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Efficient algorithms with performance guarantees for some problems of finding several discrete disjoint subgraphs in complete weighted graph. (English) Zbl 1338.05263
Summary: Some hard-to solve combinatorial problems of finding several disjoint discrete structures in complete weighted graph are considered. Efficient algorithms with performance guarantees are constructed for the Euclidean \(m\)-Peripatetic Salesman Problem, \( m\)-Weighted Clique Problem and \( m\)-Layer Planar three-index Assignment Problem.
05C85 Graph algorithms (graph-theoretic aspects)
90C35 Programming involving graphs or networks
Full Text: DOI
[1] Baburin, A. E.; Gimadi, E. Kh., On the asymptotic optimality of an algorithm for solving the maximum m-PSP in a multidimensional Euclidean space, Proc. Steklov Inst. Math., 272, Suppl. 1, 1-13, (2011) · Zbl 1230.65065
[2] Burkard, R.; Dell’Amico, M.; Martello, S., Assignments, (2009), SIAM Philadelphia, 382 p
[3] De Brey, M. J.D.; Volgenant, A., Well-solved cases of the 2-PSP, Optimization, 39, 3, 275-293, (1997) · Zbl 0886.90120
[4] De Kort, J. B.J. M., Lower bounds for symmetric K-PSP, Optimization, 22, 1, 113-122, (1991) · Zbl 0717.90048
[5] De Kort, J. B.J. M., Upper bounds for the symmetric 2-PSP, Optimization, 23, 4, 357-367, (1992) · Zbl 0814.05066
[6] De Kort, J. B.J. M., A branch and bound algorithm for symmetric 2-PSP, EJOR, 70, 229-243, (1993) · Zbl 0799.90114
[7] Dinits, E. A.; Kronrod, M. A., One algorithm for solving assignment problem, Dokl. AN SSSR, 189, 1, 23-25, (1969)
[8] Duchenne, E.; Laporte, G.; Semet, F., Branch-and-cut algorithms for the undirected m-PSP, EJOR, 162, 700-712, (2005) · Zbl 1067.90136
[9] Eremin, Ivan; Gimadi, Edward; Kelmanov, Alexander; Pyatkin, Artem; Khachay, Mikhail, 2-approximation algorithm for finding a clique with minimum weight of vertices and edges, Proc. Steklov Inst., 284, Supp. l, (2014)
[10] Fon-Der-Flaass, D. G., Array of distinct representatives - a very simple NP-complete problem, Discrete Math., 171, 1-3, 295-298, (1997) · Zbl 0879.68040
[11] Frieze, A. M., Complexity of a 3-dimensional assignment problem, Eur. J. Oper. Res., 13, 2, 161-164, (1983) · Zbl 0507.90057
[12] Gimadi, E. Kh., Asymptotically optimal algorithm for finding one and two edge-disjoint traveling salesman routes of maximal weight in Euclidean space, Proc. Steklov Inst. Math., 263, Suppl. 2, 56-67, (2008) · Zbl 1178.90335
[13] Gimadi, E. Kh.; Kelmanov, A. V.; Pyatrin, A. V.; Khachay, M. Y., Efficient algorithms with performance estimates for some problems of finding several disjoint cliques in complete undirected weighted graph, Trudy IMM UrO RAN, 20, 2, 99-112, (2014), (in Russian)
[14] E.Kh. Gimadi, Approximation efficient algorithms with performance guarantees for some hard routing problems, in; Proc. of the II International Conference ‘Optimzation and Applications’ OPTIMA-2011, Petrovac/Montenegro, 2011, pp. 98-101.
[15] Gimadi, Edward Kh., On some probability inequalities for some discrete optimization problems, (Operations Research Proceedings, Selected Papers. International Conference OR 2005, (2006), Springer Bremen, Berlin), 283-289 · Zbl 1114.90454
[16] E.Kh. Gimadi, N.M. Korkishko, On some modifications of three index planar assignment problem, in: Proc. Discrete optimization methods in production and logistics (DOM’2004), The second international workshop, Omsk, 2004, pp. 161-165.
[17] Gimadi, E. Kh.; Glazkov, Yu. V., An asymptotically optimal algorithm for one modification of planar three-index assignment problem, J. Appl. Ind. Math., 1, 4, 442-452, (2007), (Pleiades Publ. Ltd)
[18] Gimadi, E. Kh.; Glazkov, Y. V.; Tsidulko, O. Y., Probabilistic analysis of an algorithm for the m-planar 3-index assignment problem on single-cycle permutations, J. Appl. Ind. Math., 8, 2, 208-217, (2014) · Zbl 1324.90106
[19] Hopcroft, J. E.; Karp, R. M., An \(n^{5 / 2}\) algorithm for maximum matchings in bipartite graphs, SIAM J. Comput., 2, 4, 225-231, (1973) · Zbl 0266.05114
[20] Kleinschmidt, P.; Schannath, H., A strongly polynomial algorithm for the transportation problem, Math. Prog., 68, 1-13, (1995) · Zbl 0833.90084
[21] Krarup, J., The peripatetic salesman and some related unsolved problems, (Combinatorial Programming: Methods and Applications (Proc. NATO Advanced Study Inst., Versailles, 1974), (1975), Reidel Dordrecht), 173-178
[22] Ore, O., Theory of graphs, (1962), AMS Providence, 279 p
[23] (Petrov, V. V., Limit Theorems of Probability Theory, Sequences of Independent Random Variables, (1995), Clarendon Press Oxford), 304
[24] Roskind, J.; Tarjan, R. E., A note on finding minimum-cost edge-disjoint spanning trees, Math. Oper. Res., 10, 4, 701-708, (1985) · Zbl 0581.90093
[25] A.I. Serdukov, An asymptotically optimal algorithm for solving Max Euclidean TSP. Metodi Celochislennoi Optimizacii (Upravliaemie Sistemi), Novosibirsk. n. 27, 1987, p. 79-87 (in Russian).
[26] Spieksma, F. C.R., Multi-index assignment problems: complexity, approximation, applications, Nonlinear Assignment Problems, Algorithms and Applications, (2000), Kluwer Acad. Publ. Dordrecht, pp. 1-12 · Zbl 1029.90036
[27] (Gutin, G.; Punnen, A. P., The Traveling Salesman Problem and its Variations, (2002), Kluwer Academic Publishers Dordrecht, Boston, London), 830 · Zbl 0996.00026
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