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77
Effective lattice point counting in rational convex polytopes
 JOURNAL OF SYMBOLIC COMPUTATION
, 2003
"... 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 [8]. We report on computational experiments ..."
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Cited by 65 (11 self)
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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 [8]. We report on computational experiments with multiway contingency tables, knapsack type problems, rational polygons, and flow polytopes. We prove that this kind of symbolicalgebraic ideas surpasses the traditional branchandbound 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 "algebraicanalytic" algorithms, including a "polar" variation of Barvinok's algorithm which is very fast when the number of facetdefining inequalities is much smaller compared to the number of vertices.
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.
Sparsity in Sums of Squares of Polynomials
, 2004
"... Representation of a given nonnegative multivariate polynomial in terms of a sum of squares of polynomials has become an essential subject in recent developments of sums of squares optimization and SDP (semidefinite programming) relaxation of polynomial optimization problems. We discuss effective met ..."
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Cited by 24 (14 self)
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Representation of a given nonnegative multivariate polynomial in terms of a sum of squares of polynomials has become an essential subject in recent developments of sums of squares optimization and SDP (semidefinite programming) relaxation of polynomial optimization problems. We discuss effective methods to obtain a simpler representation of a “sparse” polynomial as a sum of squares of sparse polynomials by eliminating redundancy.
Aspectoriented inlined reference monitors
 In Proc. ACM Workshop on Prog. Languages and Analysis for Security (PLAS
, 2008
"... Abstract. A technique for elegantly expressing Inlined Reference Monitor (IRM) certification as modelchecking is presented and implemented. Inlined Reference Monitors (IRM’s) enforce software security policies by inlining dynamic security guards into untrusted binary code. Certifying IRM systems ..."
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Cited by 22 (12 self)
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Abstract. A technique for elegantly expressing Inlined Reference Monitor (IRM) certification as modelchecking is presented and implemented. Inlined Reference Monitors (IRM’s) enforce software security policies by inlining dynamic security guards into untrusted binary code. Certifying IRM systems provide strong formal guarantees for such systems by verifying that the instrumented code produced by the IRM system satisfies the original policy. Expressing this certification step as modelchecking allows wellestablished modelchecking technologies to be applied to this often difficult certification task. The technique is demonstrated through the enforcement and certification of a URL antiredirection policy for ActionScript web applets. 1
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,
Euler Maclaurin with remainder for a simple integral polytope
"... Abstract. We give an Euler Maclaurin formula with remainder for the sum of the values of a smooth function on the integral points in a simple integral polytope. This formula is proved by elementary methods. 1. ..."
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Cited by 16 (4 self)
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Abstract. We give an Euler Maclaurin formula with remainder for the sum of the values of a smooth function on the integral points in a simple integral polytope. This formula is proved by elementary methods. 1.
Algebraic unimodular counting
 MATH. PROGRAM
, 2003
"... We study algebraic algorithms for expressing the number of nonnegative integer solutions to a unimodular system of linear equations as a function of the right hand side. Our methods include Todd classes of toric varieties via Gröbner bases, and rational generating functions as in Barvinok's algori ..."
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Cited by 15 (4 self)
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We study algebraic algorithms for expressing the number of nonnegative integer solutions to a unimodular system of linear equations as a function of the right hand side. Our methods include Todd classes of toric varieties via Gröbner bases, and rational generating functions as in Barvinok's algorithm. We report polyhedral and computational results for two special cases: counting contingency tables and Kostant's partition function.
Random polynomials with prescribed Newton polytope
 J. Amer. Math. Soc
, 2002
"... 1.1. Statement of results 50 ..."