27,319 research outputs found
A primal Barvinok algorithm based on irrational decompositions
We introduce variants of Barvinok's algorithm for counting lattice points in
polyhedra. The new algorithms are based on irrational signed decomposition in
the primal space and the construction of rational generating functions for
cones with low index. We give computational results that show that the new
algorithms are faster than the existing algorithms by a large factor.Comment: v3: New all-primal algorithm. v4: Extended introduction, updated
computational results. To appear in SIAM Journal on Discrete Mathematic
Software for Exact Integration of Polynomials over Polyhedra
We are interested in the fast computation of the exact value of integrals of
polynomial functions over convex polyhedra. We present speed ups and extensions
of the algorithms presented in previous work. We present the new software
implementation and provide benchmark computations. The computation of integrals
of polynomials over polyhedral regions has many applications; here we
demonstrate our algorithmic tools solving a challenge from combinatorial voting
theory.Comment: Major updat
Spectrum optimization in multi-user multi-carrier systems with iterative convex and nonconvex approximation methods
Several practical multi-user multi-carrier communication systems are
characterized by a multi-carrier interference channel system model where the
interference is treated as noise. For these systems, spectrum optimization is a
promising means to mitigate interference. This however corresponds to a
challenging nonconvex optimization problem. Existing iterative convex
approximation (ICA) methods consist in solving a series of improving convex
approximations and are typically implemented in a per-user iterative approach.
However they do not take this typical iterative implementation into account in
their design. This paper proposes a novel class of iterative approximation
methods that focuses explicitly on the per-user iterative implementation, which
allows to relax the problem significantly, dropping joint convexity and even
convexity requirements for the approximations. A systematic design framework is
proposed to construct instances of this novel class, where several new
iterative approximation methods are developed with improved per-user convex and
nonconvex approximations that are both tighter and simpler to solve (in
closed-form). As a result, these novel methods display a much faster
convergence speed and require a significantly lower computational cost.
Furthermore, a majority of the proposed methods can tackle the issue of getting
stuck in bad locally optimal solutions, and hence improve solution quality
compared to existing ICA methods.Comment: 33 pages, 7 figures. This work has been submitted for possible
publicatio
A Euclid style algorithm for MacMahon's partition analysis
Solutions to a linear Diophantine system, or lattice points in a rational
convex polytope, are important concepts in algebraic combinatorics and
computational geometry. The enumeration problem is fundamental and has been
well studied, because it has many applications in various fields of
mathematics. In algebraic combinatorics, MacMahon's partition analysis has
become a general approach for linear Diophantine system related problems. Many
algorithms have been developed, but "bottlenecks" always arise when dealing
with complex problems. While in computational geometry, Barvinok's important
result asserts the existence of a polynomial time algorithm when the dimension
is fixed. However, the implementation by the LattE package of De Loera et. al.
does not perform well in many situations. By combining excellent ideas in the
two fields, we generalize Barvinok's result by giving a polynomial time
algorithm for MacMahon's partition analysis in a suitable condition. We also
present an elementary Euclid style algorithm, which might not be polynomial but
is easy to implement and performs well. As applications, we contribute the
generating series for magic squares of order 6.Comment: 28 pages, some modification, add the link to the Maple package
CTEucli
Complexity Estimates for Two Uncoupling Algorithms
Uncoupling algorithms transform a linear differential system of first order
into one or several scalar differential equations. We examine two approaches to
uncoupling: the cyclic-vector method (CVM) and the
Danilevski-Barkatou-Z\"urcher algorithm (DBZ). We give tight size bounds on the
scalar equations produced by CVM, and design a fast variant of CVM whose
complexity is quasi-optimal with respect to the output size. We exhibit a
strong structural link between CVM and DBZ enabling to show that, in the
generic case, DBZ has polynomial complexity and that it produces a single
equation, strongly related to the output of CVM. We prove that algorithm CVM is
faster than DBZ by almost two orders of magnitude, and provide experimental
results that validate the theoretical complexity analyses.Comment: To appear in Proceedings of ISSAC'13 (21/01/2013
How to Integrate a Polynomial over a Simplex
This paper settles the computational complexity of the problem of integrating
a polynomial function f over a rational simplex. We prove that the problem is
NP-hard for arbitrary polynomials via a generalization of a theorem of Motzkin
and Straus. On the other hand, if the polynomial depends only on a fixed number
of variables, while its degree and the dimension of the simplex are allowed to
vary, we prove that integration can be done in polynomial time. As a
consequence, for polynomials of fixed total degree, there is a polynomial time
algorithm as well. We conclude the article with extensions to other polytopes,
discussion of other available methods and experimental results.Comment: Tables added with new experimental results. References adde
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