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On hardly linearly provable systems. (English) Zbl 0551.68042
A well-known theorem of Rabin yields a ’dimensional’ lower bound on the width of complete polynomial proofs of a system of linear algebraic inequalities. In this note we investigate a practically motivated class of systems where the same lower bound can be obtained on the width of ’almost all’ ’non-complete’ linear proofs. The proof of our result is based on the Helly Theorem.
MSC:
68Q25 Analysis of algorithms and problem complexity
15A39 Linear inequalities of matrices
90C05 Linear programming
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References:
[1] M. O. Rabin: Proving simultaneous positivity of linear forms. JCSS 6 : 639-650 (1972). · Zbl 0274.68022 · doi:10.1016/S0022-0000(72)80034-5
[2] Joseph Stoer, Christoph Witzgall: Convexity and Optimization in Finite Dimensions I. Springer-Verlag, Berlin-Heidelberg-New York, J 970. · Zbl 0203.52203
[3] Ky Fan: On systems of linear inequalities. in ’Linear Inequalities and Related Systems’ (H. W. Kuhn and A. W. Tucker. Princeton Univ. Press, 1956. · Zbl 0072.37602
[4] David G. Luenberger: Introduction to Linear and Nonlinear Programming. Addison-Wesley Publishing Company, 1973. · Zbl 0297.90044
[5] J. W. Jaromczyk: An extension of Rabin’s complete proof concept. MFCS 1981, Lecture Notes in Computer Science 118, Springer-Verlag 1981, 321 - 326. · Zbl 0471.68026
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