Computation and Manipulation of Enumerators of Integer Projections of (original) (raw)
Related papers
Approximations of polytope enumerators using linear expansions
Several scientific problems are represented as sets of linear (or affine) con-straints over a set of variables and symbolic constants. When solutions of inter-est are integers, the number of such integer solutions is generally a meaningful information. Ehrhart polynomials are functions of the symbolic constants that count these solutions. Unfortunately, they have a complex mathematical struc-ture (resembling polynomials, hence the name), making it hard for other tools to manipulate them. Furthermore, their use may imply exponential computational complexity. This paper presents two contributions towards the useability of Ehrhart polynomials, by showing how to compute the following polynomial functions: an approximation and an upper (and a lower) bound of an Ehrhart polynomial. The computational complexity of this polynomial is less than or equal to that of * The original version of this report was submitted to the HiPEAC 2005 conference, under the title Approximating Ehrhart Polynomi...
Effective lattice point counting in rational convex polytopes
Journal of Symbolic Computation, 2004
This paper discusses algorithms and software for the enumeration of all lattice points inside a rational convex polytope: we describe LattE, a computer package for lattice point enumeration which contains the first implementation of A. Barvinok's algorithm (Math. Oper. Res. 19 (1994)
Journal of the Operations Research Society of Japan
An interesting combinatorial (enumeration) problem arises in the initial phase of the polyhedral homotopy continuation method for computing a11 so1utions of a polynomial equation system in complex variables. It is formulated as a problem of finding all solutions of a specially structured system of linear inequalities with a certain additional combinatorial condition. This paper presents a computational method for the problem fully utilizing the duality theory and the simplex method for linear programs] and report numerical results on a single cpu implementation and a parallel cpu implementation of the method.
ee. byu. edu/~ wilde/pubs. html, 1997
Optimizing parallel compilers need to be able to analyze nested loop programs with parametric a ne loop bounds, in order to derive e cient parallel programs. The iteration spaces of nested loop programs can be modeled by polyhedra and systems of linear constraints. Using this model, important program analyses such as computing the number of ops executed by a loop, computing the number of memory locations or cache lines touched by a loop, and computing the amount of processor to processor communication needed during the execution of a loop| all reduce to the same mathematical problem: nding the formula for number of integer solutions to a system of parameterized linear constraints, as a function of the parameters.
Valid integer polytope (VIP) penalties for branch-and-bound enumeration
2000
We introduce new penalties, called valid integer polytope (VIP) penalties, that tighten the bound of an integer-linear program during branch-and-bound enumeration. Early commercial codes for branch and bound commonly employed penalties developed from the dual simplicial lower bound on the cost of restricting fractional integer variables to proximate integral values. VIP penalties extend and tighten these for ubiquitous k-pack, k-partition, and k-cover constraints. In real-world problems, VIP penalties occasionally tighten the bound by more than an order of magnitude, but they usually o er small bound improvement. Their ease of implementation, speed of execution, and occasional, overwhelming success make them an attractive addition during branch-and-bound enumeration. : S 0 1 6 7 -6 3 7 7 ( 9 9 ) 0 0 0 7 2 -3
Memory optimization by counting points in integer transformations of parametric polytopes
Proceedings of the 2006 international conference on Compilers, architecture and synthesis for embedded systems - CASES '06, 2006
Memory size reduction and memory accesses optimization are crucial issues for embedded systems. In the context of affine programs, these two challenges are classically tackled by array linearization, cache access optimization and memory size computation. Their formalization in the polyhedral model reduce to solving the following problem: count the number of solutions of a Presburger formula.
Effective lattice point counting in rational convex
2004
This paper discusses algorithms and software for the enumeration of all lattice points inside a rational convex polytope: we describe LattE, a computer package for lattice point enumeration which contains the first implementation of A. Barvinok's algorithm (Math. Oper. Res. 19 (1994) 769). We report on computational experiments with multiway contingency tables, knapsack type problems, rational polygons, and flow polytopes. We prove that these kinds of symbolic-algebraic ideas surpass the traditional branch-and-bound enumeration and in some instances LattE is the only software capable of counting. Using LattE, we have also computed new formulas of Ehrhart (quasi-)polynomials for interesting families of polytopes (hypersimplices, truncated cubes, etc). We end with a survey of other "algebraic-analytic" algorithms, including a "homogeneous" variation of Barvinok's algorithm which is very fast when the number of facet-defining inequalities is much smaller compared to the number of vertices.
Integer Programming, Lattices, and Results in Fixed Dimension
Handbooks in Operations Research and Management Science, 2005
We review and describe several results regarding integer programming problems in fixed dimension. First, we describe various lattice basis reduction algorithms that are used as auxiliary algorithms when solving integer feasibility and optimization problems. Next, we review three algorithms for solving the integer feasibility problem. These algorithms are based on the idea of branching on lattice hyperplanes, and their running time is polynomial in fixed dimension. We also briefly describe an algorithm, based on a different principle, to count integer points in an integer polytope. We then turn the attention to integer optimization. Again, we describe three algorithms: binary search, a linear algorithm for a fixed number of constraints, and a randomized algorithm for a varying number of constraints. The topic of the next part of our chapter is how to use lattice basis reduction in problem reformulation. Finally, we review cutting plane results when the dimension is fixed.
On lattice point counting in Δ-modular polyhedra
arXiv (Cornell University), 2020
Let a polyhedron P be defined by one of the following ways: (i) P = {x ∈ R n : Ax ≤ b}, where A ∈ Z (n+k)×n , b ∈ Z (n+k) and rank A = n, (ii) P = {x ∈ R n + : Ax = b}, where A ∈ Z k×n , b ∈ Z k and rank A = k, and let all rank order minors of A be bounded by ∆ in absolute values. We show that the short rational generating function for the power series m∈P ∩Z n x m can be computed with the arithmetical complexity O T SNF (d) • d k • d log 2 ∆ , where k and ∆ are fixed, d = dim P , and T SNF (m) is the complexity of computing the Smith Normal Form for m × m integer matrices. In particular, d = n, for the case (i), and d = n − k, for the case (ii). The simplest examples of polyhedra that meet the conditions (i) or (ii) are the simplices, the subset sum polytope and the knapsack or multidimensional knapsack polytopes. Previously, the existence of a polynomial time algorithm in varying dimension for the considered class of problems was unknown already for simplicies (k = 1). We apply these results to parametric polytopes and show that the step polynomial representation of the function c P (y) = |P y ∩ Z n |, where P y is a parametric polytope, whose structure is close to the cases (i) or (ii), can be computed in polynomial time even if the dimension of P y is not fixed. As