60,392 research outputs found
The Distribution of the Domination Number of a Family of Random Interval Catch Digraphs
We study a new kind of proximity graphs called proportional-edge proximity
catch digraphs (PCDs)in a randomized setting. PCDs are a special kind of random
catch digraphs that have been developed recently and have applications in
statistical pattern classification and spatial point pattern analysis. PCDs are
also a special type of intersection digraphs; and for one-dimensional data, the
proportional-edge PCD family is also a family of random interval catch
digraphs. We present the exact (and asymptotic) distribution of the domination
number of this PCD family for uniform (and non-uniform) data in one dimension.
We also provide several extensions of this random catch digraph by relaxing the
expansion and centrality parameters, thereby determine the parameters for which
the asymptotic distribution is non-degenerate. We observe sudden jumps (from
degeneracy to non-degeneracy or from a non-degenerate distribution to another)
in the asymptotic distribution of the domination number at certain parameter
values.Comment: 29 pages, 3 figure
A domination algorithm for -instances of the travelling salesman problem
We present an approximation algorithm for -instances of the
travelling salesman problem which performs well with respect to combinatorial
dominance. More precisely, we give a polynomial-time algorithm which has
domination ratio . In other words, given a
-edge-weighting of the complete graph on vertices, our
algorithm outputs a Hamilton cycle of with the following property:
the proportion of Hamilton cycles of whose weight is smaller than that of
is at most . Our analysis is based on a martingale approach.
Previously, the best result in this direction was a polynomial-time algorithm
with domination ratio for arbitrary edge-weights. We also prove a
hardness result showing that, if the Exponential Time Hypothesis holds, there
exists a constant such that cannot be replaced by in the result above.Comment: 29 pages (final version to appear in Random Structures and
Algorithms
Hamilton decompositions of regular expanders: a proof of Kelly's conjecture for large tournaments
A long-standing conjecture of Kelly states that every regular tournament on n
vertices can be decomposed into (n-1)/2 edge-disjoint Hamilton cycles. We prove
this conjecture for large n. In fact, we prove a far more general result, based
on our recent concept of robust expansion and a new method for decomposing
graphs. We show that every sufficiently large regular digraph G on n vertices
whose degree is linear in n and which is a robust outexpander has a
decomposition into edge-disjoint Hamilton cycles. This enables us to obtain
numerous further results, e.g. as a special case we confirm a conjecture of
Erdos on packing Hamilton cycles in random tournaments. As corollaries to the
main result, we also obtain several results on packing Hamilton cycles in
undirected graphs, giving e.g. the best known result on a conjecture of
Nash-Williams. We also apply our result to solve a problem on the domination
ratio of the Asymmetric Travelling Salesman problem, which was raised e.g. by
Glover and Punnen as well as Alon, Gutin and Krivelevich.Comment: new version includes a standalone version of the `robust
decomposition lemma' for application in subsequent paper
Domination Analysis of Greedy Heuristics For The Frequency Assignment Problem
We introduce the greedy expectation algorithm for the
fixed spectrum version of the frequency assignment problem. This
algorithm was previously studied for the travelling salesman
problem. We show that the domination number of this algorithm is
at least where is the
available span and the number of vertices in the constraint
graph. In contrast to this we show that the standard greedy
algorithm has domination number strictly less than
for large n and fixed
Extension of One-Dimensional Proximity Regions to Higher Dimensions
Proximity maps and regions are defined based on the relative allocation of
points from two or more classes in an area of interest and are used to
construct random graphs called proximity catch digraphs (PCDs) which have
applications in various fields. The simplest of such maps is the spherical
proximity map which maps a point from the class of interest to a disk centered
at the same point with radius being the distance to the closest point from the
other class in the region. The spherical proximity map gave rise to class cover
catch digraph (CCCD) which was applied to pattern classification. Furthermore
for uniform data on the real line, the exact and asymptotic distribution of the
domination number of CCCDs were analytically available. In this article, we
determine some appealing properties of the spherical proximity map in compact
intervals on the real line and use these properties as a guideline for defining
new proximity maps in higher dimensions. Delaunay triangulation is used to
partition the region of interest in higher dimensions. Furthermore, we
introduce the auxiliary tools used for the construction of the new proximity
maps, as well as some related concepts that will be used in the investigation
and comparison of them and the resulting graphs. We characterize the geometry
invariance of PCDs for uniform data. We also provide some newly defined
proximity maps in higher dimensions as illustrative examples
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