3,876 research outputs found
Locating Two Transfer Points on a Network with a Trip Covering Criterion and Mixed Distances
In this paper we consider a set of origin-destination pairs in a mixed model in which a
network embedded in the plane represents an alternative high-speed transportation system,
and study a trip covering problem which consists on locating two points in the network
which maximize the number of covered pairs, that is, the number of pairs which use the
network by acceding and exiting through such points. To deal with the absence of convexity
of this mixed distance function we propose a decomposition method based on formulating
a collection of subproblems and solving each of them via discretization of the solution set.Ministerio de Educación, Ciencia e Innovación MTM2009-14243Ministerio de Economía y Competitividad MTM2012-37048Junta de Andalucía P09-TEP-5022Junta de Andalucía P10-FQM-584
TDMA is Optimal for All-unicast DoF Region of TIM if and only if Topology is Chordal Bipartite
The main result of this work is that an orthogonal access scheme such as TDMA
achieves the all-unicast degrees of freedom (DoF) region of the topological
interference management (TIM) problem if and only if the network topology graph
is chordal bipartite, i.e., every cycle that can contain a chord, does contain
a chord. The all-unicast DoF region includes the DoF region for any arbitrary
choice of a unicast message set, so e.g., the results of Maleki and Jafar on
the optimality of orthogonal access for the sum-DoF of one-dimensional convex
networks are recovered as a special case. The result is also established for
the corresponding topological representation of the index coding problem
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
- …