Enclosing Depth and Other Depth Measures

We study families of depth measures defined by natural sets of axioms. We show that any such depth measure is a constant factor approximation of Tukey depth. We further investigate the dimensions of depth regions, showing that the Cascade conjecture, introduced by Kalai for Tverberg depth, holds for all depth measures which satisfy our most restrictive set of axioms, which includes Tukey depth. Along the way, we introduce and study a new depth measure called enclosing depth, which we believe to be of independent interest, and show its relation to a constant-fraction Radon theorem on certain two-colored point sets

The Complexity of Sharing a Pizza

Assume you have a 2-dimensional pizza with 2n ingredients that you want to share with your friend. For this you are allowed to cut the pizza using several straight cuts, and then give every second piece to your friend. You want to do this fairly, that is, your friend and you should each get exactly half of each ingredient. How many cuts do you need? It was recently shown using topological methods that n cuts always suffice. In this work, we study the computational complexity of finding such n cuts. Our main result is that this problem is PPA-complete when the ingredients are represented as point sets. For this, we give a new proof that for point sets n cuts suffice, which does not use any topological methods. We further prove several hardness results as well as a higher-dimensional variant for the case where the ingredients are well-separated

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 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 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 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 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 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

Combinatorial Depth Measures for Hyperplane Arrangements

Regression depth, introduced by Rousseeuw and Hubert in 1999, is a notion that measures how good of a regression hyperplane a given query hyperplane is with respect to a set of data points. Under projective duality, this can be interpreted as a depth measure for query points with respect to an arrangement of data hyperplanes. The study of depth measures for query points with respect to a set of data points has a long history, and many such depth measures have natural counterparts in the setting of hyperplane arrangements. For example, regression depth is the counterpart of Tukey depth. Motivated by this, we study general families of depth measures for hyperplane arrangements and show that all of them must have a deep point. Along the way we prove a Tverberg-type theorem for hyperplane arrangements, giving a positive answer to a conjecture by Rousseeuw and Hubert from 1999. We also get three new proofs of the centerpoint theorem for regression depth, all of which are either stronger or more general than the original proof by Amenta, Bern, Eppstein, and Teng. Finally, we prove a version of the center transversal theorem for regression depth

Efficiently stabbing convex polygons and variants of the Hadwiger-Debrunner (p,q)(p, q)-theorem

Hadwiger and Debrunner showed that for families of convex sets in $\mathbb{R}^d$ with the property that among any $p$ of them some $q$ have a common point, the whole family can be stabbed with $p-q+1$ points if $p \geq q \geq d+1$ and $(d-1)p < d(q-1)$. This generalizes a classical result by Helly. We show how such a stabbing set can be computed for a family of convex polygons in the plane with a total of $n$ vertices in $O((p-q+1)n^{4/3}\log^{8} n(\log\log n)^{1/3} + np^2)$ expected time. For polyhedra in $\mathbb{R}^3$, we get an algorithm running in $O((p-q+1)n^{5/2}\log^{10} n(\log\log n)^{1/6} + np^3)$ expected time. We also investigate other conditions on convex polygons for which our algorithm can find a fixed number of points stabbing them. Finally, we show that analogous results of the Hadwiger and Debrunner $(p,q)$-theorem hold in other settings, such as convex sets in $\mathbb{R}^d\times\mathbb{Z}^k$ or abstract convex geometries

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 R^d, there is a point c, not necessarily from P, such that each halfspace containing c contains at least 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 epsilon-nets and weak epsilon-approximations, where it is stronger than the former but weaker than the latter. We show that for any point set of size n in R^d and for any positive alpha_1,...,alpha_k where alpha_1 <= alpha_2 <= ... <= alpha_k and for every i,j with i+j <= k+1 we have that (d-1)alpha_k+alpha_i+alpha_j <= 1, we can find Q of size k such that each halfspace containing j points of Q contains least alpha_j n points of P. For two-dimensional point sets we further show that for every alpha and beta with alpha <= beta and alpha+beta <= 2/3 we can find Q with |Q|=3 such that each halfplane containing one point of Q contains at least alpha n of the points of P and each halfplane containing all of Q contains at least beta n points of P. All these results generalize to the setting where P is any mass distribution. For the case where P is a point set in R^2 and |Q|=2, we provide algorithms to find such points in time O(n log^3 n)

A Topological Version of Schaefer's Dichotomy Theorem

Schaefer's dichotomy theorem [Schaefer, STOC'78] states that a boolean constraint satisfaction problem (CSP) is polynomial-time solvable if one of six given conditions holds for every type of constraint allowed in its instances. Otherwise, it is NP-complete. In this paper, we analyze boolean CSPs in terms of their topological complexity, instead of their computational complexity. We attach a natural topological space to the set of solutions of a boolean CSP and introduce the notion of projection-universality. We prove that a boolean CSP is projection-universal if and only if it is categorized as NP-complete by Schaefer's dichotomy theorem, showing that the dichotomy translates exactly from computational to topological complexity. We show a similar dichotomy for SAT variants and homotopy-universality.Comment: 18 pages, 1 figur