Abstract
This paper deals with a geometric construction of algebraic non-realizability proofs for certain oriented matroids. As main result we obtain an algorithm which generates a (bi-quadratic) final polynomial [3], [5] for any non-euclidean oriented matroid. Here we apply the results of Edmonds, Fukuda and Mandel [6], [7] concerning non-degenerate cycling of linear programs in non-euclidean oriented matroids.
Similar content being viewed by others
References
A. Björner. M. Las Vergnas, B. Sturmfels, N. White, andG. M. Ziegler:Oriented Matroids, Cambridge University Press, Encyclopedia of Mathematics and its Applications,46, 1993.
R. G. Bland: A combinatorial abstraction of linear programming,J. Combinatorial Theory B 23 (1977), 33–57.
J. Bokowski, andJ. Richter: On the finding of final polynomials,Europ. J. Combinatorics 11 (1990), 21–34.
J. Bokowski, andJ. Richter: On the classification of non-realizable oriented Matroids, Part II: Properties. Preprint, TH-Darmstadt (1990).
J. Bokowski, andB. Sturmfels:Computational Synthetic Geometry, Lecture Notes in Mathematics1355, Springer, Berlin, 1989.
K. Fukuda:Oriented matroid programming, Ph.D. Thesis, University of Waterloo, 1982.
A. Mandel:Topology of Oriented Matroids, Ph.D. Thesis, University of Waterloo, 1982.
J. Richter-Gebert:On the realizability problem of combinatorial geometries — decision methods, Dissertation, TH-Darmstadt, 1991.
P. W. Shor: Stretchability of Pseudolines is NP-Hard, in:Applied Geometry and Discrete Mathematics — The Victor Klee Festschrift (P. Gritzmann, B. Sturmfels, eds.), DIMACS Series in Discrete Mathematics and Theoretical Computer Science, Amer. Math. Soc., Providence, RI,4, 531–554.