FPTAS for optimizing polynomials over the mixed-integer points of polytopes in fixed dimension |
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Authors: | Jesús A. De Loera Raymond Hemmecke Matthias Köppe Robert Weismantel |
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Affiliation: | 1. Department of Mathematics, University of California, Davis, CA, 95616, USA 2. Otto-von-Guericke-Universit?t Magdeburg, FMA/IMO, Universit?tsplatz 2, 39106, Magdeburg, Germany
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Abstract: | We show the existence of a fully polynomial-time approximation scheme (FPTAS) for the problem of maximizing a non-negative polynomial over mixed-integer sets in convex polytopes, when the number of variables is fixed. Moreover, using a weaker notion of approximation, we show the existence of a fully polynomial-time approximation scheme for the problem of maximizing or minimizing an arbitrary polynomial over mixed-integer sets in convex polytopes, when the number of variables is fixed. A conference version of this article, containing a part of the results presented here, appeared in Proceedings of the 17th Annual ACM-SIAM Symposium on Discrete Algorithms, Miami, FL, January 22–24, 2006, pp. 743–748. The first author gratefully acknowledges support from NSF grant DMS-0608785, a 2003 UC-Davis Chancellor’s fellow award, the Alexander von Humboldt foundation, and IMO Magdeburg. The remaining authors were supported by the European TMR network ADONET 504438. |
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Keywords: | Mixed-integer nonlinear programming Integer programming in fixed dimension Computational complexity Approximation algorithms FPTAS |
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