky, C.M., Lin, G.: Automated Protein NMR Resonance Assignments. In: Proceedings of the 2nd IEEE Computer Society Conference on Bioinformatics, pp. 197–208 (2003)Google Scholar
14. 14.
Wüthrich, K.: NMR of Proteins and Nucleic Acids. John Wiley & Sons, New York (1986)Google Scholar
15. 15.
Xu, Y., Xu, D., Kim, D., Olman, V., Razumovskaya, J., Jiang, T.: Automated Assignment of Backbone NMR Peaks Using Constrained Bipartite Matching. Computing in Science and Engineering 4, 50–62 (2002)Google Scholar