1,493 research outputs found
On the Combinatorial Complexity of Approximating Polytopes
Approximating convex bodies succinctly by convex polytopes is a fundamental
problem in discrete geometry. A convex body of diameter
is given in Euclidean -dimensional space, where is a constant. Given an
error parameter , the objective is to determine a polytope of
minimum combinatorial complexity whose Hausdorff distance from is at most
. By combinatorial complexity we mean the
total number of faces of all dimensions of the polytope. A well-known result by
Dudley implies that facets suffice, and a dual
result by Bronshteyn and Ivanov similarly bounds the number of vertices, but
neither result bounds the total combinatorial complexity. We show that there
exists an approximating polytope whose total combinatorial complexity is
, where conceals a
polylogarithmic factor in . This is a significant improvement
upon the best known bound, which is roughly .
Our result is based on a novel combination of both old and new ideas. First,
we employ Macbeath regions, a classical structure from the theory of convexity.
The construction of our approximating polytope employs a new stratified
placement of these regions. Second, in order to analyze the combinatorial
complexity of the approximating polytope, we present a tight analysis of a
width-based variant of B\'{a}r\'{a}ny and Larman's economical cap covering.
Finally, we use a deterministic adaptation of the witness-collector technique
(developed recently by Devillers et al.) in the context of our stratified
construction.Comment: In Proceedings of the 32nd International Symposium Computational
Geometry (SoCG 2016) and accepted to SoCG 2016 special issue of Discrete and
Computational Geometr
Lower bounds on the size of semidefinite programming relaxations
We introduce a method for proving lower bounds on the efficacy of
semidefinite programming (SDP) relaxations for combinatorial problems. In
particular, we show that the cut, TSP, and stable set polytopes on -vertex
graphs are not the linear image of the feasible region of any SDP (i.e., any
spectrahedron) of dimension less than , for some constant .
This result yields the first super-polynomial lower bounds on the semidefinite
extension complexity of any explicit family of polytopes.
Our results follow from a general technique for proving lower bounds on the
positive semidefinite rank of a matrix. To this end, we establish a close
connection between arbitrary SDPs and those arising from the sum-of-squares SDP
hierarchy. For approximating maximum constraint satisfaction problems, we prove
that SDPs of polynomial-size are equivalent in power to those arising from
degree- sum-of-squares relaxations. This result implies, for instance,
that no family of polynomial-size SDP relaxations can achieve better than a
7/8-approximation for MAX-3-SAT
On polyhedral approximations of the positive semidefinite cone
Let be the set of positive semidefinite matrices of trace
equal to one, also known as the set of density matrices. We prove two results
on the hardness of approximating with polytopes. First, we show that if and is an arbitrary matrix of trace equal to one, any
polytope such that must have
linear programming extension complexity at least where is a constant that depends on . Second, we show that any polytope
such that and such that the Gaussian width of is at most
twice the Gaussian width of must have extension complexity at least
. The main ingredient of our proofs is hypercontractivity of
the noise operator on the hypercube.Comment: 12 page
Separation-Sensitive Collision Detection for Convex Objects
We develop a class of new kinetic data structures for collision detection
between moving convex polytopes; the performance of these structures is
sensitive to the separation of the polytopes during their motion. For two
convex polygons in the plane, let be the maximum diameter of the polygons,
and let be the minimum distance between them during their motion. Our
separation certificate changes times when the relative motion of
the two polygons is a translation along a straight line or convex curve,
for translation along an algebraic trajectory, and for
algebraic rigid motion (translation and rotation). Each certificate update is
performed in time. Variants of these data structures are also
shown that exhibit \emph{hysteresis}---after a separation certificate fails,
the new certificate cannot fail again until the objects have moved by some
constant fraction of their current separation. We can then bound the number of
events by the combinatorial size of a certain cover of the motion path by
balls.Comment: 10 pages, 8 figures; to appear in Proc. 10th Annual ACM-SIAM
Symposium on Discrete Algorithms, 1999; see also
http://www.uiuc.edu/ph/www/jeffe/pubs/kollide.html ; v2 replaces submission
with camera-ready versio
Enumerating a subset of the integer points inside a Minkowski sum
AbstractSparse elimination exploits the structure of algebraic equations in order to obtain tighter bounds on the number of roots and better complexity in numerically approximating them. The model of sparsity is of combinatorial nature, thus leading to certain problems in general-dimensional convex geometry. This work addresses one such problem, namely the computation of a certain subset of integer points in the interior of integer convex polytopes. These polytopes are Minkowski sums, but avoiding their explicit construction is precisely one of the main features of the algorithm. Complexity bounds for our algorithm are derived under certain hypotheses, in terms of output-size and the sparsity parameters. A public domain implementation is described and its performance studied. Linear optimization lies at the inner loop of the algorithm, hence we analyze the structure of the linear programs and compare different implementations
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