769 research outputs found
A QPTAS for the Base of the Number of Triangulations of a Planar Point Set
The number of triangulations of a planar n point set is known to be ,
where the base lies between and The fastest known algorithm
for counting triangulations of a planar n point set runs in time.
The fastest known arbitrarily close approximation algorithm for the base of the
number of triangulations of a planar n point set runs in time subexponential in
We present the first quasi-polynomial approximation scheme for the base of
the number of triangulations of a planar point set
Approximation Schemes for Partitioning: Convex Decomposition and Surface Approximation
We revisit two NP-hard geometric partitioning problems - convex decomposition
and surface approximation. Building on recent developments in geometric
separators, we present quasi-polynomial time algorithms for these problems with
improved approximation guarantees.Comment: 21 pages, 6 figure
Space-time discontinuous Galerkin approximation of acoustic waves with point singularities
We develop a convergence theory of space-time discretizations for the linear,
2nd-order wave equation in polygonal domains ,
possibly occupied by piecewise homogeneous media with different propagation
speeds. Building on an unconditionally stable space-time DG formulation
developed in [Moiola, Perugia 2018], we (a) prove optimal convergence rates for
the space-time scheme with local isotropic corner mesh refinement on the
spatial domain, and (b) demonstrate numerically optimal convergence rates of a
suitable \emph{sparse} space-time version of the DG scheme. The latter scheme
is based on the so-called \emph{combination formula}, in conjunction with a
family of anisotropic space-time DG-discretizations. It results in
optimal-order convergent schemes, also in domains with corners, with a number
of degrees of freedom that scales essentially like the DG solution of one
stationary elliptic problem in on the finest spatial grid. Numerical
experiments for both smooth and singular solutions support convergence rate
optimality on spatially refined meshes of the full and sparse space-time DG
schemes.Comment: 38 pages, 8 figure
Space-time discontinuous Galerkin approximation of acoustic waves with point singularities
We develop a convergence theory of space-time discretizations for the linear,
2nd-order wave equation in polygonal domains ,
possibly occupied by piecewise homogeneous media with different propagation
speeds. Building on an unconditionally stable space-time DG formulation
developed in [Moiola, Perugia 2018], we (a) prove optimal convergence rates for
the space-time scheme with local isotropic corner mesh refinement on the
spatial domain, and (b) demonstrate numerically optimal convergence rates of a
suitable \emph{sparse} space-time version of the DG scheme. The latter scheme
is based on the so-called \emph{combination formula}, in conjunction with a
family of anisotropic space-time DG-discretizations. It results in
optimal-order convergent schemes, also in domains with corners, with a number
of degrees of freedom that scales essentially like the DG solution of one
stationary elliptic problem in on the finest spatial grid. Numerical
experiments for both smooth and singular solutions support convergence rate
optimality on spatially refined meshes of the full and sparse space-time DG
schemes.Comment: 38 pages, 8 figure
Implicitization of curves and (hyper)surfaces using predicted support
We reduce implicitization of rational planar parametric curves and (hyper)surfaces to linear algebra, by interpolating the coefficients of the implicit equation.
For predicting the implicit support, we focus on methods that exploit input and output structure in the sense of sparse (or toric) elimination theory, namely by computing the Newton polytope of the implicit polynomial, via sparse resultant theory.
Our algorithm works even in the presence of base points but, in this case, the implicit equation shall be obtained as a factor of the produced polynomial.
We implement our methods on Maple, and some on Matlab as well, and study their numerical stability and efficiency on several classes of curves and surfaces.
We apply our approach to approximate implicitization,
and quantify the accuracy of the approximate output,
which turns out to be satisfactory on all tested examples; we also relate our measures to Hausdorff distance.
In building a square or rectangular matrix, an important issue is (over)sampling the given curve or surface: we conclude that unitary complexes offer the best tradeoff between speed and accuracy when numerical methods are employed, namely SVD, whereas for exact kernel computation random integers is the method of choice.
We compare our prototype to existing software and find that it is rather competitive
Optimality program in segment and string graphs
Planar graphs are known to allow subexponential algorithms running in time
or for most of the paradigmatic
problems, while the brute-force time is very likely to be
asymptotically best on general graphs. Intrigued by an algorithm packing curves
in by Fox and Pach [SODA'11], we investigate which
problems have subexponential algorithms on the intersection graphs of curves
(string graphs) or segments (segment intersection graphs) and which problems
have no such algorithms under the ETH (Exponential Time Hypothesis). Among our
results, we show that, quite surprisingly, 3-Coloring can also be solved in
time on string graphs while an algorithm running
in time for 4-Coloring even on axis-parallel segments (of unbounded
length) would disprove the ETH. For 4-Coloring of unit segments, we show a
weaker ETH lower bound of which exploits the celebrated
Erd\H{o}s-Szekeres theorem. The subexponential running time also carries over
to Min Feedback Vertex Set but not to Min Dominating Set and Min Independent
Dominating Set.Comment: 19 pages, 15 figure
QPTAS for Weighted Geometric Set Cover on Pseudodisks and Halfspaces
International audienceWeighted geometric set-cover problems arise naturally in several geometric and non-geometric settings (e.g. the breakthrough of Bansal and Pruhs (FOCS 2010) reduces a wide class of machine scheduling problems to weighted geometric set-cover). More than two decades of research has succeeded in settling the (1 + status for most geometric set-cover problems, except for some basic scenarios which are still lacking. One is that of weighted disks in the plane for which, after a series of papers, Varadarajan (STOC 2010) presented a clever quasi-sampling technique, which together with improvements by Chan et al. (SODA 2012), yielded an O(1)-approximation algorithm. Even for the unweighted case, a PTAS for a fundamental class of objects called pseudodisks (which includes half-spaces, disks, unit-height rectangles, translates of convex sets etc.) is currently unknown. Another fundamental case is weighted halfspaces in R 3 , for which a PTAS is currently lacking. In this paper, we present a QPTAS for all of these remaining problems. Our results are based on the separator framework of Adamaszek and Wiese (FOCS 2013, SODA 2014), who recently obtained a QPTAS for weighted independent set of polygonal regions. This rules out the possibility that these problems are APX-hard, assuming NP DTIME(2 polylog(n)). Together with the recent work of Chan and Grant (CGTA 2014), this settles the APX-hardness status for all natural geometric set-cover problems
Approximation Algorithms for Polynomial-Expansion and Low-Density Graphs
We study the family of intersection graphs of low density objects in low
dimensional Euclidean space. This family is quite general, and includes planar
graphs. We prove that such graphs have small separators. Next, we present
efficient -approximation algorithms for these graphs, for
Independent Set, Set Cover, and Dominating Set problems, among others. We also
prove corresponding hardness of approximation for some of these optimization
problems, providing a characterization of their intractability in terms of
density
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