10 research outputs found
Computing parametric rational generating functions with a primal Barvinok algorithm
Computations with Barvinok's short rational generating functions are
traditionally being performed in the dual space, to avoid the combinatorial
complexity of inclusion--exclusion formulas for the intersecting proper faces
of cones. We prove that, on the level of indicator functions of polyhedra,
there is no need for using inclusion--exclusion formulas to account for
boundary effects: All linear identities in the space of indicator functions can
be purely expressed using half-open variants of the full-dimensional polyhedra
in the identity. This gives rise to a practically efficient, parametric
Barvinok algorithm in the primal space.Comment: 16 pages, 1 figure; v2: Minor corrections, new example and summary of
algorithm; submitted to journa
Minimizing the number of lattice points in a translated polygon
The parametric lattice-point counting problem is as follows: Given an integer
matrix , compute an explicit formula parameterized by that determines the number of integer points in the polyhedron . In the last decade, this counting problem has received
considerable attention in the literature. Several variants of Barvinok's
algorithm have been shown to solve this problem in polynomial time if the
number of columns of is fixed.
Central to our investigation is the following question: Can one also
efficiently determine a parameter such that the number of integer points in
is minimized? Here, the parameter can be chosen
from a given polyhedron .
Our main result is a proof that finding such a minimizing parameter is
-hard, even in dimension 2 and even if the parametrization reflects a
translation of a 2-dimensional convex polygon. This result is established via a
relationship of this problem to arithmetic progressions and simultaneous
Diophantine approximation.
On the positive side we show that in dimension 2 there exists a polynomial
time algorithm for each fixed that either determines a minimizing
translation or asserts that any translation contains at most times
the minimal number of lattice points
Complexity of short Presburger arithmetic
We study complexity of short sentences in Presburger arithmetic (Short-PA).
Here by "short" we mean sentences with a bounded number of variables,
quantifiers, inequalities and Boolean operations; the input consists only of
the integers involved in the inequalities. We prove that assuming Kannan's
partition can be found in polynomial time, the satisfiability of Short-PA
sentences can be decided in polynomial time. Furthermore, under the same
assumption, we show that the numbers of satisfying assignments of short
Presburger sentences can also be computed in polynomial time
On lattice point counting in -modular polyhedra
Let a polyhedron be defined by one of the following ways:
(i) , where ,
and ;
(ii) , where , and .
And let all rank order minors of be bounded by in absolute
values. We show that the short rational generating function for the power
series can be computed with the
arithmetic complexity where and are fixed, , and
is the complexity to compute the Smith Normal Form for integer matrix. In particular, for the case (i) and for
the case (ii).
The simplest examples of polyhedra that meet conditions (i) or (ii) are the
simplicies, the subset sum polytope and the knapsack or multidimensional
knapsack polytopes.
We apply these results to parametric polytopes, and show that the step
polynomial representation of the function , where
is parametric polytope, can be computed by a polynomial time even in
varying dimension if has a close structure to the cases (i) or (ii). As
another consequence, we show that the coefficients of the Ehrhart
quasi-polynomial can be computed by a polynomial time algorithm for fixed and
Three Ehrhart Quasi-polynomials
Let be a semi-rational parametric polytope, where
is a real multi-parameter. We study intermediate sums of
polynomial functions on , where we
integrate over the intersections of with the subspaces parallel to a
fixed rational subspace through all lattice points, and sum the integrals.
The purely discrete sum is of course a particular case (), so counts the integer points in the parametric polytopes.
The chambers are the open conical subsets of such that the shape of
does not change when runs over a chamber. We first prove that on
every chamber of , is given by a quasi-polynomial function
of . A key point of our paper is an analysis of the interplay between
two notions of degree on quasi-polynomials: the usual polynomial degree and a
filtration, called the local degree.
Then, for a fixed , we consider a particular linear combination of
such intermediate weighted sums, which was introduced by Barvinok in order to
compute efficiently the highest coefficients of the Ehrhart
quasi-polynomial which gives the number of points of a dilated rational
polytope. Thus, for each chamber, we obtain a quasi-polynomial function of ,
which we call Barvinok's patched quasi-polynomial (at codimension level ).
Finally, for each chamber, we introduce a new quasi-polynomial function of
, the cone-by-cone patched quasi-polynomial (at codimension level ),
defined in a refined way by linear combinations of intermediate generating
functions for the cones at vertices of .
We prove that both patched quasi-polynomials agree with the discrete weighted
sum in the terms corresponding to the highest
polynomial degrees.Comment: 41 pages, 13 figures; v2: changes to introduction, new graphics; v3:
add more detailed references, move example to introduction; v4: fix
reference
Decomposition Methods for Nonlinear Optimization and Data Mining
We focus on two central themes in this dissertation. The first one is on
decomposing polytopes and polynomials in ways that allow us to perform
nonlinear optimization. We start off by explaining important results on
decomposing a polytope into special polyhedra. We use these decompositions and
develop methods for computing a special class of integrals exactly. Namely, we
are interested in computing the exact value of integrals of polynomial
functions over convex polyhedra. We present prior work and new extensions of
the integration algorithms. Every integration method we present requires that
the polynomial has a special form. We explore two special polynomial
decomposition algorithms that are useful for integrating polynomial functions.
Both polynomial decompositions have strengths and weaknesses, and we experiment
with how to practically use them.
After developing practical algorithms and efficient software tools for
integrating a polynomial over a polytope, we focus on the problem of maximizing
a polynomial function over the continuous domain of a polytope. This
maximization problem is NP-hard, but we develop approximation methods that run
in polynomial time when the dimension is fixed. Moreover, our algorithm for
approximating the maximum of a polynomial over a polytope is related to
integrating the polynomial over the polytope. We show how the integration
methods can be used for optimization.
The second central topic in this dissertation is on problems in data science.
We first consider a heuristic for mixed-integer linear optimization. We show
how many practical mixed-integer linear have a special substructure containing
set partition constraints. We then describe a nice data structure for finding
feasible zero-one integer solutions to systems of set partition constraints.
Finally, we end with an applied project using data science methods in medical
research.Comment: PHD Thesis of Brandon Dutr