Abstract
We consider the traveling salesperson problem in a directed graph. The pyramidal tours with step-backs are a special class of Hamiltonian tours for which the traveling salesperson problem is solved by dynamic programming in polynomial time. The polytope of pyramidal tours with step-backs \(\mathrm{{PSB}}(n)\) is defined as the convex hull of the characteristic vectors of all possible pyramidal tours with step-backs in a complete directed graph. The skeleton of \({\mathrm{{PSB}}} (n)\) is the graph whose vertex set is the vertex set of \({\mathrm{{PSB}}} (n)\) and the edge set is the set of geometric edges or one-dimensional faces of \({\mathrm{{PSB}}} (n)\). The main result of the paper is a necessary and sufficient condition for vertex adjacencies in the skeleton of the polytope \({\mathrm{{PSB}}} (n)\) that can be verified in polynomial time.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
References
Aguilera, N.E., Katz, R.D., Tolomei, P.B.: Vertex adjacencies in the set covering polyhedron. Discrete Appl. Math. 218, 40–56 (2017). https://doi.org/10.1016/j.dam.2016.10.024
Arthanari, T.S.: Study of the pedigree polytope and a sufficiency condition for nonadjacency in the tour polytope. Discrete Optim. 10(3), 224–232 (2013). https://doi.org/10.1016/j.disopt.2013.07.001
Balinski, M.L.: Signature methods for the assignment problem. Oper. Res. 33(3), 527–536 (1985). https://doi.org/10.1287/opre.33.3.527
Bondarenko, V.A., Maksimenko, A.N.: Geometricheskie konstruktsii i slozhnost’ v kombinatornoy optimizatsii (Geometric constructions and complexity in combinatorial optimization). LKI, Moscow (2008). [in Russian]
Bondarenko, V.A., Nikolaev, A.V.: On graphs of the cone decompositions for the min-cut and max-cut problems. Int. J. Math. Sci. 2016 (2016). Article ID 7863650, 6 p. https://doi.org/10.1155/2016/7863650
Bondarenko, V.A., Nikolaev, A.V.: Some properties of the skeleton of the pyramidal tours polytope. Electron. Notes Discrete Math. 61, 131–137 (2017). https://doi.org/10.1016/j.endm.2017.06.030
Bondarenko, V.A., Nikolaev, A.V.: On the skeleton of the polytope of pyramidal tours. J. Appl. Ind. Math. 12, 9–18 (2018). https://doi.org/10.1134/S1990478918010027
Bondarenko, V.A., Nikolaev, A.V., Shovgenov, D.A.: 1-skeletons of the spanning tree problems with additional constraints. Autom. Control Comput. Sci. 51(7), 682–688 (2017). https://doi.org/10.3103/s0146411617070033
Bondarenko, V.A., Nikolaev, A.V., Shovgenov, D.A.: Polyhedral characteristics of balanced and unbalanced bipartite subgraph problems. Autom. Control Comput. Sci. 51(7), 576–585 (2017). https://doi.org/10.3103/s0146411617070276
Chegireddy, C.R., Hamacher, H.W.: Algorithms for finding K-best perfect matchings. Discrete Appl. Math. 18, 155–165 (1987). https://doi.org/10.1016/0166-218X(87)90017-5
Combarro, E.F., Miranda, P.: Adjacency on the order polytope with applications to the theory of fuzzy measures. Fuzzy Sets Syst. 161, 619–641 (2010). https://doi.org/10.1016/j.fss.2009.05.004
Enomoto, H., Oda, Y., Ota, K.: Pyramidal tours with step-backs and the asymmetric traveling salesman problem. Discrete Appl. Math. 87, 57–65 (1998). https://doi.org/10.1016/S0166-218X(98)00048-1
Gabow, H.N.: Two algorithms for generating weighted spanning trees in order. SIAM J. Comput. 6, 139–150 (1977). https://doi.org/10.1137/0206011
Gilmore, P.C., Lawler, E.L., Shmoys, D.B.: Well-solved special cases. In: Lawler, E., Lenstra, J.K., Rinnooy Kan, A., Shmoys, D. (eds.) The Traveling Salesman Problem: A Guided Tour of Combinatorial Optimization, pp. 87–143. Wiley, Chichester (1985)
Grötschel, M., Padberg, M.: Polyhedral theory. In: Lawler, E., Lenstra, J.K., Rinnooy Kan, A., Shmoys, D. (eds.) The Traveling Salesman Problem: A Guided Tour of Combinatorial Optimization, pp. 251–305. Wiley, Chichester (1985)
Khachay, M., Neznakhina, K.: Generalized pyramidal tours for the generalized traveling salesman problem. In: Gao, X., Du, H., Han, M. (eds.) COCOA 2017. LNCS, vol. 10627, pp. 265–277. Springer, Cham (2017). https://doi.org/10.1007/978-3-319-71150-8_23
Matsui, T.: NP-completeness of non-adjacency relations on some 0–1 polytopes. In: Proceedings of ISORA 1995. Lecture Notes in Operations Research, vol. 1, pp. 249–258 (1995)
Oda, Y.: An asymmetric analogue of van der Veen conditions and the traveling salesman problem. Discrete Appl. Math. 109, 279–292 (2001). https://doi.org/10.1016/S0166-218X(00)00273-0
Papadimitriou, C.H.: The adjacency relation on the traveling salesman polytope is NP-Complete. Math. Program. 14, 312–324 (1978). https://doi.org/10.1007/BF01588973
Sierksma, G.: The skeleton of the symmetric traveling salesman polytope. Discrete Appl. Math. 43, 63–74 (1993). https://doi.org/10.1016/0166-218X(93)90169-O
Sierksma, G., Teunter, R.H., Tijssen, G.A.: Faces of diameter two on the Hamiltonian cycle polytype. Oper. Res. Lett. 18(2), 59–64 (1995). https://doi.org/10.1016/0167-6377(95)00035-6
Simanchev, R.Yu.: On the vertex adjacency in a polytope of connected k-factors. Trudy Inst. Mat. i Mekh. UrO RAN 24(2), 235–242 (2018). https://doi.org/10.21538/0134-4889-2018-24-2-235-242
Acknowledgments
The research is supported by the grant of the President of the Russian Federation MK-2620.2018.1 (agreement no. 075-015-2019-746).
Author information
Authors and Affiliations
Corresponding author
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2019 Springer Nature Switzerland AG
About this paper
Cite this paper
Nikolaev, A. (2019). On Vertex Adjacencies in the Polytope of Pyramidal Tours with Step-Backs. In: Khachay, M., Kochetov, Y., Pardalos, P. (eds) Mathematical Optimization Theory and Operations Research. MOTOR 2019. Lecture Notes in Computer Science(), vol 11548. Springer, Cham. https://doi.org/10.1007/978-3-030-22629-9_18
Download citation
DOI: https://doi.org/10.1007/978-3-030-22629-9_18
Published:
Publisher Name: Springer, Cham
Print ISBN: 978-3-030-22628-2
Online ISBN: 978-3-030-22629-9
eBook Packages: Computer ScienceComputer Science (R0)