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On the symmetric travelling salesman problem I: inequalities.(English)Zbl 0413.90048


MSC:

90C10 Integer programming
52Bxx Polytopes and polyhedra
52A40 Inequalities and extremum problems involving convexity in convex geometry

Cite

References:

[1]C. Berge,Graphs et hypergraphs (Dunod, Paris, 1970). ·Zbl 0334.05117
[2]V. Chvátal, ”Edmonds polytopes and weakly Hamiltonian graphs”,Mathematical Programming 5 (1973) 29–40. ·Zbl 0267.05118 ·doi:10.1007/BF01580109
[3]G.B. Dantzig, D.R. Fulkerson and S.M. Johnson, ”Solution of a large-scale travelling salesman problem”,Operations Research 2 (1954) 393–410. ·doi:10.1287/opre.2.4.393
[4]J. Edmonds, ”Maximum matching and a polyhedron with 0, 1 vertices”,Journal of Research of the National Buruea of Standards Sect. B 69 (1965) 125–130. ·Zbl 0141.21802
[5]J. Edmonds and E.L. Johnson, ”Matching: A well-solved class of integer linear programs”, in R.K. Guy, H. Hanani, N. Sauer and J. Schonheim, eds.,Combinatorial structure and their applications (Gordon and Breach, New York, 1970) pp. 89–92. ·Zbl 0258.90032
[6]R. Garfinkel and G.L. Nemhauser,Integer programming (Wiley, New York, 1972). ·Zbl 0259.90022
[7]R.E. Gomory, ”The travelling salesman problem”, in:Proceedings of the IBM Scientific Computing Symposium on Combinatorial Problems, IBM Data Processing Division (White Plains, NY, 1964). ·Zbl 0116.25001
[8]M. Grötschel, ”Polyedrische Charakterisierungen kombinatorischer Optimierungsprobleme”, Dissertation, Universität Bonn, 1977 (Verlag A. Hain, Meisenheim, 1977). ·Zbl 0392.90045
[9]M. Grötschel, ”Further results on the facial structure of the travelling salesman polytope”, IX International Symposium on Mathematical Programming (Budapest, August 1976).
[10]M. Grötschel and M.W. Padberg, ”Linear characterizations of travelling salesman problems”, EURO-I, First European Congress on Operations Research (Brussels, January 1975). ·Zbl 0348.90146
[11]M. Grötschel and M.W. Padberg, ”Zur Oberflächenstruktur des Travelling Salesman Polytopen”, in: H.J. Zimmermann, A. Schub, H. Späth and J. Stoer, eds.,Proceedings in Operations Research, 4 (Physica-Verlag, Würzburg, 1974) pp. 207–211. ·Zbl 0314.90061
[12]M. Grötschel and M.W. Padberg: ”On the travelling salesman problem”, Joint National Meeting of ORSA and TIMS (Las Vegas, November 1975). ·Zbl 0348.90146
[13]M. Grötschel and M.W. Padberg, ”Lineare Charakterisierungen von Travelling Salesman Problemen”,Zeitschrift für Operations Research 21 (1977) 33–64. ·Zbl 0348.90147 ·doi:10.1007/BF01918456
[14]M. Grötschel and M.W. Padberg, ”Partial linear characterizations of the asymmetric travelling salesman polytope”,Mathematical Programming 8 (1975) 378–381. ·Zbl 0348.90146 ·doi:10.1007/BF01580454
[15]M. Grötschel and M.W. Padberg, ”On the symmetric travelling salesman problem”, Report No. 7536-OR, Inst. für Ökonometrie und Operations Research, Universität Bonn (Bonn, 1975). ·Zbl 0348.90146
[16]M. Grötschel and M.W. Padberg, ”On the symmetric travelling salesman problem II: Lifting theorems and facets”,Mathematical Programming 16 (1979) 281–302 (this issue). ·Zbl 0413.90049 ·doi:10.1007/BF01582117
[17]F. Harary,Graph theory (Addison-Wesley, Reading, MA, 1972). ·Zbl 0235.05105
[18]I. Heller, ”On the problem of shortest paths between point”, I and II (Abstract),Bulletin of the American Mathematical Society 59 (1953) 551–552.
[19]I. Heller, ”Neighbor relations on the convex of cyclic permutations”,Pacific Journal of Mathematics (1956) 467–477. ·Zbl 0074.38204
[20]S. Hong, ”A linear programming approach for the travelling salesman problem”, Ph.D. Thesis, The Johns Hopkins University (Baltimore, 1971).
[21]R.M. Karp, ”Reducibility among combinatorial problems”, in: R.E. Miller and J.W. Thatcher, eds.,Complexity of computer computations (Plenum Press, New York, 1972) pp. 83–103.
[22]H.W. Kuhn, ”On certain convex polyhedra” (Abstract),Bulletin of the American Mathematical Society 61 (1955) 557–558.
[23]J.F. Maurras, ”Polytopes à sommets dans [0, 1] n ”, Thèse, University of Paris (Paris, 1976).
[24]K. Menger, ”Botenproblem”, in: K. Menger, ed.,Ergebnisse eines mathematischen Kolloquiums (Wien 1930), Heft 2 (Leipzig, 1932) pp. 11–12.
[25]E. Netto,Lehrbuch der Combinatorik (Chelsea Publishing Company, New York). ·JFM 53.0073.09
[26]R.Z. Norman, ”On the convex polyhedra of the symmetric travelling salesman problem” (Abstract),Bulletin of the American Mathematical Society 61 (1955) 559.
[27]M.W. Padberg and S. Hong, ”On the symmetric travelling salesman problem: A computational study”, T.J. Watson Research Report, IBM Research (Yorktown Heights, 1977). ·Zbl 0388.90054
[28]M.W. Padberg and M.R. Rao, ”The travelling salesman problem and a class of polyhedra of diameter two”,Mathematical Programming 7 (1974) 32–45. ·Zbl 0318.90042 ·doi:10.1007/BF01585502
[29]J. Stoer and Ch. Witzgall,Convexity and optimization in finite dimensions I (Springer, Berlin 1970). ·Zbl 0203.52203
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.
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