Abstract
We shall now exploit the ellipsoid method (the central-cut and the shallow-cut version) described in Chapter 3. In Sections 4.2, 4.3, and 4.4 we study the algorithmic relations between problems (2.1.10),..., (2.1.14), and we will prove that — under certain assumptions — these problems are equivalent with respect to polynomial time solvability. Section 4.5 serves to show that these assumptions cannot be weakened. In Section 4.6 we investigate various other basic questions of convex geometry from an algorithmic point of view and prove algorithmic analogues of some well-known theorems. Finally, in Section 4.7 we discuss to what extent algorithmic properties of convex bodies are preserved when they are subjected to operations like sum, intersection etc.
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© 1988 Springer-Verlag Berlin Heidelberg
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Grötschel, M., Lovász, L., Schrijver, A. (1988). Algorithms for Convex Bodies. In: Geometric Algorithms and Combinatorial Optimization. Algorithms and Combinatorics, vol 2. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-97881-4_5
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DOI: https://doi.org/10.1007/978-3-642-97881-4_5
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-642-97883-8
Online ISBN: 978-3-642-97881-4
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