34,046 research outputs found
Spanning trees short or small
We study the problem of finding small trees. Classical network design
problems are considered with the additional constraint that only a specified
number of nodes are required to be connected in the solution. A
prototypical example is the MST problem in which we require a tree of
minimum weight spanning at least nodes in an edge-weighted graph. We show
that the MST problem is NP-hard even for points in the Euclidean plane. We
provide approximation algorithms with performance ratio for the
general edge-weighted case and for the case of points in the
plane. Polynomial-time exact solutions are also presented for the class of
decomposable graphs which includes trees, series-parallel graphs, and bounded
bandwidth graphs, and for points on the boundary of a convex region in the
Euclidean plane. We also investigate the problem of finding short trees, and
more generally, that of finding networks with minimum diameter. A simple
technique is used to provide a polynomial-time solution for finding -trees
of minimum diameter. We identify easy and hard problems arising in finding
short networks using a framework due to T. C. Hu.Comment: 27 page
Euclidean Greedy Drawings of Trees
Greedy embedding (or drawing) is a simple and efficient strategy to route
messages in wireless sensor networks. For each source-destination pair of nodes
s, t in a greedy embedding there is always a neighbor u of s that is closer to
t according to some distance metric. The existence of greedy embeddings in the
Euclidean plane R^2 is known for certain graph classes such as 3-connected
planar graphs. We completely characterize the trees that admit a greedy
embedding in R^2. This answers a question by Angelini et al. (Graph Drawing
2009) and is a further step in characterizing the graphs that admit Euclidean
greedy embeddings.Comment: Expanded version of a paper to appear in the 21st European Symposium
on Algorithms (ESA 2013). 24 pages, 20 figure
Spanners for Geometric Intersection Graphs
Efficient algorithms are presented for constructing spanners in geometric
intersection graphs. For a unit ball graph in R^k, a (1+\epsilon)-spanner is
obtained using efficient partitioning of the space into hypercubes and solving
bichromatic closest pair problems. The spanner construction has almost
equivalent complexity to the construction of Euclidean minimum spanning trees.
The results are extended to arbitrary ball graphs with a sub-quadratic running
time.
For unit ball graphs, the spanners have a small separator decomposition which
can be used to obtain efficient algorithms for approximating proximity problems
like diameter and distance queries. The results on compressed quadtrees,
geometric graph separators, and diameter approximation might be of independent
interest.Comment: 16 pages, 5 figures, Late
Additive functions on trees
The motivation of considering positive additive functions on
trees was the characterization of extended Dynkin graphs (see I. Reiten [R])
and the application of additive functions in the representation theory (see
H. Lenzing and I. Reiten [LR] and T. H¨ubner [H]).
We consider graphs equipped with functions of integer values, i.e.valued
graphs (see also [DR]). Methods are given for the construction of additive
functions on valued trees (in particular on Euclidean graphs) and for the
characterization of their structure. We introduce the concept of almost additive
functions, which are additive on each vertex of a graph except for one
(called exceptional vertex). On (valued) trees (with fixed exceptional vertex)
the almost additive functions are unique up to rational multiples. For valued
trees a necessary and sufficient condition is given for the existence of positive
almost additive functions
Localization game on geometric and planar graphs
The main topic of this paper is motivated by a localization problem in
cellular networks. Given a graph we want to localize a walking agent by
checking his distance to as few vertices as possible. The model we introduce is
based on a pursuit graph game that resembles the famous Cops and Robbers game.
It can be considered as a game theoretic variant of the \emph{metric dimension}
of a graph. We provide upper bounds on the related graph invariant ,
defined as the least number of cops needed to localize the robber on a graph
, for several classes of graphs (trees, bipartite graphs, etc). Our main
result is that, surprisingly, there exists planar graphs of treewidth and
unbounded . On a positive side, we prove that is bounded
by the pathwidth of . We then show that the algorithmic problem of
determining is NP-hard in graphs with diameter at most .
Finally, we show that at most one cop can approximate (arbitrary close) the
location of the robber in the Euclidean plane
The sum of edge lengths in random linear arrangements
Spatial networks are networks where nodes are located in a space equipped
with a metric. Typically, the space is two-dimensional and until recently and
traditionally, the metric that was usually considered was the Euclidean
distance. In spatial networks, the cost of a link depends on the edge length,
i.e. the distance between the nodes that define the edge. Hypothesizing that
there is pressure to reduce the length of the edges of a network requires a
null model, e.g., a random layout of the vertices of the network. Here we
investigate the properties of the distribution of the sum of edge lengths in
random linear arrangement of vertices, that has many applications in different
fields. A random linear arrangement consists of an ordering of the elements of
the nodes of a network being all possible orderings equally likely. The
distance between two vertices is one plus the number of intermediate vertices
in the ordering. Compact formulae for the 1st and 2nd moments about zero as
well as the variance of the sum of edge lengths are obtained for arbitrary
graphs and trees. We also analyze the evolution of that variance in Erdos-Renyi
graphs and its scaling in uniformly random trees. Various developments and
applications for future research are suggested
Gap-ETH-Tight Approximation Schemes for Red-Green-Blue Separation and Bicolored Noncrossing Euclidean Travelling Salesman Tours
In this paper, we study problems of connecting classes of points via
noncrossing structures. Given a set of colored terminal points, we want to find
a graph for each color that connects all terminals of its color with the
restriction that no two graphs cross each other. We consider these problems
both on the Euclidean plane and in planar graphs.
On the algorithmic side, we give a Gap-ETH-tight EPTAS for the two-colored
traveling salesman problem as well as for the red-blue-green separation problem
(in which we want to separate terminals of three colors with two noncrossing
polygons of minimum length), both on the Euclidean plane. This improves the
work of Arora and Chang (ICALP 2003) who gave a slower PTAS for the simpler
red-blue separation problem. For the case of unweighted plane graphs, we also
show a PTAS for the two-colored traveling salesman problem. All these results
are based on our new patching procedure that might be of independent interest.
On the negative side, we show that the problem of connecting terminal pairs
with noncrossing paths is NP-hard on the Euclidean plane, and that the problem
of finding two noncrossing spanning trees is NP-hard in plane graphs.Comment: 36 pages, 15 figures (colored
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