294,765 research outputs found
The complexity of approximating the matching polynomial in the complex plane
We study the problem of approximating the value of the matching polynomial on
graphs with edge parameter , where takes arbitrary values in
the complex plane.
When is a positive real, Jerrum and Sinclair showed that the problem
admits an FPRAS on general graphs. For general complex values of ,
Patel and Regts, building on methods developed by Barvinok, showed that the
problem admits an FPTAS on graphs of maximum degree as long as
is not a negative real number less than or equal to
. Our first main result completes the picture for the
approximability of the matching polynomial on bounded degree graphs. We show
that for all and all real less than ,
the problem of approximating the value of the matching polynomial on graphs of
maximum degree with edge parameter is #P-hard.
We then explore whether the maximum degree parameter can be replaced by the
connective constant. Sinclair et al. showed that for positive real it
is possible to approximate the value of the matching polynomial using a
correlation decay algorithm on graphs with bounded connective constant (and
potentially unbounded maximum degree). We first show that this result does not
extend in general in the complex plane; in particular, the problem is #P-hard
on graphs with bounded connective constant for a dense set of values
on the negative real axis. Nevertheless, we show that the result does extend
for any complex value that does not lie on the negative real axis. Our
analysis accounts for complex values of using geodesic distances in
the complex plane in the metric defined by an appropriate density function
Covariant un-reduction for curve matching
The process of un-reduction, a sort of reversal of reduction by the Lie group
symmetries of a variational problem, is explored in the setting of field
theories. This process is applied to the problem of curve matching in the
plane, when the curves depend on more than one independent variable. This
situation occurs in a variety of instances such as matching of surfaces or
comparison of evolution between species. A discussion of the appropriate
Lagrangian involved in the variational principle is given, as well as some
initial numerical investigations.Comment: Conference paper for MFCA201
Inter-plane satellite matching in dense LEO constellations
Dense constellations of Low Earth Orbit (LEO) small satellites are envisioned
to make extensive use of the inter-satellite link (ISL). Within the same
orbital plane, the inter-satellite distances are preserved and the links are
rather stable. In contrast, the relative motion between planes makes the
inter-plane ISL challenging. In a dense set-up, each spacecraft has several
satellites in its coverage volume, but the time duration of each of these links
is small and the maximum number of active connections is limited by the
hardware. We analyze the matching problem of connecting satellites using the
inter-plane ISL for unicast transmissions. We present and evaluate the
performance of two solutions to the matching problem with any number of orbital
planes and up to two transceivers: a heuristic solution with the aim of
minimizing the total cost; and a Markovian solution to maintain the on-going
connections as long as possible. The Markovian algorithm reduces the time
needed to solve the matching up to 1000x and 10x with respect to the optimal
solution and to the heuristic solution, respectively, without compromising the
total cost. Our model includes power adaptation and optimizes the network
energy consumption as the exemplary cost in the evaluations, but any other
QoS-oriented KPI can be used instead
A Constant Approximation for Colorful k-Center
In this paper, we consider the colorful k-center problem, which is a generalization of the well-known k-center problem. Here, we are given red and blue points in a metric space, and a coverage requirement for each color. The goal is to find the smallest radius rho, such that with k balls of radius rho, the desired number of points of each color can be covered. We obtain a constant approximation for this problem in the Euclidean plane. We obtain this result by combining a "pseudo-approximation" algorithm that works in any metric space, and an approximation algorithm that works for a special class of instances in the plane. The latter algorithm uses a novel connection to a certain matching problem in graphs
Partial-Matching and Hausdorff RMS Distance Under Translation: Combinatorics and Algorithms
We consider the RMS distance (sum of squared distances between pairs of
points) under translation between two point sets in the plane, in two different
setups. In the partial-matching setup, each point in the smaller set is matched
to a distinct point in the bigger set. Although the problem is not known to be
polynomial, we establish several structural properties of the underlying
subdivision of the plane and derive improved bounds on its complexity. These
results lead to the best known algorithm for finding a translation for which
the partial-matching RMS distance between the point sets is minimized. In
addition, we show how to compute a local minimum of the partial-matching RMS
distance under translation, in polynomial time. In the Hausdorff setup, each
point is paired to its nearest neighbor in the other set. We develop algorithms
for finding a local minimum of the Hausdorff RMS distance in nearly linear time
on the line, and in nearly quadratic time in the plane. These improve
substantially the worst-case behavior of the popular ICP heuristics for solving
this problem.Comment: 31 pages, 6 figure
- …