4,363 research outputs found
Analysis of Farthest Point Sampling for Approximating Geodesics in a Graph
A standard way to approximate the distance between any two vertices and
on a mesh is to compute, in the associated graph, a shortest path from
to that goes through one of sources, which are well-chosen vertices.
Precomputing the distance between each of the sources to all vertices of
the graph yields an efficient computation of approximate distances between any
two vertices. One standard method for choosing sources, which has been used
extensively and successfully for isometry-invariant surface processing, is the
so-called Farthest Point Sampling (FPS), which starts with a random vertex as
the first source, and iteratively selects the farthest vertex from the already
selected sources.
In this paper, we analyze the stretch factor of
approximate geodesics computed using FPS, which is the maximum, over all pairs
of distinct vertices, of their approximated distance over their geodesic
distance in the graph. We show that can be bounded in terms
of the minimal value of the stretch factor obtained using an
optimal placement of sources as , where is the ratio of the lengths of
the longest and the shortest edges of the graph. This provides some evidence
explaining why farthest point sampling has been used successfully for
isometry-invariant shape processing. Furthermore, we show that it is
NP-complete to find sources that minimize the stretch factor.Comment: 13 pages, 4 figure
Total Curvature of Graphs after Milnor and Euler
We define a new notion of total curvature, called net total curvature, for
finite graphs embedded in Rn, and investigate its properties. Two guiding
principles are given by Milnor's way of measuring the local crookedness of a
Jordan curve via a Crofton-type formula, and by considering the double cover of
a given graph as an Eulerian circuit. The strength of combining these ideas in
defining the curvature functional is (1) it allows us to interpret the
singular/non-eulidean behavior at the vertices of the graph as a superposition
of vertices of a 1-dimensional manifold, and thus (2) one can compute the total
curvature for a wide range of graphs by contrasting local and global properties
of the graph utilizing the integral geometric representation of the curvature.
A collection of results on upper/lower bounds of the total curvature on
isotopy/homeomorphism classes of embeddings is presented, which in turn
demonstrates the effectiveness of net total curvature as a new functional
measuring complexity of spatial graphs in differential-geometric terms.Comment: Most of the results contained in "Total curvature and isotopy of
graphs in ."(arXiv:0806.0406) have been incorporated into the current
articl
Approximation Algorithms for the Capacitated Domination Problem
We consider the {\em Capacitated Domination} problem, which models a
service-requirement assignment scenario and is also a generalization of the
well-known {\em Dominating Set} problem. In this problem, given a graph with
three parameters defined on each vertex, namely cost, capacity, and demand, we
want to find an assignment of demands to vertices of least cost such that the
demand of each vertex is satisfied subject to the capacity constraint of each
vertex providing the service. In terms of polynomial time approximations, we
present logarithmic approximation algorithms with respect to different demand
assignment models for this problem on general graphs, which also establishes
the corresponding approximation results to the well-known approximations of the
traditional {\em Dominating Set} problem. Together with our previous work, this
closes the problem of generally approximating the optimal solution. On the
other hand, from the perspective of parameterization, we prove that this
problem is {\it W[1]}-hard when parameterized by a structure of the graph
called treewidth. Based on this hardness result, we present exact
fixed-parameter tractable algorithms when parameterized by treewidth and
maximum capacity of the vertices. This algorithm is further extended to obtain
pseudo-polynomial time approximation schemes for planar graphs
Almost-Smooth Histograms and Sliding-Window Graph Algorithms
We study algorithms for the sliding-window model, an important variant of the
data-stream model, in which the goal is to compute some function of a
fixed-length suffix of the stream. We extend the smooth-histogram framework of
Braverman and Ostrovsky (FOCS 2007) to almost-smooth functions, which includes
all subadditive functions. Specifically, we show that if a subadditive function
can be -approximated in the insertion-only streaming model, then
it can be -approximated also in the sliding-window model with
space complexity larger by factor , where is the
window size.
We demonstrate how our framework yields new approximation algorithms with
relatively little effort for a variety of problems that do not admit the
smooth-histogram technique. For example, in the frequency-vector model, a
symmetric norm is subadditive and thus we obtain a sliding-window
-approximation algorithm for it. Another example is for streaming
matrices, where we derive a new sliding-window
-approximation algorithm for Schatten -norm. We then
consider graph streams and show that many graph problems are subadditive,
including maximum submodular matching, minimum vertex-cover, and maximum
-cover, thereby deriving sliding-window -approximation algorithms for
them almost for free (using known insertion-only algorithms). Finally, we
design for every an artificial function, based on the
maximum-matching size, whose almost-smoothness parameter is exactly
Non-Uniform Robust Network Design in Planar Graphs
Robust optimization is concerned with constructing solutions that remain
feasible also when a limited number of resources is removed from the solution.
Most studies of robust combinatorial optimization to date made the assumption
that every resource is equally vulnerable, and that the set of scenarios is
implicitly given by a single budget constraint. This paper studies a robustness
model of a different kind. We focus on \textbf{bulk-robustness}, a model
recently introduced~\cite{bulk} for addressing the need to model non-uniform
failure patterns in systems.
We significantly extend the techniques used in~\cite{bulk} to design
approximation algorithm for bulk-robust network design problems in planar
graphs. Our techniques use an augmentation framework, combined with linear
programming (LP) rounding that depends on a planar embedding of the input
graph. A connection to cut covering problems and the dominating set problem in
circle graphs is established. Our methods use few of the specifics of
bulk-robust optimization, hence it is conceivable that they can be adapted to
solve other robust network design problems.Comment: 17 pages, 2 figure
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