References
Alevras, D., G. Cramer and M. Padberg (1994). Double Description Algorithm for Calculating a Basis of the Lineality Space and the Extreme Rays of a Polyhedral Cone. Software available via ftp at elib.zib-berlin.de, directory pub/mathprog/polyth/dda.
Alevras, D. (1995). personal communication.
Applegate, D., R.E. Bixby, V. Chvátal and W. Cook (1994). Finding cuts in the TSP (A preliminary report). Research Report, Rice University.
Avis, D. (1993). AC Implementation of the Reverse Search Vertex Enumeration Algorithm. Software available via ftp at mutt.cs.mcgiil.ca, directory pub/C.
Avis, D. and K. Fukuda (1992). A Pivoting Algorithm for Convex Hulls and Vertex Enumeration of Arrangements and Polyhedra.Discrete and Computational Geometry 8, 295–313.
Balas, E., S. Ceria and C. Cornuéjols (1993). A Lift-and-Project Cutting Plane Algorithm for Mixed 0–1 Programs.Mathematical Programming 58, 295–324.
Borchers, B. (1992). Improved Branch and Bound Algorithms for Integer Programming. Troy, NY.
Boyd, S.C. and W.H. Cunningham (1991). Small Traveling Salesman Polytopes.Mathematics of Operations Research 16, 259–271.
Bruger, E. (1956). Über homogene lineare Ungleichungssysteme.Zeitschrift für Angewandte Mathematik und Mechanik 36, 135–139.
Chernikov, S.N. (1971).Lineare Ungleichungen. Berlin: Deutscher Verlag der Wissenschaften.
Chernikova, N.V. (1965). Algorithm for Finding a General Formula for the Nonnegative Solutions of a System of Linear Inequalities.U.S.S.R. Computational Mathematics and Mathematical Physics 5, 228–233.
Christof, T. (1991).Ein Verfahren zur Transformation zwischen Polyederdarstellungen. Diploma Thesis, Universität Augsburg.
Christof, T., M. Jünger and G. Reinelt (1991). A Complete Description of the Traveling Salesman Polytope on 8 Nodes.Operations Research Letters 10, 497–500.
Christof, T. and G. Reinelt (1995). Parallel Cutting Plane Generation for the TSP, in P. Fritzson and L. Finmo, eds.Parallel Programming and Applications. IOS Press.
Dantzig, G.B. and B.C. Eaves (1973). Fourier-Motzkin Elimination and its Dual.J. Combinatorial Theory Ser. A 14, 228–297.
Dantzig, G.B., R. Fulkerson and S.M. Johnson (1954). Solution of a Large-Scale Traveling Salesman Problem.Operations Research 2, 393–410.
Duffin, R.J. (1974). On Fourier's Analysis of Linear Inequality Systems. Mathematical Programming Study 1.
Edmonds, J. (1965). Maximum Matching and a Polyhedron with (0,1) Vertices.Journal Res. Nat. Bur. Stand. 69B, 125–130.
Edelsbrunner, H. (1987).Algorithms in Combinatorial Geometry. Berlin: Springer.
Euler, R. and H. Le Verge (1992). A Complete and Irredundant Linear Description of the Asymmetric Traveling Salesman Polytope on 6 Nodes. Research Report, University of Brest.
Fishburn, P.C. (1990). Binary Probabilities Induced by Rankings. SIAM Journal of Discrete Mathematics3, 478–488.
Fishburn, P.C. (1991). Induced Binary Probabilities and the Linear Ordering Polytope: A Status Report. Report AT&T Bell Laboratories.
Fukuda, K. (1993). cdd.c.:C Implementation of the Double Desription Method for Computing All Vertices and Extremal Rays of a Convex Polyhedron Given by a System of Linear Inequalities. Technical Report, DMA-EPFL Lausanne, software available via ftp at ifor13.ethz.ch, directory pub/fukuda/cdd.
Fukuda, K. and A. Prodon (1995). Double Description Method Revisited. Report, ETHZ Zürich.
Galperin, A.M. (1976). The General Solution of a Finite Set of Linear Inequalities.Math. Oper. Res,1, 185–196.
Gomory, R.E. (1963). An Algorithm for Integer Solutions to Linear Programs, in R.L. Graves and P. Wolfe, eds.Recent Advances in Mathematical Programming. New York: McGraw Hill.
Grötschel, M. and O. Holland (1985). Solving Matching Problems with Linear Programming.Mathematical Programming 33, 243–259.
Grötschel, M., M. Jünger and G. Reinelt (1984). A Cutting Plane Algorithm for the Linear Ordering Problem.Operations Research 32, 1195–1220.
Grötschel, M., M. Jünger and G. Reinelt (1985). Facets of the Linear Ordering Polytope.Mathematical Programming 33, 43–60.
Grötschel, M., L. Lovász and A. Schrijver (1988).Geometric Algorithms and Combinatorial Optimization. Berlin: Springer.
Grötschel, M. and M.W. Padberg (1985). Polyhedral theory, in E.L. Lawler, J.K. Lenstra, A.H.G. Rinnooy Kan and D.B. Shmoys, eds.The Traveling Salesman Problem. Chichester: Wiley & Sons.
