1,450 research outputs found
Local Maximum Stable Sets Greedoids Stemmed from Very Well-Covered Graphs
A maximum stable set in a graph G is a stable set of maximum cardinality. S
is called a local maximum stable set of G if S is a maximum stable set of the
subgraph induced by the closed neighborhood of S. A greedoid (V,F) is called a
local maximum stable set greedoid if there exists a graph G=(V,E) such that its
family of local maximum stable sets coinsides with (V,F). It has been shown
that the family local maximum stable sets of a forest T forms a greedoid on its
vertex set. In this paper we demonstrate that if G is a very well-covered
graph, then its family of local maximum stable sets is a greedoid if and only
if G has a unique perfect matching.Comment: 12 pages, 12 figure
Grid classes and the Fibonacci dichotomy for restricted permutations
We introduce and characterise grid classes, which are natural generalisations
of other well-studied permutation classes. This characterisation allows us to
give a new, short proof of the Fibonacci dichotomy: the number of permutations
of length n in a permutation class is either at least as large as the nth
Fibonacci number or is eventually polynomial
On some Graphs with a Unique Perfect Matching
We show that deciding whether a given graph of size has a unique
perfect matching as well as finding that matching, if it exists, can be done in
time if is either a cograph, or a split graph, or an interval graph,
or claw-free. Furthermore, we provide a constructive characterization of the
claw-free graphs with a unique perfect matching
Cover-Encodings of Fitness Landscapes
The traditional way of tackling discrete optimization problems is by using
local search on suitably defined cost or fitness landscapes. Such approaches
are however limited by the slowing down that occurs when the local minima that
are a feature of the typically rugged landscapes encountered arrest the
progress of the search process. Another way of tackling optimization problems
is by the use of heuristic approximations to estimate a global cost minimum.
Here we present a combination of these two approaches by using cover-encoding
maps which map processes from a larger search space to subsets of the original
search space. The key idea is to construct cover-encoding maps with the help of
suitable heuristics that single out near-optimal solutions and result in
landscapes on the larger search space that no longer exhibit trapping local
minima. We present cover-encoding maps for the problems of the traveling
salesman, number partitioning, maximum matching and maximum clique; the
practical feasibility of our method is demonstrated by simulations of adaptive
walks on the corresponding encoded landscapes which find the global minima for
these problems.Comment: 15 pages, 4 figure
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