2,000 research outputs found
Planar Support for Non-piercing Regions and Applications
Given a hypergraph H=(X,S), a planar support for H is a planar graph G with vertex set X, such that for each hyperedge S in S, the sub-graph of G induced by the vertices in S is connected. Planar supports for hypergraphs have found several algorithmic applications, including several packing and covering problems, hypergraph coloring, and in hypergraph visualization.
The main result proved in this paper is the following: given two families of regions R and B in the plane, each of which consists of connected, non-piercing regions, the intersection hypergraph H_R(B) = (B, {B_r}_{r in R}), where B_r = {b in B: b cap r != empty set} has a planar support. Further, such a planar support can be computed in time polynomial in |R|, |B|, and the number of vertices in the arrangement of the regions in R cup B. Special cases of this result include the setting where either the family R, or the family B is a set of points.
Our result unifies and generalizes several previous results on planar supports, PTASs for packing and covering problems on non-piercing regions in the plane and coloring of intersection hypergraph of non-piercing regions
Approximation Algorithm for Line Segment Coverage for Wireless Sensor Network
The coverage problem in wireless sensor networks deals with the problem of
covering a region or parts of it with sensors. In this paper, we address the
problem of covering a set of line segments in sensor networks. A line segment `
is said to be covered if it intersects the sensing regions of at least one
sensor distributed in that region. We show that the problem of finding the
minimum number of sensors needed to cover each member in a given set of line
segments in a rectangular area is NP-hard. Next, we propose a constant factor
approximation algorithm for the problem of covering a set of axis-parallel line
segments. We also show that a PTAS exists for this problem.Comment: 16 pages, 5 figures
On Hypergraph Supports
Let be a hypergraph. A support is a graph
on such that for each , the subgraph of induced on the
elements in is connected. In this paper, we consider hypergraphs defined on
a host graph. Given a graph , with ,
and a collection of connected subgraphs of , a primal support
is a graph on such that for each , the
induced subgraph on vertices is connected. A \emph{dual support} is a graph on
s.t. for each , the induced subgraph
is connected, where . We present
sufficient conditions on the host graph and hyperedges so that the resulting
support comes from a restricted family.
We primarily study two classes of graphs: If the host graph has genus
and the hypergraphs satisfy a topological condition of being
\emph{cross-free}, then there is a primal and a dual support of genus at most
. If the host graph has treewidth and the hyperedges satisfy a
combinatorial condition of being \emph{non-piercing}, then there exist primal
and dual supports of treewidth . We show that this exponential blow-up
is sometimes necessary. As an intermediate case, we also study the case when
the host graph is outerplanar. Finally, we show applications of our results to
packing and covering, and coloring problems on geometric hypergraphs
Improved Approximation Algorithm for Set Multicover with Non-Piercing Regions
In the Set Multicover problem, we are given a set system (X,?), where X is a finite ground set, and ? is a collection of subsets of X. Each element x ? X has a non-negative demand d(x). The goal is to pick a smallest cardinality sub-collection ?\u27 of ? such that each point is covered by at least d(x) sets from ?\u27. In this paper, we study the set multicover problem for set systems defined by points and non-piercing regions in the plane, which includes disks, pseudodisks, k-admissible regions, squares, unit height rectangles, homothets of convex sets, upward paths on a tree, etc.
We give a polynomial time (2+?)-approximation algorithm for the set multicover problem (P, ?), where P is a set of points with demands, and ? is a set of non-piercing regions, as well as for the set multicover problem (?, P), where ? is a set of pseudodisks with demands, and P is a set of points in the plane, which is the hitting set problem with demands
The ?-t-Net Problem
We study a natural generalization of the classical ?-net problem (Haussler - Welzl 1987), which we call the ?-t-net problem: Given a hypergraph on n vertices and parameters t and ? ? t/n, find a minimum-sized family S of t-element subsets of vertices such that each hyperedge of size at least ? n contains a set in S. When t=1, this corresponds to the ?-net problem.
We prove that any sufficiently large hypergraph with VC-dimension d admits an ?-t-net of size O((1+log t)d/? log 1/?). For some families of geometrically-defined hypergraphs (such as the dual hypergraph of regions with linear union complexity), we prove the existence of O(1/?)-sized ?-t-nets.
We also present an explicit construction of ?-t-nets (including ?-nets) for hypergraphs with bounded VC-dimension. In comparison to previous constructions for the special case of ?-nets (i.e., for t=1), it does not rely on advanced derandomization techniques. To this end we introduce a variant of the notion of VC-dimension which is of independent interest
Flow and air-entrainment around partially submerged vertical cylinders
In this study, a partially submerged vertical cylinder is moved at constant
velocity through water, which is initially at rest. During the motion, the wake
behind the cylinder induces free-surface deformation. Eleven cylinders, with
diameters from to 16 cm, were tested at two different conditions: (i)
constant immersed height and (ii) constant . The range of translation
velocities and diameters are in the regime of turbulent wake with experiments
carried out for and , where and are
the Reynolds and Froude numbers based on . The focus here is on drag force
measurements and relatively strong free-surface deformation up to
air-entrainment. Specifically, two modes of air-entraiment have been uncovered:
(i) in the cavity along the cylinder wall and (ii) in the wake of the cylinder.
A scaling for the critical velocity for air-entrainment in the cavity has been
observed in agreement with a simple model. Furthermore, for , the drag
force varies linearly with
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