Abstract
We examine a generalization of the symmetric bipartite traveling salesman problem (TSP) with quadrangle inequality, by extending the cost function of a Hamiltonian tour to include a bias factor β ≥ 1. The bias factor is known and given as a part of the input. We propose a novel heuristic procedure for building Hamiltonian cycles in bipartite graphs, and show that it is an approximation algorithm for the generalized problem with an approximation ratio of \(1+\frac{1+\lambda}{\beta+\lambda}\), where λ is a real parameter dependent on the problem instance. This expression is bounded above by a constant 2, for any positive real λ and β ≥ 1, which improves a previously reported approximation ratio of 16/7. As a part of a composite heuristic, the proposed procedure can contribute to an approximation ratio of \(1+\frac{2}{\zeta+\beta(2-\zeta)}\), where ζ is an approximation ratio for the metric TSP.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Preview
Unable to display preview. Download preview PDF.
Similar content being viewed by others
References
Akiyama, T., Nishizeki, T., Saito, N.: NP-completeness of the Hamiltonian cycle problem for bipartite graphs. Journal of Information Processing 3(2), 73–76 (1980)
Anily, S., Hassin, R.: The swapping problem. Networks 22(4), 419–433 (1992)
Baltz, A., Srivastav, A.: Approximation algorithms for the Euclidean bipartite TSP. Operations Research Letters 33(4), 403–410 (2005)
Chalasani, P., Motwani, R.: Approximating capacitated routing and delivery problems. SIAM Journal on Computing 28(6), 2133–2149 (1999)
Chalasani, P., Motwani, R., Rao, A.: Algorithms for robot grasp and delivery. In: Proceedings of 2nd International Workshop on Algorithmic Foundations of Robotics, pp. 347–362 (1996)
Christofides, N.: Worst-case analysis of a new heuristic for the travelling salesman problem. Technical Report 388, Graduate School of Industrial Administration, Carnegie-Mellon University, Pittsburgh (1976)
Cook, W.J.: In Pursuit of the Traveling Salesman: Mathematics at the Limits of Computation. Princeton University Press (2012)
Frank, A.: Connections in Combinatorial Optimization. Oxford Lecture Series in Mathematics and Its Applications. OUP Oxford (2011)
Frank, A.: A weighed matroid intersection algorithm. Journal of Algorithms 2, 328–336 (1981)
Frank, A., Triesch, E., Korte, B., Vygen, J.: On the bipartite traveling salesman problem. Technical Report 98866-OR, University of Bonn (1998)
Karuno, Y., Nagamochi, H., Shurbevski, A.: An approximation algorithm with factor two for a repetitive routing problem of grasp-and-delivery robots. Journal of Advanced Computational Intelligence and Intelligent Informatics 15(8), 1103–1108 (2011)
Korte, B., Vygen, J.: Combinatorial Optimization: Theory and Algorithms, 3rd edn. Springer (2005)
Krishnamoorthy, M.: An NP-hard problem in bipartite graphs. SIGACT News 7, 26–26 (1975)
Langston, M.A.: A study of composite heuristic algorithms. The Journal of the Operational Research Society 38(6), 539–544 (1987)
Sahni, S., Gonzalez, T.: P-complete approximation problems. Journal of the ACM 23(3), 555–565 (1976)
Schrijver, A.: Combinatorial Optimization - Polyhedra and Efficiency. Springer (2003)
Shurbevski, A., Nagamochi, H., Karuno, Y.: Heuristics for a repetitive routing problem of a single grasp-and-delivery robot with an asymmetric edge cost function. In: 10th International Conference of the Society for Electronics, Telecommunications, Automatics and Informatics (ETAI 2011), CD-ROM Proceedings, pp. A1–1 (2011)
Srivastav, A., Schroeter, H., Michel, C.: Approximation algorithms for pick-and-place robots. Annals of Operations Research 107(3), 321–338 (2001)
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2014 Springer International Publishing Switzerland
About this paper
Cite this paper
Shurbevski, A., Nagamochi, H., Karuno, Y. (2014). Approximating the Bipartite TSP and Its Biased Generalization. In: Pal, S.P., Sadakane, K. (eds) Algorithms and Computation. WALCOM 2014. Lecture Notes in Computer Science, vol 8344. Springer, Cham. https://doi.org/10.1007/978-3-319-04657-0_8
Download citation
DOI: https://doi.org/10.1007/978-3-319-04657-0_8
Publisher Name: Springer, Cham
Print ISBN: 978-3-319-04656-3
Online ISBN: 978-3-319-04657-0
eBook Packages: Computer ScienceComputer Science (R0)