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• W1669761370 abstract "The main theme of this dissertation is the study of the lattice points in a rational convex polyhedron and their encoding in terms of Barvinok's short rational functions. The first part of this thesis looks into theoretical applications of these rational functions to Optimization, Statistics, and Computational Algebra. The main theorem on Chapter 2 concerns the computation of the toric ideal IA of an integral n x d matrix A. We encode the binomials belonging to the toric ideal IA associated with A using Barvinok's rational functions. If we fix d and n, this representation allows us to compute a universal Grobner basis and the reduced Grobner basis of the ideal IA, with respect to any term order, in polynomial time. We also derive a polynomial time algorithm for normal form computations which replaces in this new encoding the usual reductions of the division algorithm. Chapter 3 presents three ways to use Barvinok's rational functions to solve Integer Programs: The (A, b, c)-test set algorithm , the Barvinok's binary search algorithm, and the digging algorithm. The second part of the thesis is experimental and consists mainly of the software package LattE, the first implementation of Barvinok's algorithm to compute short rational functions which encode the lattice points in a rational convex polytope. In Chapter 4 we report on experiments with families of well-known rational polytopes: multiway contingency tables, knapsack type problems, and rational polygons, and we present formulas for the Ehrhart quasi-polynomials of several hypersimplices and truncations of cubes. We also developed a new algorithm, the homogenized Barvinok's algorithm to compute the generating function for a rational polytope. We showed that it runs in polynomial time in fixed dimension. With the homogenized Barvinok's algorithm, we obtained new combinatorial formulas: the generating function for the number of 5 x 5 magic squares and the generating function for the number of 3 x 3 x 3 x 3 magic cubes as rational functions." @default.
• W1669761370 created "2016-06-24" @default.
• W1669761370 creator A5055755657 @default.
• W1669761370 creator A5079338715 @default.
• W1669761370 date "2004-01-01" @default.
• W1669761370 modified "2023-10-03" @default.
• W1669761370 title "Barvinok's rational functions: algorithms and applications to optimization, statistics, and algebra" @default.
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