105,228 research outputs found
Trees with Given Stability Number and Minimum Number of Stable Sets
We study the structure of trees minimizing their number of stable sets for
given order and stability number . Our main result is that the
edges of a non-trivial extremal tree can be partitioned into stars,
each of size or , so that every vertex is included in at most two
distinct stars, and the centers of these stars form a stable set of the tree.Comment: v2: Referees' comments incorporate
Tur\'an Graphs, Stability Number, and Fibonacci Index
The Fibonacci index of a graph is the number of its stable sets. This
parameter is widely studied and has applications in chemical graph theory. In
this paper, we establish tight upper bounds for the Fibonacci index in terms of
the stability number and the order of general graphs and connected graphs.
Tur\'an graphs frequently appear in extremal graph theory. We show that Tur\'an
graphs and a connected variant of them are also extremal for these particular
problems.Comment: 11 pages, 3 figure
A Framework for Algorithm Stability
We say that an algorithm is stable if small changes in the input result in
small changes in the output. This kind of algorithm stability is particularly
relevant when analyzing and visualizing time-varying data. Stability in general
plays an important role in a wide variety of areas, such as numerical analysis,
machine learning, and topology, but is poorly understood in the context of
(combinatorial) algorithms. In this paper we present a framework for analyzing
the stability of algorithms. We focus in particular on the tradeoff between the
stability of an algorithm and the quality of the solution it computes. Our
framework allows for three types of stability analysis with increasing degrees
of complexity: event stability, topological stability, and Lipschitz stability.
We demonstrate the use of our stability framework by applying it to kinetic
Euclidean minimum spanning trees
Learning Membership Functions in a Function-Based Object Recognition System
Functionality-based recognition systems recognize objects at the category
level by reasoning about how well the objects support the expected function.
Such systems naturally associate a ``measure of goodness'' or ``membership
value'' with a recognized object. This measure of goodness is the result of
combining individual measures, or membership values, from potentially many
primitive evaluations of different properties of the object's shape. A
membership function is used to compute the membership value when evaluating a
primitive of a particular physical property of an object. In previous versions
of a recognition system known as Gruff, the membership function for each of the
primitive evaluations was hand-crafted by the system designer. In this paper,
we provide a learning component for the Gruff system, called Omlet, that
automatically learns membership functions given a set of example objects
labeled with their desired category measure. The learning algorithm is
generally applicable to any problem in which low-level membership values are
combined through an and-or tree structure to give a final overall membership
value.Comment: See http://www.jair.org/ for any accompanying file
Maximum Performance at Minimum Cost in Network Synchronization
We consider two optimization problems on synchronization of oscillator
networks: maximization of synchronizability and minimization of synchronization
cost. We first develop an extension of the well-known master stability
framework to the case of non-diagonalizable Laplacian matrices. We then show
that the solution sets of the two optimization problems coincide and are
simultaneously characterized by a simple condition on the Laplacian
eigenvalues. Among the optimal networks, we identify a subclass of hierarchical
networks, characterized by the absence of feedback loops and the normalization
of inputs. We show that most optimal networks are directed and
non-diagonalizable, necessitating the extension of the framework. We also show
how oriented spanning trees can be used to explicitly and systematically
construct optimal networks under network topological constraints. Our results
may provide insights into the evolutionary origin of structures in complex
networks for which synchronization plays a significant role.Comment: 29 pages, 9 figures, accepted for publication in Physica D, minor
correction
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