Polyhedra Circuits and Their Applications

To better compute the volume and count the lattice points in geometric objects, we propose polyhedral circuits. Each polyhedral circuit characterizes a geometric region in Rd . They can be applied to represent a rich class of geometric objects, which include all polyhedra and the union of a finite number of polyhedron. They can be also used to approximate a large class of d-dimensional manifolds in Rd . Barvinok [3] developed polynomial time algorithms to compute the volume of a rational polyhedron, and to count the number of lattice points in a rational polyhedron in Rd with a fixed dimensional number d. Let d be a fixed dimensional number, TV(d,n) be polynomial time in n to compute the volume of a rational polyhedron, TL(d,n) be polynomial time in n to count the number of lattice points in a rational polyhedron, where n is the total number of linear inequalities from input polyhedra, and TI(d,n) be polynomial time in n to solve integer linear programming problem with n be the total number of input linear inequalities. We develop algorithms to count the number of lattice points in geometric region determined by a polyhedral circuit in O(nd⋅rd(n)⋅TV(d,n)) time and to compute the volume of geometric region determined by a polyhedral circuit in O(n⋅rd(n)⋅TI(d,n)+rd(n)TL(d,n)) time, where rd(n) is the maximum number of atomic regions that n hyperplanes partition Rd . The applications to continuous polyhedra maximum coverage problem, polyhedra maximum lattice coverage problem, polyhedra (1−β) -lattice set cover problem, and (1−β) -continuous polyhedra set cover problem are discussed. We also show the NP-hardness of the geometric version of maximum coverage problem and set cover problem when each set is represented as union of polyhedra

Barvinok's Rational Functions: Algorithms and Applications to Optimization, Statistics, and Algebra

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 \emph{toric ideal} $I_A$ of an integral $n \times d$ matrix $A$. We encode the binomials belonging to the toric ideal $I_A$ associated with $A$ using Barvinok's rational functions. If we fix $d$ and $n$, this representation allows us to compute a universal Gr\"obner basis and the reduced Gr\"obner basis of the ideal $I_A$, with respect to any term order, in polynomial time. We 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 second part of the thesis is experimental and consists mainly of the software package {\tt LattE}, the first implementation of Barvinok's algorithm. We report on experiments with families of well-known rational polytopes: multiway contingency tables, knapsack type problems, and rational polygons. We also developed a new algorithm, {\em 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 $5\times 5$ magic squares and the generating function for the number of $3\times 3 \times 3 \times 3$ magic cubes as rational functions.Comment: Thesi