1,996 research outputs found
Positional Games
Positional games are a branch of combinatorics, researching a variety of
two-player games, ranging from popular recreational games such as Tic-Tac-Toe
and Hex, to purely abstract games played on graphs and hypergraphs. It is
closely connected to many other combinatorial disciplines such as Ramsey
theory, extremal graph and set theory, probabilistic combinatorics, and to
computer science. We survey the basic notions of the field, its approaches and
tools, as well as numerous recent advances, standing open problems and
promising research directions.Comment: Submitted to Proceedings of the ICM 201
Computational Geometry Column 42
A compendium of thirty previously published open problems in computational
geometry is presented.Comment: 7 pages; 72 reference
Forward stagewise regression and the monotone lasso
We consider the least angle regression and forward stagewise algorithms for
solving penalized least squares regression problems. In Efron, Hastie,
Johnstone & Tibshirani (2004) it is proved that the least angle regression
algorithm, with a small modification, solves the lasso regression problem. Here
we give an analogous result for incremental forward stagewise regression,
showing that it solves a version of the lasso problem that enforces
monotonicity. One consequence of this is as follows: while lasso makes optimal
progress in terms of reducing the residual sum-of-squares per unit increase in
-norm of the coefficient , forward stage-wise is optimal per unit
arc-length traveled along the coefficient path. We also study a condition
under which the coefficient paths of the lasso are monotone, and hence the
different algorithms coincide. Finally, we compare the lasso and forward
stagewise procedures in a simulation study involving a large number of
correlated predictors.Comment: Published at http://dx.doi.org/10.1214/07-EJS004 in the Electronic
Journal of Statistics (http://www.i-journals.org/ejs/) by the Institute of
Mathematical Statistics (http://www.imstat.org
On Upward Drawings of Trees on a Given Grid
Computing a minimum-area planar straight-line drawing of a graph is known to
be NP-hard for planar graphs, even when restricted to outerplanar graphs.
However, the complexity question is open for trees. Only a few hardness results
are known for straight-line drawings of trees under various restrictions such
as edge length or slope constraints. On the other hand, there exist
polynomial-time algorithms for computing minimum-width (resp., minimum-height)
upward drawings of trees, where the height (resp., width) is unbounded.
In this paper we take a major step in understanding the complexity of the
area minimization problem for strictly-upward drawings of trees, which is one
of the most common styles for drawing rooted trees. We prove that given a
rooted tree and a grid, it is NP-hard to decide whether
admits a strictly-upward (unordered) drawing in the given grid.Comment: Appears in the Proceedings of the 25th International Symposium on
Graph Drawing and Network Visualization (GD 2017
Three Puzzles on Mathematics, Computation, and Games
In this lecture I will talk about three mathematical puzzles involving
mathematics and computation that have preoccupied me over the years. The first
puzzle is to understand the amazing success of the simplex algorithm for linear
programming. The second puzzle is about errors made when votes are counted
during elections. The third puzzle is: are quantum computers possible?Comment: ICM 2018 plenary lecture, Rio de Janeiro, 36 pages, 7 Figure
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