7,715 research outputs found
Hamilton cycles in graphs and hypergraphs: an extremal perspective
As one of the most fundamental and well-known NP-complete problems, the
Hamilton cycle problem has been the subject of intensive research. Recent
developments in the area have highlighted the crucial role played by the
notions of expansion and quasi-randomness. These concepts and other recent
techniques have led to the solution of several long-standing problems in the
area. New aspects have also emerged, such as resilience, robustness and the
study of Hamilton cycles in hypergraphs. We survey these developments and
highlight open problems, with an emphasis on extremal and probabilistic
approaches.Comment: to appear in the Proceedings of the ICM 2014; due to given page
limits, this final version is slightly shorter than the previous arxiv
versio
Extremal points of high dimensional random walks and mixing times of a Brownian motion on the sphere
We derive asymptotics for the probability of the origin to be an extremal
point of a random walk in R^n. We show that in order for the probability to be
roughly 1/2, the number of steps of the random walk should be between e^{c n /
log n}$ and e^{C n log n}. As a result, we attain a bound for the
?pi/2-covering time of a spherical brownian motion.Comment: 22 Page
On densities of lattice arrangements intersecting every i-dimensional affine subspace
In 1978, Makai Jr. established a remarkable connection between the
volume-product of a convex body, its maximal lattice packing density and the
minimal density of a lattice arrangement of its polar body intersecting every
affine hyperplane. Consequently, he formulated a conjecture that can be seen as
a dual analog of Minkowski's fundamental theorem, and which is strongly linked
to the well-known Mahler-conjecture.
Based on the covering minima of Kannan & Lov\'asz and a problem posed by
Fejes T\'oth, we arrange Makai Jr.'s conjecture into a wider context and
investigate densities of lattice arrangements of convex bodies intersecting
every i-dimensional affine subspace. Then it becomes natural also to formulate
and study a dual analog to Minkowski's second fundamental theorem. As our main
results, we derive meaningful asymptotic lower bounds for the densities of such
arrangements, and furthermore, we solve the problems exactly for the special,
yet important, class of unconditional convex bodies.Comment: 19 page
Space Exploration via Proximity Search
We investigate what computational tasks can be performed on a point set in
, if we are only given black-box access to it via nearest-neighbor
search. This is a reasonable assumption if the underlying point set is either
provided implicitly, or it is stored in a data structure that can answer such
queries. In particular, we show the following: (A) One can compute an
approximate bi-criteria -center clustering of the point set, and more
generally compute a greedy permutation of the point set. (B) One can decide if
a query point is (approximately) inside the convex-hull of the point set.
We also investigate the problem of clustering the given point set, such that
meaningful proximity queries can be carried out on the centers of the clusters,
instead of the whole point set
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