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59
Short rational generating functions for lattice point problems
 JOURNAL OF THE AMERICAN MATHEMATICAL SOCIETY
, 2003
"... We prove that for any fixed d the generating function of the projection of the set of integer points in a rational ddimensional polytope can be computed in polynomial time. As a corollary, we deduce that various interesting sets of lattice points, notably integer semigroups and (minimal) Hilbert ..."
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Cited by 40 (5 self)
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We prove that for any fixed d the generating function of the projection of the set of integer points in a rational ddimensional polytope can be computed in polynomial time. As a corollary, we deduce that various interesting sets of lattice points, notably integer semigroups and (minimal) Hilbert bases of rational cones, have short rational generating functions provided certain parameters (the dimension and the number of generators) are fixed. It follows then that many computational problems for such sets (for example, finding the number of positive integers not representable as a nonnegative integer combination of given coprime positive integers a1,..., ad) admit polynomial time algorithms. We also discuss a related problem of computing the Hilbert series of a ring generated by monomials.
Analytical Computation of Ehrhart Polynomials: Enabling more Compiler Analyses and Optimizations
 In CASES
, 2004
"... 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 ..."
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Cited by 29 (10 self)
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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 worstcase computation time for a single Ehrhart polynomial is exponential in the input size, even for fixed dimensions. The worstcase 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.
Counting integer points in parametric polytopes using Barvinok’s rational functions
 Algorithmica
, 2007
"... 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 ..."
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Cited by 18 (6 self)
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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 quasipolynomials each associated with a chamber in the parameter space. Previously, interpolation was used to obtain these quasipolynomials, but this technique has several disadvantages. Its worstcase computation time for a single quasipolynomial is exponential in the input size, even for fixed dimensions. The worstcase size of such a quasipolynomial (measured in bits needed to represent the quasipolynomial) 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,
Uniform bounds on multigraded regularity
 MR 2005g:14098 Zbl 1070.14006
, 2003
"... 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 homogen ..."
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Cited by 16 (2 self)
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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.
Experiences with Enumeration of Integer Projections of Parametric Polytopes
 in Compiler Construction: 14th Int. Conf
, 2005
"... 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 probl ..."
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Cited by 16 (5 self)
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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...
Coefficients and roots of Ehrhart polynomials
 In Integer points in polyhedra—geometry, number theory, algebra, optimization, volume 374 of Contemp. Math
, 2005
"... 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 ..."
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Cited by 13 (3 self)
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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 dpolytopes, 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 0/1polytopes. 1.
Computing the Ehrhart quasipolynomial of a rational simplex
 Math. Comp
, 2005
"... Abstract. We present a polynomial time algorithm to compute any fixed number of the highest coefficients of the Ehrhart quasipolynomial 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 f ..."
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Cited by 10 (1 self)
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Abstract. We present a polynomial time algorithm to compute any fixed number of the highest coefficients of the Ehrhart quasipolynomial 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 quasipolynomial of a rational polytope to volumes of sections of the polytope by affine lattice subspaces parallel to kdimensional faces of the polytope. We discuss possible extensions andopenquestions. 1. Introduction and
Integer polynomial optimization in fixed dimension
 MATHEMATICS OF OPERATIONS RESEARCH
, 2006
"... 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 ..."
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Cited by 10 (4 self)
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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 nonnegative over the polytope, these sequences of bounds lead to a fully polynomialtime approximation scheme for the optimization problem.
Computation of storage requirements for multidimensional signal processing applications
 IEEE TRANS. ON VLSI SYSTEMS
, 2007
"... 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 c ..."
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Cited by 9 (7 self)
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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 systemlevel 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 highlevel 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.
On the computation of ClebschGordan coefficients and the dilation effect, Experiment
 Math
"... We investigate the problem of computing tensor product multiplicities for complex semisimple Lie algebras. Even though computing these numbers is #Phard in general, we show that if the rank of the Lie algebra is assumed fixed, then there is a polynomial time algorithm, based on counting the lattice ..."
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Cited by 9 (0 self)
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We investigate the problem of computing tensor product multiplicities for complex semisimple Lie algebras. Even though computing these numbers is #Phard in general, we show that if the rank of the Lie algebra is assumed fixed, then there is a polynomial time algorithm, based on counting the lattice points in polytopes. In fact, for Lie algebras of type Ar, there is an algorithm, based on the ellipsoid algorithm, to decide when the coefficients are nonzero in polynomial time for arbitrary rank. Our experiments show that the lattice point algorithm is superior in practice to the standard techniques for computing multiplicities when the weights have large entries but small rank. Using an implementation of this algorithm, we provide experimental evidence for conjectured generalizations of the saturation property of Littlewood–Richardson coefficients. One of these conjectures seems to be valid for types Bn, Cn, and Dn. 1