Jünger, M., G. Reinelt and G. Rinaldi (1995). The Traveling Salesman Problem, in M. Ball, T. Magnanti, C.L. Monma and G.L. Nemhauser, eds.Handbooks in Operations Research and Management Sciences: Networks. North-Holland.
Jünger, M., G. Reinelt and S. Thienel (1994). Optimal and Provably Good Solutions for the Symmetric Traveling Salesman Problem.Zeitschrift für Operations Research 40, 183–217.
Jünger, M., G. Reinelt and S. Thienel (1995). Practical Problem Solving with Cutting Plane Algorithms in Combinatorial Optimization, in W. Cook, L. Lovász and P. Seymour, eds.DIMACS Series in Discrete Mathematics and Theoretical Computer Science, Volume 20: Combinatorial Optimization. American Mathematical Society.
Jünger, M. and P. Störmer (1995). Solving Large-Scale Traveling Salesman Problems with parallel Branch-and-Cut. Technical Report No. 95.191, Universität zu Köln.
Kirkpatrick, C.D., Gelatt Jr. and M.P. Vecchi (1983). Optimization by Simulated Annealing.Science 222, 671–680.
Leung, J. and J. Lee (1994). “More Facets from Fences for Linear Ordering and Acyclic Subgraph Polytopes”. Discrete Appl. Math.50, No. 2, 185–200.
Mathies, T.H. and D.S. Rubin (1980). A Survey and Comparison of Methods for Finding all Vertices of Convex Polyhedral Sets.Mathematics of Operations Research 5, 167–184.
Mitchell, J.E. (1994). Interior Point Algorithms for Integer Programming. Rensselaer Polytechnic Institute, TR 215.
Motzkin, T.S., H. Raiffa, G.L. Thompson and R.M. Thrall (1953). The Double Description Method, in H.W. Kuhn and A.W. Tucker, eds.Contributions to the Theory of Games (Vol. 2). Princeton University Press, Princeton, N. J.
Müller, R. (1992). On the Transitive Acyclic Subdigraph Polytope. Report No. 337, TU Berlin.
Naddef, D. and G. Rinaldi (1992). The Crown Inequalities for the Symmetric Traveling Salesman Polytope.Mathematics of Operations Research 17, 308–326.
Naddef D. and G. Rinaldi (1993). The Graphical Relaxation: A New Framework for the Symmetric Traveling Salesman Polytope.Mathematical Programming 58, 53–87.
Padberg, M. (1995).Linear Optimization and Extensions, Berlin: Springer.
Padberg, M.W. and G. Rinaldi (1987). Optimization of a 532 City Symmetric Traveling Salesman Problem by Branch and Cut.Operations Research Letters 6, 1–7.
Padberg, M.W. and G. Rinaldi (1990). Facet Identification for the Symmetric Traveling Salesman Polytope.Mathematical Programming 47, 219–257.
Padberg, M.W. and G. Rinaldi (1991). A Branch and Cut Algorithm for the Resolution of Large-scale Symmetric Traveling Salesman Problems.SIAM Review 33, 60–100.
Pulleyblank, W.R. (1989). Polyhedral Combinatorics, in G.L. Nemhauser, A.H.G. Rinnoy Kan and M.J. Todd, eds.Handbooks in Operations Research and Management Sciences: Optimization. North-Holland.
Queyranne, M. and Y. Wang (1993). Hamiltonian Path and Symmetric Traveling Salesman Polytopes.Mathematical Programming 58, 89–110.
Reinelt, G. (1985).The Linear Ordering Problem: Algorithms and Applications. Research and exposition in mathematics,8, Berlin: Heldermann.
Reinelt, G. (1991). TSPLIB—A Traveling Salesman Problem Library.ORSA Journal on Computing 3, 376–384. (WWW-access: http:// www.iwr.uni-heidelberg.de/iwr/comopt/soft/TSPLIB/TSPLIB.htlm).
Reinelt, G. (1993). A Note on Small Linear Ordering Polytopes.Discrete & Computational Geometry 10, 67–78.
Rubin, D. (1975). Vertex Generation and Cardinality Constrained Linear Programs.Operations Research 23, 555–565.
Schrijver, A. (1986).Theory of Linear and Integer Programming. Chichester: John Wiley & Sons.
Suck, R. (1991). Geometric and Combinatorial Properties of the Polytope of Binary Choice Probabilities. Report Universität Osnabrück.
Thienel, S. (1995).ABACUS—A Branch and Cut System. PhD Thesis, Universität zu Köln.
Wilde, D. (1993). A Library for Doing Polyhedral Operations. Report 785, IRISA, Rennes, France, Software available via ftp at ftp.irisa.fr, directory: local/API.
Verge Le, H. (1992). A Note on Chernikova's Algorithm. Report 785, IRISA, Rennes, France.
Zigler, G. (1995). Lectures on Polytopes, Berlin: Springer.
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Christof, T., Reinelt, G. Combinatorial optimization and small polytopes. Top 4, 1–53 (1996). https://doi.org/10.1007/BF02568602
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DOI: https://doi.org/10.1007/BF02568602