61,997 research outputs found
Approximating the Maximum Overlap of Polygons under Translation
Let and be two simple polygons in the plane of total complexity ,
each of which can be decomposed into at most convex parts. We present an
-approximation algorithm, for finding the translation of ,
which maximizes its area of overlap with . Our algorithm runs in
time, where is a constant that depends only on and .
This suggest that for polygons that are "close" to being convex, the problem
can be solved (approximately), in near linear time
RRR: Rank-Regret Representative
Selecting the best items in a dataset is a common task in data exploration.
However, the concept of "best" lies in the eyes of the beholder: different
users may consider different attributes more important, and hence arrive at
different rankings. Nevertheless, one can remove "dominated" items and create a
"representative" subset of the data set, comprising the "best items" in it. A
Pareto-optimal representative is guaranteed to contain the best item of each
possible ranking, but it can be almost as big as the full data. Representative
can be found if we relax the requirement to include the best item for every
possible user, and instead just limit the users' "regret". Existing work
defines regret as the loss in score by limiting consideration to the
representative instead of the full data set, for any chosen ranking function.
However, the score is often not a meaningful number and users may not
understand its absolute value. Sometimes small ranges in score can include
large fractions of the data set. In contrast, users do understand the notion of
rank ordering. Therefore, alternatively, we consider the position of the items
in the ranked list for defining the regret and propose the {\em rank-regret
representative} as the minimal subset of the data containing at least one of
the top- of any possible ranking function. This problem is NP-complete. We
use the geometric interpretation of items to bound their ranks on ranges of
functions and to utilize combinatorial geometry notions for developing
effective and efficient approximation algorithms for the problem. Experiments
on real datasets demonstrate that we can efficiently find small subsets with
small rank-regrets
Vertex Sparsifiers: New Results from Old Techniques
Given a capacitated graph and a set of terminals ,
how should we produce a graph only on the terminals so that every
(multicommodity) flow between the terminals in could be supported in
with low congestion, and vice versa? (Such a graph is called a
flow-sparsifier for .) What if we want to be a "simple" graph? What if
we allow to be a convex combination of simple graphs?
Improving on results of Moitra [FOCS 2009] and Leighton and Moitra [STOC
2010], we give efficient algorithms for constructing: (a) a flow-sparsifier
that maintains congestion up to a factor of , where , (b) a convex combination of trees over the terminals that maintains
congestion up to a factor of , and (c) for a planar graph , a
convex combination of planar graphs that maintains congestion up to a constant
factor. This requires us to give a new algorithm for the 0-extension problem,
the first one in which the preimages of each terminal are connected in .
Moreover, this result extends to minor-closed families of graphs.
Our improved bounds immediately imply improved approximation guarantees for
several terminal-based cut and ordering problems.Comment: An extended abstract appears in the 13th International Workshop on
Approximation Algorithms for Combinatorial Optimization Problems (APPROX),
2010. Final version to appear in SIAM J. Computin
Probabilistic Matching of Planar Regions
We analyze a probabilistic algorithm for matching shapes modeled by planar
regions under translations and rigid motions (rotation and translation). Given
shapes and , the algorithm computes a transformation such that with
high probability the area of overlap of and is close to maximal. In
the case of polygons, we give a time bound that does not depend significantly
on the number of vertices
Minimizing the stabbing number of matchings, trees, and triangulations
The (axis-parallel) stabbing number of a given set of line segments is the
maximum number of segments that can be intersected by any one (axis-parallel)
line. This paper deals with finding perfect matchings, spanning trees, or
triangulations of minimum stabbing number for a given set of points. The
complexity of these problems has been a long-standing open question; in fact,
it is one of the original 30 outstanding open problems in computational
geometry on the list by Demaine, Mitchell, and O'Rourke. The answer we provide
is negative for a number of minimum stabbing problems by showing them NP-hard
by means of a general proof technique. It implies non-trivial lower bounds on
the approximability. On the positive side we propose a cut-based integer
programming formulation for minimizing the stabbing number of matchings and
spanning trees. We obtain lower bounds (in polynomial time) from the
corresponding linear programming relaxations, and show that an optimal
fractional solution always contains an edge of at least constant weight. This
result constitutes a crucial step towards a constant-factor approximation via
an iterated rounding scheme. In computational experiments we demonstrate that
our approach allows for actually solving problems with up to several hundred
points optimally or near-optimally.Comment: 25 pages, 12 figures, Latex. To appear in "Discrete and Computational
Geometry". Previous version (extended abstract) appears in SODA 2004, pp.
430-43
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