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EN\|RU About Journal Editorial Board Publishing Ethics Author Guide Submission Prepare Your Manuscript Peer review Archive English version: Journal of Applied and Industrial Mathematics, 2017, 11:3, 354-361
	Volume 24, No 3, 2017, P. 5-19 UDC 519.8 E. Kh. Gimadi and O. Yu. Tsidulko An asymptotically optimal algorithm for the $m$ -Peripatetic Salesman Problem on random inputs with discrete distribution Abstract: We consider the $m$ -Peripatetic Salesman Problem ( $m$ -PSP) on random inputs with discrete distribution function. In this paper we present a polynomial approximation algorithm which, under certain conditions, with high probability (w.h.p.) gives optimal solution for both the $m$ -PSP on random inputs with identical weight functions and the $m$ -PSP with different weight functions. Illustr. 1, bibliogr. 27. Keywords: $m$ -Peripatetic Salesman Problem, asymptotically optimal algorithm, random inputs, discrete distribution. DOI: 10.17377/daio.2017.24.551 Edward Kh. Gimadi ^1,2 Oxana Yu. Tsidulko ^1,2 1. Sobolev Institute of Mathematics, 4 Koptyug Ave., 630090 Novosibirsk, Russia 2. Novosibirsk State University, 2 Pirogov St., 630090 Novosibirsk, Russia e-mail: gimadi@math.nsc.ru, tsidulko.ox@gmail.com Received 3 August 2016 Revised 13 October 2016 References [1] A. A. Ageev, A. E. Baburin, and E. Kh. Gimadi, A 3/4-approximation algorithm for finding two disjoint Hamiltonian cycles of maximum weight, Diskretn. Anal. Issled. Oper., Ser. 1, 13, No. 2, 11–20, 2006 [Russian]. Translated in J. Appl. Ind. Math., 1, No. 2, 142–147, 2007. [2] A. A. Ageev and A. V. Pyatkin, A 2-approximation algorithm for the metric 2-peripatetic salesman problem, Diskretn. Anal. Issled. Oper., 16, No. 4, 3–20, 2009 [Russian]. [3] A. E. Baburin and E. Kh. Gimadi, On the asymptotic optimality of an algorithm for solving the maximum $m$ -PSP in a multidimensional Euclidean space, Tr. Inst. Mat. Mekh., 16, No. 3, 12–24, 2010 [Russian]. Translated in Proc. Steklov Inst. Math., 272, Suppl. 1, S1–S13, 2011. [4] A. E. Baburin, E. Kh. Gimadi, and N. M. 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Bollobás, T. I. Fenner, and A. M. Frieze, An algorithm for finding Hamilton paths and cycles in random graphs, Combinatorica, 7, No. 4, 327–341, 1987. [16] M. J. D. De Brey and A. Volgenant, Well-solved cases of the 2-peripatetic salesman problem, Optimization, 39, No. 3, 275–293, 1997. [17] J. B. J. M. De Kort, Lower bounds for symmetric $K$ -peripatetic salesman problems, Optimization, 22, No. 1, 113–122, 1991. [18] J. B. J. M. De Kort, Bounds for the symmetric $K$ -peripatetic salesman problem, Optimization, 23, No. 4, 357–367, 1992. [19] J. B. J. M. De Kort, A branch and bound algorithm for symmetric 2-peripatetic salesman problems, Eur. J. Oper. Res., 70, No. 2, 229–243, 1993. [20] P. Erdös and A. Rényi, On random graphs I, Publ. Math., 6, 290–297, 1959. [21] A. M. Frieze, An algorithm for finding Hamilton cycles in random directedgraphs, J. Algorithms, 9, No. 2, 181–204, 1988. [22] A. M. Frieze and S. Haber, An almost linear time algorithm for finding Hamilton cycles in sparse random graphs with minimum degree at least three, Random Struct. Algorithms, 47, No. 1, 73–98, 2015. [23] E. Kh. Gimadi, A. N. Glebov, A. A. Skretneva, O. Yu. Tsidulko, and D. Zh. Zambalaeva, Combinatorial algorithms with performance guarantees for finding several Hamiltonian circuits in a complete directed weighted graph, Discrete Appl. Math., 196, No. 11, 54–61, 2015. [24] A. N. Glebov and A. V. Gordeeva, An algorithm with approximation ratio 5/6 for the metric maximum $m$ -PSP, in Discrete Optimization and Operations Research (Proc. 9th Int. Conf. DOOR, Vladivostok, Russia, Sept. 19–23, 2016), pp. 159–170, Springer, Cham, Switzerland, 2016 (Lect. Notes Comput. Sci., Vol. 9869). [25] J. Komlós and E. Szemerédi, Limit distributions for the existence of Hamilton circuits in a random graph, Discrete Math., 43, No. 1, 55–63, 1983. [26] J. 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