arXiv subjects

Patrick Schnider

Publications and source records attributed to Patrick Schnider.

41 records · Page 3Linked to original sources

arXiv2019

Ham-Sandwich cuts and center transversals in subspaces

The Ham-Sandwich theorem is a well-known result in geometry. It states that any $d$ mass distributions in $\mathbb{R}^d$ can be simultaneously bisected by a hyperplane. The result is tight, that is, there are examples of $d+1$ mass distributions that cannot be simultaneously bisected by a single hyperplane. In this abstract we will study the following question: given a continuous assignment of mass distributions to certain subsets of $\mathbb{R}^d$, is there a subset on which we can bisect more masses than what is guaranteed by the Ham-Sandwich theorem? We investigate two types of subsets. The first type are linear subspaces of $\mathbb{R}^d$, i.e., $k$-dimensional flats containing the origin. We show that for any continuous assignment of $d$ mass distributions to the $k$-dimensional linear subspaces of $\mathbb{R}^d$, there is always a subspace on which we can simultaneously bisect the images of all $d$ assignments. We extend this result to center transversals, a generalization of Ham-Sandwich cuts. As for Ham-Sandwich cuts, we further show that for $d-k+2$ masses, we can choose $k-1$ of the vectors defining the $k$-dimensional subspace in which the solution lies. The second type of subsets we consider are subsets that are determined by families of $n$ hyperplanes in $\mathbb{R}^d$. Also in this case, we find a Ham-Sandwich-type result. In an attempt to solve a conjecture by Langerman about bisections with several cuts, we show that our underlying topological result can be used to prove this conjecture in a relaxed setting.

cs.CG↗

arXiv2018

Extending the centerpoint theorem to multiple points

The centerpoint theorem is a well-known and widely used result in discrete geometry. It states that for any point set $P$ of $n$ points in $\mathbb{R}^d$, there is a point $c$, not necessarily from $P$, such that each halfspace containing $c$ contains at least $\frac{n}{d+1}$ points of $P$. Such a point $c$ is called a centerpoint, and it can be viewed as a generalization of a median to higher dimensions. In other words, a centerpoint can be interpreted as a good representative for the point set $P$. But what if we allow more than one representative? For example in one-dimensional data sets, often certain quantiles are chosen as representatives instead of the median. We present a possible extension of the concept of quantiles to higher dimensions. The idea is to find a set $Q$ of (few) points such that every halfspace that contains one point of $Q$ contains a large fraction of the points of $P$ and every halfspace that contains more of $Q$ contains an even larger fraction of $P$. This setting is comparable to the well-studied concepts of weak $\varepsilon$-nets and weak $\varepsilon$-approximations, where it is stronger than the former but weaker than the latter.

cs.CG↗

arXiv2018

The Partition Spanning Forest Problem

Given a set of colored points in the plane, we ask if there exists a crossing-free straight-line drawing of a spanning forest, such that every tree in the forest contains exactly the points of one color class. We show that the problem is NP-complete, even if every color class contains at most five points, but it is solvable in $O(n^2)$ time when each color class contains at most three points. If we require that the spanning forest is a linear forest, then the problem becomes NP-complete even if every color class contains at most four points.

cs.CG↗

arXiv2018

Even flying cops should think ahead

We study the entanglement game, which is a version of cops and robbers, on sparse graphs. While the minimum degree of a graph G is a lower bound for the number of cops needed to catch a robber in G, we show that the required number of cops can be much larger, even for graphs with small maximum degree. In particular, we show that there are 3-regular graphs where a linear number of cops are needed.

cs.DM↗

arXiv2017

A generalization of crossing families

For a set of points in the plane, a \emph{crossing family} is a set of line segments, each joining two of the points, such that any two line segments cross. We investigate the following generalization of crossing families: a \emph{spoke set} is a set of lines drawn through a point set such that each unbounded region of the induced line arrangement contains at least one point of the point set. We show that every point set has a spoke set of size $\sqrt{\frac{n}{8}}$. We also characterize the matchings obtained by selecting exactly one point in each unbounded region and connecting every such point to the point in the antipodal unbounded region.

cs.CG↗