Many optimization techniques, including several targeted specifically at embedded systems, depend on the ability to calculate the number of elements that satisfy certain conditions. If these conditions can be represented by linear constraints, then such problems are equivalent to counting the number of integer points in (possibly) parametric polytopes. It is well known that this parametric count can be represented by a set of Ehrhart polynomials. Previously, interpolation was used to obtain these polynomials, but this technique has several disadvantages. Its worst-case computation time for a single Ehrhart polynomial is exponential in the input size, even for fixed dimensions. The worst-case size of such an Ehrhart polynomial (measured in bits needed to represent the polynomial) is also exponential in the input size. Under certain conditions this technique even fails to produce a solution.

We give an effective uniform bound on the multigraded regularity of a subscheme of a smooth projective toric variety X with a given multigraded Hilbert polynomial. To establish this bound, we introduce a new combinatorial tool, called a Stanley filtration, for studying monomial ideals in the homogeneous coordinate ring of X. As a special case, we obtain a new proof of Gotzmann’s regularity theorem. We also discuss applications of this bound to the construction of multigraded Hilbert schemes. 1.

Many compiler optimization techniques depend on the ability to calculate the number of integer values that satisfy a given set of linear constraints. This count (the enumerator of a parametric polytope) is a function of the symbolic parameters that may appear in the constraints. In an extended problem (the "integer projection" of a parametric polytope), some of the variables that appear in the constraints may be existentially quantified and then the enumerated set corresponds to the projection of the integer points in a parametric polytope. This paper shows how to...

Abstract Many compiler optimization techniques depend on the ability to calculate the number of elements that satisfy certain conditions. If these conditions can be represented by linear constraints, then such problems are equivalent to counting the number of integer points in (possibly) parametric polytopes. It is well known that the enumerator of such a set can be represented by an explicit function consisting of a set of quasi-polynomials each associated with a chamber in the parameter space. Previously, interpolation was used to obtain these quasi-polynomials, but this technique has several disadvantages. Its worstcase computation time for a single quasi-polynomial is exponential in the input size, even for fixed dimensions. The worst-case size of such a quasi-polynomial (measured in bits needed to represent the quasi-polynomial) is also exponential in the input size. Under certain conditions this technique even fails to produce a solution. Our main contribution is a novel method for calculating the required quasipolynomials analytically. It extends an existing method, based on Barvinok’s decomposition,

Abstract. The Ehrhart polynomial of a convex lattice polytope counts integer points in integral dilates of the polytope. We present new linear inequalities satisfied by the coefficients of Ehrhart polynomials and relate them to known inequalities. We also investigate the roots of Ehrhart polynomials. We prove that for fixed d, there exists a bounded region of C containing all roots of Ehrhart polynomials of d-polytopes, and that all real roots of these polynomials lie in [−d, ⌊d/2⌋). In contrast, we prove that when the dimension d is not fixed the positive real roots can be arbitrarily large. We finish with an experimental investigation of the Ehrhart polynomials of cyclic polytopes and 0/1-polytopes. 1.

Abstract. We present a polynomial time algorithm to compute any fixed number of the highest coefficients of the Ehrhart quasi-polynomial of a rational simplex. Previously such algorithms were known for integer simplices and for rational polytopes of a fixed dimension. The algorithm is based on the formularelatingthekth coefficient of the Ehrhart quasi-polynomial of a rational polytope to volumes of sections of the polytope by affine lattice subspaces parallel to k-dimensional faces of the polytope. We discuss possible extensions andopenquestions. 1. Introduction and

We classify, according to their computational complexity, integer optimization problems whose constraints and objective functions are polynomials with integer coefficients and the number of variables is fixed. For the optimization of an integer polynomial over the lattice points of a convex polytope, we show an algorithm to compute lower and upper bounds for the optimal value. For polynomials that are non-negative over the polytope, these sequences of bounds lead to a fully polynomial-time approximation scheme for the optimization problem.

Many integrated circuit systems, particularly in the multimedia and telecom domains, are inherently data dominant. For this class of systems, a large part of the power consumption is due to the data storage and data transfer. Moreover, a significant part of the chip area is occupied by memory. The computation of the memory size is an important step in the system-level exploration, in the early stage of designing an optimized (for area and/or power) memory architecture for this class of systems. This paper presents a novel nonscalar approach for computing exactly the minimum size of the data memory for high-level procedural specifications of multidimensional signal processing applications. In contrast with all the previous works which are estimation methods, this approach can perform exact memory computations even for applications with numerous and complex array references, and also with large numbers of scalars.

Abstract – In real-time multimedia processing systems a very large part of the power consumption is due to the data storage and data transfer. Moreover, the area cost is often largely dominated by the memory modules. The computation of the memory size is an important step in the process of designing an optimized (for area and/or power) memory architecture for multimedia processing systems. This paper presents a novel non-scalar approach for computing exactly the memory size in real-time multimedia algorithms. This methodology uses both algebraic techniques specific to the data-flow analysis used in modern compilers, and also recent advances in the theory of integral polyhedra. In contrast with all the previous works which are only estimation methods, this approach performs exact memory computations even for applications with a large number of scalar signals. 1

We present an algorithm for counting the number of integer solutions to selected free variables of a Presburger formula. We represent the Presburger formula as a deterministic finite automaton (DFA) whose accepting paths encode the standard binary representations of satisfying free variable values. We count the number of accepting paths in such a DFA to obtain the number of solutions without enumerating the actual solutions. We demonstrate our algorithm on a suite of eight problems to show that it is universal, robust, fast, and scalable.