Abstract
Traveling salesman problem (TSP) is well-known in combinatorial optimization. Recent research demonstrates that there are competitive algorithms for TSP on the bounded degree or genus graphs. We present an iterative algorithm to convert the complete graph of a TSP instance into a bounded degree graph based on frequency quadrilaterals. First, the frequency of each edge is computed with N frequency quadrilaterals. Second, \(\frac{1}{3}\) of the edges having the smallest frequencies are cut for each vertex. The two steps are repeated until there are no quadrilaterals for most edges in the residual graph. At the \(k^{th}\ge 1\) iteration, the maximum vertex degree is smaller than \(\left\lceil \left( \frac{2}{3}\right) ^k(n-1)\right\rceil \). Two theorems illustrate that the long edges are cut and the short edges will be preserved. Thus, the optimal solution or a good approximation will be found in the bounded degree graphs. The iterative algorithm is tested with the real-life TSP instances.
The author acknowledge the funds supported by the Fundamental Research Funds for the Central Universities (No. 2018MS039 and No. 2018ZD09).
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References
Gutin, G., Punnen, A.P.: The Traveling Salesman Problem and its Variations. Springer, New York (2007). https://doi.org/10.1007/b101971
Karp, R.M.: On the computational complexity of combinatorial problems. Networks (USA) 5(1), 45–68 (1975)
Held, M., Karp, R.M.: A dynamic programming approach to sequencing problems. J. Soc. Ind. Appl. Math. 10(1), 196–210 (1962)
Bellman, R.E.: Dynamic programming treatment of the travelling salesman problem. J. ACM 9(1), 61–63 (1962)
Carpaneto, G., Dell’Amico, M., Toth, P.: Exact solution of large-scale, asymmetric traveling salesman problems. ACM Trans. Math. Softw. (TOMS) 21(4), 394–409 (1995)
de Klerk, E., Dobre, C.: A comparison of lower bounds for the symmetric circulant traveling salesman problem. Discret. Appl. Math. 159(16), 1815–1826 (2011)
Levine, M.S.: Finding the right cutting planes for the TSP. ACM J. Exp. Algorithmics (JEA) 5, 1–20 (2000)
Applegate, D.L., et al.: Certification of an optimal TSP tour through 85900 cities. Oper. Res. Lett. 37(1), 11–15 (2009)
Sharir, M., Welzl, E.: On the number of crossing-free matchings, cycles, and partitions. SIAM J. Comput. 36(3), 695–720 (2006)
Gebauer, H.: Enumerating all Hamilton cycles and bounding the number of Hamiltonian cycles in 3-regular graphs. Electr. J. Comb. 18(1), 1–28 (2011)
Björklund, A., Husfeldt, T., Kaski, P., Koivisto, M.: The traveling salesman problem in bounded degree graphs. ACM Trans. Algorithms 8(2), 1–18 (2012)
Borradaile, G., Demaine, E.D., Tazari, S.: Polynomial-time approximation schemes for subset-connectivity problems in bounded-genus graphs. Algorithmica 68(2), 287–311 (2014)
Gharan, S.O., Saberi, A.: The asymmetric traveling salesman problem on graphs with bounded genus. In: Proceedings of the Twenty-Second Annual ACM-SIAM Symposium on Discrete Algorithms (SODA 2011), pp. 1–12. ACM, NY, USA (2011)
Hougardy, S., Schroeder, R.T.: Edge elimination in TSP instances. In: Kratsch, D., Todinca, I. (eds.) WG 2014. LNCS, vol. 8747, pp. 275–286. Springer, Cham (2014). https://doi.org/10.1007/978-3-319-12340-0_23
Wang, Y.: An approximate method to compute a sparse graph for traveling salesman problem. Expert Syst. Appl. 42(12), 5150–5162 (2015)
Wang, Y., Remmel, J.B.: A binomial distribution model for the traveling salesman problem based on frequency quadrilaterals. J. Graph Algorithms Appl. 20(2), 411–434 (2016)
Wang, Y., Remmel, J.: A method to compute the sparse graphs for traveling salesman problem based on frequency quadrilaterals. In: Chen, J., Lu, P. (eds.) FAW 2018. LNCS, vol. 10823, pp. 286–299. Springer, Cham (2018). https://doi.org/10.1007/978-3-319-78455-7_22
Reinelt, G.: TSPLIB (1995). http://comopt.ifi.uni-heidelberg.de/software/TSPLIB95/
Mittelmann, H.: NEOS Server for Concorde (2018). http://neos-server.org/neos/solvers/co:concorde/TSP.html
Nešetřil, J., de Mendez, P.O.: Sparsity Graphs. Structures and Algorithms. Springer, Heidelberg (2012). https://doi.org/10.1007/978-3-642-27875-4
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Wang, Y. (2019). Bounded Degree Graphs Computed for Traveling Salesman Problem Based on Frequency Quadrilaterals. In: Li, Y., Cardei, M., Huang, Y. (eds) Combinatorial Optimization and Applications. COCOA 2019. Lecture Notes in Computer Science(), vol 11949. Springer, Cham. https://doi.org/10.1007/978-3-030-36412-0_43
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DOI: https://doi.org/10.1007/978-3-030-36412-0_43
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