12,290 research outputs found
Search for the end of a path in the d-dimensional grid and in other graphs
We consider the worst-case query complexity of some variants of certain
\cl{PPAD}-complete search problems. Suppose we are given a graph and a
vertex . We denote the directed graph obtained from by
directing all edges in both directions by . is a directed subgraph of
which is unknown to us, except that it consists of vertex-disjoint
directed paths and cycles and one of the paths originates in . Our goal is
to find an endvertex of a path by using as few queries as possible. A query
specifies a vertex , and the answer is the set of the edges of
incident to , together with their directions. We also show lower bounds for
the special case when consists of a single path. Our proofs use the theory
of graph separators. Finally, we consider the case when the graph is a grid
graph. In this case, using the connection with separators, we give
asymptotically tight bounds as a function of the size of the grid, if the
dimension of the grid is considered as fixed. In order to do this, we prove a
separator theorem about grid graphs, which is interesting on its own right
On the inducibility of small trees
The quantity that captures the asymptotic value of the maximum number of
appearances of a given topological tree (a rooted tree with no vertices of
outdegree ) with leaves in an arbitrary tree with sufficiently large
number of leaves is called the inducibility of . Its precise value is known
only for some specific families of trees, most of them exhibiting a symmetrical
configuration. In an attempt to answer a recent question posed by Czabarka,
Sz\'ekely, and the second author of this article, we provide bounds for the
inducibility of the -leaf binary tree whose branches are a
single leaf and the complete binary tree of height . It was indicated before
that appears to be `close' to . We can make this precise by
showing that . Furthermore, we
also consider the problem of determining the inducibility of the tree ,
which is the only tree among -leaf topological trees for which the
inducibility is unknown
Ternary numbers and algebras. Reflexive numbers and Berger graphs
The Calabi-Yau spaces with SU(m) holonomy can be studied by the algebraic way
through the integer lattice where one can construct the Newton reflexive
polyhedra or the Berger graphs. Our conjecture is that the Berger graphs can be
directly related with the -ary algebras. To find such algebras we study the
n-ary generalization of the well-known binary norm division algebras, , , , , which helped to discover the
most important "minimal" binary simple Lie groups, U(1), SU(2) and G(2). As the
most important example, we consider the case , which gives the ternary
generalization of quaternions and octonions, , , respectively. The
ternary generalization of quaternions is directly related to the new ternary
algebra and group which are related to the natural extensions of the binary
algebra and SU(3) group. Using this ternary algebra we found the
solution for the Berger graph: a tetrahedron.Comment: Revised version with minor correction
Tag-Aware Recommender Systems: A State-of-the-art Survey
In the past decade, Social Tagging Systems have attracted increasing
attention from both physical and computer science communities. Besides the
underlying structure and dynamics of tagging systems, many efforts have been
addressed to unify tagging information to reveal user behaviors and
preferences, extract the latent semantic relations among items, make
recommendations, and so on. Specifically, this article summarizes recent
progress about tag-aware recommender systems, emphasizing on the contributions
from three mainstream perspectives and approaches: network-based methods,
tensor-based methods, and the topic-based methods. Finally, we outline some
other tag-related works and future challenges of tag-aware recommendation
algorithms.Comment: 19 pages, 3 figure
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