×

Backtracking algorithms for constructing the Hamiltonian decomposition of a 4-regular multigraph. (Russian. English summary) Zbl 1503.05119

MSC:

05C85 Graph algorithms (graph-theoretic aspects)
05C70 Edge subsets with special properties (factorization, matching, partitioning, covering and packing, etc.)
52B05 Combinatorial properties of polytopes and polyhedra (number of faces, shortest paths, etc.)

Software:

Stony Brook

References:

[1] J. Krarup, “The peripatetic salesman and some related unsolved problems”, Combinatorial programming: methods and applications, NATO Advanced Study Institutes Series, 19, Springer Netherlands, 1995, 173-178 · doi:10.1007/978-94-011-7557-9_8
[2] M. M. Bae, B. Bose, “Edge disjoint hamiltonian cycles in \(k\)-ary \(n\)-cubes and hypercubes”, IEEE Transactions on Computers, 52:10 (2003), 1271-1284 · doi:10.1109/TC.2003.1234525
[3] R. F. Bailey, “Error-correcting codes from permutation groups”, Discrete Mathematics, 309:13 (2009), 4253-4265 · Zbl 1184.94273 · doi:10.1016/j.disc.2008.12.027
[4] C. Clifton, M. Kantarcioglu, J. Vaidya, X. Lin, M. Y. Zhu, “Tools for privacy preserving distributed data mining”, SIGKDD Explor. Newsl., 4:2 (2002), 28-34 · doi:10.1145/772862.772867
[5] R. W. Hung, “Embedding two edge-disjoint hamiltonian cycles into locally twisted cubes”, Theoretical Computer Science, 412:35 (2011), 4747-4753 · Zbl 1223.68081 · doi:10.1016/j.tcs.2011.05.004
[6] R. Glebov, Z. Luria, B. Sudakov, “The number of hamiltonian decompositions of regular graphs”, Israel Journal of Mathematics, 222:1 (2017), 91-108 · Zbl 1400.05139 · doi:10.1007/s11856-017-1583-y
[7] G. Dantzig, R. Fulkerson, S. Johnson, “Solution of a large-scale traveling-salesman problem”, Journal of the Operations Research Society of America, 2:4 (1954), 393-410 · Zbl 1414.90372 · doi:10.1287/opre.2.4.393
[8] D. L. Applegate, R. E. Bixby, V. Chvatál, W. J. Cook, The traveling salesman problem: a computational study, Princeton University Press, 2006 · Zbl 1130.90036
[9] N. E. Aguilera, R. D. Katz, P. B. Tolomei, “Vertex adjacencies in the set covering polyhedron”, Discrete Applied Mathematics, 218 (2017), 40-56 · Zbl 1352.05179 · doi:10.1016/j.dam.2016.10.024
[10] M. L. Balinski, “Signature methods for the assignment problem”, Operations Research, 33:3 (1985), 527-536 · Zbl 0583.90064 · doi:10.1287/opre.33.3.527
[11] C. R. Chegireddy, H. W. Hamacher, “Algorithms for finding \(k\)-best perfect matchings”, Discrete Applied Mathematics, 18:2 (1987), 155-165 · Zbl 0621.05026 · doi:10.1016/0166-218X(87)90017-5
[12] E. F. Combarro, P. Miranda, “Adjacency on the order polytope with applications to the theory of fuzzy measures”, Fuzzy Sets and Systems, 161:5 (2010), 619-641 · Zbl 1188.28017 · doi:10.1016/j.fss.2009.05.004
[13] H. N. Gabow, “Two algorithms for generating weighted spanning trees in order”, SIAM Journal on Computing, 6:1 (1977), 139-150 · Zbl 0346.68021 · doi:10.1137/0206011
[14] V. A. Bondarenko, “Nonpolynomial lower bounds for the complexity of the traveling salesman problem in a class of algorithms”, Automation and Remote Control, 44:9 (1983), 1137-1142 · Zbl 0571.90089
