138 research outputs found
Proximity results and faster algorithms for Integer Programming using the Steinitz Lemma
We consider integer programming problems in standard form where , and . We show that such an integer program can be solved in time , where is an upper bound on each
absolute value of an entry in . This improves upon the longstanding best
bound of Papadimitriou (1981) of , where in addition,
the absolute values of the entries of also need to be bounded by .
Our result relies on a lemma of Steinitz that states that a set of vectors in
that is contained in the unit ball of a norm and that sum up to zero can
be ordered such that all partial sums are of norm bounded by . We also use
the Steinitz lemma to show that the -distance of an optimal integer and
fractional solution, also under the presence of upper bounds on the variables,
is bounded by . Here is again an
upper bound on the absolute values of the entries of . The novel strength of
our bound is that it is independent of . We provide evidence for the
significance of our bound by applying it to general knapsack problems where we
obtain structural and algorithmic results that improve upon the recent
literature.Comment: We achieve much milder dependence of the running time on the largest
entry in $b
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
Max-sum diversity via convex programming
Diversity maximization is an important concept in information retrieval,
computational geometry and operations research. Usually, it is a variant of the
following problem: Given a ground set, constraints, and a function
that measures diversity of a subset, the task is to select a feasible subset
such that is maximized. The \emph{sum-dispersion} function , which is the sum of the pairwise distances in , is
in this context a prominent diversification measure. The corresponding
diversity maximization is the \emph{max-sum} or \emph{sum-sum diversification}.
Many recent results deal with the design of constant-factor approximation
algorithms of diversification problems involving sum-dispersion function under
a matroid constraint. In this paper, we present a PTAS for the max-sum
diversification problem under a matroid constraint for distances
of \emph{negative type}. Distances of negative type are, for
example, metric distances stemming from the and norm, as well
as the cosine or spherical, or Jaccard distance which are popular similarity
metrics in web and image search
Diameter of Polyhedra: Limits of Abstraction
We investigate the diameter of a natural abstraction of the
-skeleton of polyhedra. Even if this abstraction is more general than
other abstractions previously studied in the literature,
known upper bounds on the diameter of polyhedra continue to hold
here. On the other hand, we show that this abstraction has its
limits by providing an almost quadratic lower bound
Parametric Integer Programming in Fixed Dimension
We consider the following problem: Given a rational matrix A \in \setQ^{m
\times n} and a rational polyhedron Q \subseteq\setR^{m+p}, decide if for
all vectors b \in \setR^m, for which there exists an integral z \in \setZ^p
such that , the system of linear inequalities has an
integral solution. We show that there exists an algorithm that solves this
problem in polynomial time if and are fixed. This extends a result of
Kannan (1990) who established such an algorithm for the case when, in addition
to and , the affine dimension of is fixed.
As an application of this result, we describe an algorithm to find the
maximum difference between the optimum values of an integer program \max \{c x
: A x \leq b, x \in \setZ^n \} and its linear programming relaxation over all
right-hand sides , for which the integer program is feasible. The algorithm
is polynomial if is fixed. This is an extension of a recent result of
Ho\c{s}ten and Sturmfels (2003) who presented such an algorithm for integer
programs in standard form.Comment: 23 pages, 3 figure
On largest volume simplices and sub-determinants
We show that the problem of finding the simplex of largest volume in the
convex hull of points in can be approximated with a factor
of in polynomial time. This improves upon the previously best
known approximation guarantee of by Khachiyan. On the other hand,
we show that there exists a constant such that this problem cannot be
approximated with a factor of , unless . % This improves over the
inapproximability that was previously known. Our hardness result holds
even if , in which case there exists a \bar c\,^{d}-approximation
algorithm that relies on recent sampling techniques, where is again a
constant. We show that similar results hold for the problem of finding the
largest absolute value of a subdeterminant of a matrix
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