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On a new edge function on complete weighted graphs and its application for locating Hamiltonian cycles of small weight

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Abstract

In this paper, we define a new combinatorial function on the edges of complete weighted graphs. This function assigns to each edge of the graph the sum of the weights of all Hamiltonian cycles that contain the edge. Since this function involves the factorial function, whose exact calculation is intractable due to its superexponential asymptotic rate of increase, we also introduce a normalized version of the function that is efficiently computable. From this version, we derive an upper bound to the weight of the minimum weight Hamiltonian cycle of the graph based on the weights of the graph edges. Then we investigate the possible algorithmic applications of this normalized function using the Nearest Neighbor Heuristic and a smallest edge first heuristic. As evidence for its applicability, we show that the use of this function as a criterion for the selection of the next edge, improves the performance of both heuristics for approximating the minimum weight Hamiltonian cycles in Euclidean plane graphs. Moreover, our experimental results show that the use of the function is more suitable with the structure of the smallest edge first heuristic since it provides a solution closer to the best known solution of known hard TSP instances but in \(O(n^3)\) time.

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References

  1. Burkard, R.E.: Quadratic Assignment Problems. Handbook of Combinatorial Optimization, Second Edition 5, 2741–2814 (2005)

  2. Pardalos, P.M., Wolkowicz, H., Editors, Quadratic Assignment and Related Problems AMS|DIMACS Series Vol 16 (1994)

  3. Pardalos, P.M., Pitsoulis, L.: Nonlinear assignment problems: algorithms and applications (combinatorial optimization). Kluwer Academic Publishers, Dordrecht (2000). Please check and confirm that the publisher location is updated correctly

    Book  MATH  Google Scholar 

  4. Chen, J., Xia, G.: Complexity Issues on PTAS. Handbook of combinatorial optimization, Second Edition 2, 723–746 (2005)

  5. Sahni, S., Gonzales, T.: P-complete approximation problems. JACM 23, 555–565 (1976)

    Article  MathSciNet  MATH  Google Scholar 

  6. Arora, S.: Polynomial time approximation schemes for Euclidean traveling salesman and other geometric problems. J. ACM 45(5), 753–782 (1998)

    Article  MathSciNet  MATH  Google Scholar 

  7. Christofides, N.: Worst-case analysis of a new heuristic for the traveling salesman problem. In: Traub, J.F. (ed.) Symposium on new directions and recent results in algorithms and complexity, page 441. Academic Press, NY (1976)

  8. Garey, M.R., Graham, R.L., Johnson, D.S.: Some NP-complete geometric problems. In: Proceeding 8th ACM Symposium on Theory of Computing, pp. 10–22 (1976)

  9. Glover, F., Punnen, A.: The traveling salesman problem: new solvable cases and linkages with the development of approximation algorithms. J. Oper. Res. Soc. 48, 502–510 (1997)

    Article  MATH  Google Scholar 

  10. Gutin, G., Yeo, A., Zverovich, A.: Traveling salesman should not be greedy: domination analysis of greedy-type heuristics for the TSP. Disc. Appl. Math. 117, 81–86 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  11. Karp, R.M.: Reducibility among combinatorial problems. In: Miller, R.E., Thatcher, J.W. (eds.) Complexity of computer computations, advances in cmputing research, pp. 85–103. Plenum Press, New York (1972)

    Chapter  Google Scholar 

  12. Karp, R.M.: Probabilistic analysis of partitioning algorithms for the TSP in the plane. Math. Oper. Res. 2, 209–224 (1977)

    Article  MathSciNet  MATH  Google Scholar 

  13. Lawler, E.L., Lenstra, J.K., Rinnooy Kan, A.H.G., Shmoys, D.B.: The traveling salesman problem. John Wiley, New York (1985)

    MATH  Google Scholar 

  14. Wallis, W.D., George, J.C.: Introduction to combinatorics. Chapman and Hall/CRC-Press, London (2011)

    MATH  Google Scholar 

  15. Samples of TSP Library in http://www.iwr.uni-heidelberg.de/groups/comopt/software/TSPLIB95/

  16. Mitchell, J.: Guillotine subdivisions approximate polygonal subdivisions: a simple polynomial-time approximation scheme for geometric TSP, \(k\)-MST, and related problems. SIAM J. Comp. 28, 1298–1309 (1999)

    Article  MathSciNet  MATH  Google Scholar 

  17. Papadimitriou, C.H.: Euclidean TSP is NP-complete. Theo. Comp. Sci. 4, 237–244 (1977)

    Article  MathSciNet  MATH  Google Scholar 

  18. Rao, S., Smith, W.: Approximating geometrical graphs via “spanners” and “banyans”. In: Proc. 30th Annual ACM Symposium on Theory of Computing, pp. 540–550 (1998)

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Acknowledgments

Research is partially supported by US Air Force and DTRA Grants. We also thank the anonymous reviewers for their valuable comments that helped us to improve our paper.

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Authors and Affiliations

  1. Department of Mathematics, University of the Aegean, Morali Building, 83200, Samos Island, Greece

    Panayotis E. Nastou

  2. Center for Applied Optimization, Department of Industrial and Systems Engineering, University of Florida, 303 Weil Hall, Gainesville, USA

    Panayotis E. Nastou & Panos M. Pardalos

  3. Department of Informatics, Hellenic Open University, Patras, Greece

    Vassilis Papadinas

  4. Department of Business Administration, University of Patras Computer Technology Institute and Press- “Diophantus”, 26504, Rio, Patras, Greece

    Yannis C. Stamatiou

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  1. Panayotis E. Nastou
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  2. Vassilis Papadinas
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  3. Panos M. Pardalos
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  4. Yannis C. Stamatiou
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Correspondence to Panayotis E. Nastou.

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Nastou, P.E., Papadinas, V., Pardalos, P.M. et al. On a new edge function on complete weighted graphs and its application for locating Hamiltonian cycles of small weight. Optim Lett 10, 1203–1220 (2016). https://doi.org/10.1007/s11590-015-0914-3

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  • DOI: https://doi.org/10.1007/s11590-015-0914-3

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