[15] V. Bondarenko, A. Nikolaev, “On graphs of the cone decompositions for the min-cut and max-cut problems”, International Journal of Mathematics and Mathematical Sciences, 2016 (2016), 7863650 · Zbl 1457.90165 · doi:10.1155/2016/7863650
[16] M. Grötschel, M. Padberg, “Polyhedral theory”, The traveling salesman problem: a guided tour of combinatorial optimization, John Wiley, Chichester, 1985, 251-305 · Zbl 0587.90073
[17] C. H. Papadimitriou, “The adjacency relation on the traveling salesman polytope is np-complete”, Mathematical Programming, 14:1 (1978), 312-324 · Zbl 0376.90067 · doi:10.1007/BF01588973
[18] V. A. Bondarenko, A. V. Nikolaev, “On the skeleton of the polytope of pyramidal tours”, Journal of Applied and Industrial Mathematics, 12:1 (2018), 9-18 · Zbl 1413.05074 · doi:10.1134/S1990478918010027
[19] A. Nikolaev, “On vertex adjacencies in the polytope of pyramidal tours with step-backs”, Mathematical optimization theory and operations research, Motor 2019, LNCS, 11548, Springer, 2019, 247-263 · Zbl 1443.90300 · doi:10.1007/978-3-030-22629-9_18
[20] T. S. Arthanari, “On pedigree polytopes and hamiltonian cycles”, Discrete Mathematics, 306:14 (2006), 1474-1492 · Zbl 1103.90078 · doi:10.1016/j.disc.2005.11.030
[21] T. S. Arthanari, “Study of the pedigree polytope and a sufficiency condition for nonadjacency in the tour polytope”, Discrete Optimization, 10:3 (2013), 224-232 · Zbl 1474.90290 · doi:10.1016/j.disopt.2013.07.001
[22] M. R. Rao, “Adjacency of the traveling salesman tours and \(0 - 1\) vertices”, SIAM Journal on Applied Mathematics, 30:2 (1976), 191-198 · Zbl 0346.90065 · doi:10.1137/0130021
[23] B. Péroche, “Np-completeness of some problems of partitioning and covering in graphs”, Discrete Applied Mathematics, 8:2 (1984), 195-208 · Zbl 0541.05048 · doi:10.1016/0166-218X(84)90101-X
[24] A. Kozlova, A. Nikolaev, “Simulated annealing approach to verify vertex adjacencies in the traveling salesperson polytope”, Mathematical optimization theory and operations research, Motor 2019, LNCS, 11548, Springer, 2019, 374-389 · Zbl 1444.90101 · doi:10.1007/978-3-030-22629-9_26
[25] A. Nikolaev, A. Kozlova, “Hamiltonian decomposition and verifying vertex adjacency in 1-skeleton of the traveling salesperson polytope by variable neighborhood search”, Journal of Combinatorial Optimization, 2020 · Zbl 1477.90088 · doi:10.1007/s10878-020-00652-7
[26] S. S. Skiena, The algorithm design manual, Springer, 2008 · Zbl 1149.68081 · doi:10.1007/978-1-84800-070-4
[27] A. V. Korostil, A. V. Nikolaev, “backtracking algorithm to construct the hamiltonian decomposition of a 4-regular multigraph”, Notes on Computer Science and Mathematics, 12, P.G. Demidov Yaroslavl State University, 2020, 91-97
[28] D. E. Knuth, The art of computer programming, v. 2, Seminumerical algorithms, 3rd ed., Addison-Wesley Longman Publishing Co., Inc., USA, 1997 · Zbl 0895.68055 · doi:10.5555/270146
[29] J. H. Kim, N. C. Wormald, “Random matchings which induce hamilton cycles and hamiltonian decompositions of random regular graphs”, Journal of Combinatorial Theory, Series B, 81:1 (2001), 20-44 · Zbl 1030.05107 · doi:10.1006/jctb.2000.1991
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.