9,265 research outputs found
A Characterization Theorem and An Algorithm for A Convex Hull Problem
Given and , testing if , the convex hull of , is a fundamental
problem in computational geometry and linear programming. First, we prove a
Euclidean {\it distance duality}, distinct from classical separation theorems
such as Farkas Lemma: lies in if and only if for each there exists a {\it pivot}, satisfying . Equivalently, if and only if there exists a
{\it witness}, whose Voronoi cell relative to contains
. A witness separates from and approximate to
within a factor of two. Next, we describe the {\it Triangle Algorithm}: given
, an {\it iterate}, , and , if
, it stops. Otherwise, if there exists a pivot
, it replace with and with the projection of onto the
line . Repeating this process, the algorithm terminates in arithmetic operations, where
is the {\it visibility factor}, a constant satisfying and
, over all iterates . Additionally,
(i) we prove a {\it strict distance duality} and a related minimax theorem,
resulting in more effective pivots; (ii) describe -time algorithms that may compute a witness or a good
approximate solution; (iii) prove {\it generalized distance duality} and
describe a corresponding generalized Triangle Algorithm; (iv) prove a {\it
sensitivity theorem} to analyze the complexity of solving LP feasibility via
the Triangle Algorithm. The Triangle Algorithm is practical and competitive
with the simplex method, sparse greedy approximation and first-order methods.Comment: 42 pages, 17 figures, 2 tables. This revision only corrects minor
typo
The distributions of functions related to parametric integer optimization
We consider the asymptotic distribution of the IP sparsity function, which
measures the minimal support of optimal IP solutions, and the IP to LP distance
function, which measures the distance between optimal IP and LP solutions. We
create a framework for studying the asymptotic distribution of general
functions related to integer optimization. There has been a significant amount
of research focused around the extreme values that these functions can attain,
however less is known about their typical values. Each of these functions is
defined for a fixed constraint matrix and objective vector while the right hand
sides are treated as input. We show that the typical values of these functions
are smaller than the known worst case bounds by providing a spectrum of
probability-like results that govern their overall asymptotic distributions.Comment: Accepted for journal publicatio
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
Symmetric confidence regions and confidence intervals for normal map formulations of stochastic variational inequalities
Stochastic variational inequalities (SVI) model a large class of equilibrium
problems subject to data uncertainty, and are closely related to stochastic
optimization problems. The SVI solution is usually estimated by a solution to a
sample average approximation (SAA) problem. This paper considers the normal map
formulation of an SVI, and proposes a method to build asymptotically exact
confidence regions and confidence intervals for the solution of the normal map
formulation, based on the asymptotic distribution of SAA solutions. The
confidence regions are single ellipsoids with high probability. We also discuss
the computation of simultaneous and individual confidence intervals
The intersection of two halfspaces has high threshold degree
The threshold degree of a Boolean function f:{0,1}^n->{-1,+1} is the least
degree of a real polynomial p such that f(x)=sgn p(x). We construct two
halfspaces on {0,1}^n whose intersection has threshold degree Theta(sqrt n), an
exponential improvement on previous lower bounds. This solves an open problem
due to Klivans (2002) and rules out the use of perceptron-based techniques for
PAC learning the intersection of two halfspaces, a central unresolved challenge
in computational learning. We also prove that the intersection of two majority
functions has threshold degree Omega(log n), which is tight and settles a
conjecture of O'Donnell and Servedio (2003).
Our proof consists of two parts. First, we show that for any nonconstant
Boolean functions f and g, the intersection f(x)^g(y) has threshold degree O(d)
if and only if ||f-F||_infty + ||g-G||_infty < 1 for some rational functions F,
G of degree O(d). Second, we settle the least degree required for approximating
a halfspace and a majority function to any given accuracy by rational
functions.
Our technique further allows us to make progress on Aaronson's challenge
(2008) and contribute strong direct product theorems for polynomial
representations of composed Boolean functions of the form F(f_1,...,f_n). In
particular, we give an improved lower bound on the approximate degree of the
AND-OR tree.Comment: Full version of the FOCS'09 pape
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