Covering all points except one

Covering all points except one

In many point-line geometries, to cover all points except one, more lines are needed than to cover all points. Bounds can be given by looking at the dimension of the space of functions induced by polynomials of bounded degree.

Root location in random trees: a polarity property of all sampling consistent phylogenetic models except one.

Neutral macroevolutionary models, such as the Yule model, give rise to a probability distribution on the set of discrete rooted binary trees over a given leaf set. Such models can provide a signal as to the approximate location of the root when only the unrooted phylogenetic tree is known, and this signal becomes relatively more significant as the number of leaves grows. In this short note, we ...

Covering all the bases.

Covering lattice points by subspaces

We find tight estimates for the minimum number of proper subspaces needed to cover all lattice points in an n-dimensional convex body C, symmetric about the origin 0. This enables us to prove the following statement, which settles a problem of G. Halász. The maximum number of n-wise linearly independent lattice points in the n-dimensional ball rB of radius r around 0 is O(rn/(n−1)). This bound ...

Covering points by disjoint boxes with outliers

For a set of n points in the plane, we consider the axis–aligned (p, k)-Box Covering problem: Find p axis-aligned, pairwise-disjoint boxes that together contain at least n − k points. In this paper, we consider the boxes to be either squares or rectangles, and we want to minimize the area of the largest box. For general p we show that the problem is NP-hard for both squares and rectangles. For ...