20,544 research outputs found
Maximizing Maximal Angles for Plane Straight-Line Graphs
Let be a plane straight-line graph on a finite point set
in general position. The incident angles of a vertex
of are the angles between any two edges of that appear consecutively in
the circular order of the edges incident to .
A plane straight-line graph is called -open if each vertex has an
incident angle of size at least . In this paper we study the following
type of question: What is the maximum angle such that for any finite set
of points in general position we can find a graph from a certain
class of graphs on that is -open? In particular, we consider the
classes of triangulations, spanning trees, and paths on and give tight
bounds in most cases.Comment: 15 pages, 14 figures. Apart of minor corrections, some proofs that
were omitted in the previous version are now include
Bicriteria Network Design Problems
We study a general class of bicriteria network design problems. A generic
problem in this class is as follows: Given an undirected graph and two
minimization objectives (under different cost functions), with a budget
specified on the first, find a <subgraph \from a given subgraph-class that
minimizes the second objective subject to the budget on the first. We consider
three different criteria - the total edge cost, the diameter and the maximum
degree of the network. Here, we present the first polynomial-time approximation
algorithms for a large class of bicriteria network design problems for the
above mentioned criteria. The following general types of results are presented.
First, we develop a framework for bicriteria problems and their
approximations. Second, when the two criteria are the same %(note that the cost
functions continue to be different) we present a ``black box'' parametric
search technique. This black box takes in as input an (approximation) algorithm
for the unicriterion situation and generates an approximation algorithm for the
bicriteria case with only a constant factor loss in the performance guarantee.
Third, when the two criteria are the diameter and the total edge costs we use a
cluster-based approach to devise a approximation algorithms --- the solutions
output violate both the criteria by a logarithmic factor. Finally, for the
class of treewidth-bounded graphs, we provide pseudopolynomial-time algorithms
for a number of bicriteria problems using dynamic programming. We show how
these pseudopolynomial-time algorithms can be converted to fully
polynomial-time approximation schemes using a scaling technique.Comment: 24 pages 1 figur
Matroidal Degree-Bounded Minimum Spanning Trees
We consider the minimum spanning tree (MST) problem under the restriction
that for every vertex v, the edges of the tree that are adjacent to v satisfy a
given family of constraints. A famous example thereof is the classical
degree-constrained MST problem, where for every vertex v, a simple upper bound
on the degree is imposed. Iterative rounding/relaxation algorithms became the
tool of choice for degree-bounded network design problems. A cornerstone for
this development was the work of Singh and Lau, who showed for the
degree-bounded MST problem how to find a spanning tree violating each degree
bound by at most one unit and with cost at most the cost of an optimal solution
that respects the degree bounds.
However, current iterative rounding approaches face several limits when
dealing with more general degree constraints. In particular, when several
constraints are imposed on the edges adjacent to a vertex v, as for example
when a partition of the edges adjacent to v is given and only a fixed number of
elements can be chosen out of each set of the partition, current approaches
might violate each of the constraints by a constant, instead of violating all
constraints together by at most a constant number of edges. Furthermore, it is
also not clear how previous iterative rounding approaches can be used for
degree constraints where some edges are in a super-constant number of
constraints.
We extend iterative rounding/relaxation approaches both on a conceptual level
as well as aspects involving their analysis to address these limitations. This
leads to an efficient algorithm for the degree-constrained MST problem where
for every vertex v, the edges adjacent to v have to be independent in a given
matroid. The algorithm returns a spanning tree T of cost at most OPT, such that
for every vertex v, it suffices to remove at most 8 edges from T to satisfy the
matroidal degree constraint at v
Proximity Drawings of High-Degree Trees
A drawing of a given (abstract) tree that is a minimum spanning tree of the
vertex set is considered aesthetically pleasing. However, such a drawing can
only exist if the tree has maximum degree at most 6. What can be said for trees
of higher degree? We approach this question by supposing that a partition or
covering of the tree by subtrees of bounded degree is given. Then we show that
if the partition or covering satisfies some natural properties, then there is a
drawing of the entire tree such that each of the given subtrees is drawn as a
minimum spanning tree of its vertex set
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