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تسجيل مستخدم جديدThe complexity of sharing a pizza
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Patrick Schnider
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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.
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Assume you have a pizza consisting of four ingredients (e.g., bread, tomatoes, cheese and olives) that you want to share with your friend. You want to do this fairly, meaning that you and your friend should get the same amount of each ingredient. How
many times do you need to cut the pizza so that this is possible? We will show that two straight cuts always suffice. More formally, we will show the following extension of the well-known Ham-sandwich theorem: Given four mass distributions in the plane, they can be simultaneously bisected with two lines. That is, there exist two oriented lines with the following property: let $R^+_1$ be the region of the plane that lies to the positive side of both lines and let $R^+_2$ be the region of the plane that lies to the negative side of both lines. Then $R^+=R^+_1cup R^+_2$ contains exactly half of each mass distribution.
In this article, we provide new structural results and algorithms for the Homotopy Height problem. In broad terms, this problem quantifies how much a curve on a surface needs to be stretched to sweep continuously between two positions. More precisely
, given two homotopic curves $gamma_1$ and $gamma_2$ on a combinatorial (say, triangulated) surface, we investigate the problem of computing a homotopy between $gamma_1$ and $gamma_2$ where the length of the longest intermediate curve is minimized. Such optimal homotopies are relevant for a wide range of purposes, from very theoretical questions in quantitative homotopy theory to more practical applications such as similarity measures on meshes and graph searching problems. We prove that Homotopy Height is in the complexity class NP, and the corresponding exponential algorithm is the best one known for this problem. This result builds on a structural theorem on monotonicity of optimal homotopies, which is proved in a companion paper. Then we show that this problem encompasses the Homotopic Frechet distance problem which we therefore also establish to be in NP, answering a question which has previously been considered in several different settings. We also provide an O(log n)-approximation algorithm for Homotopy Height on surfaces by adapting an earlier algorithm of Har-Peled, Nayyeri, Salvatipour and Sidiropoulos in the planar setting.
Let ${cal L}$ be an arrangement of $n$ lines in the Euclidean plane. The emph{$k$-level} of ${cal L}$ consists of all vertices $v$ of the arrangement which have exactly $k$ lines of ${cal L}$ passing below $v$. The complexity (the maximum size) of th
e $k$-level in a line arrangement has been widely studied. In 1998 Dey proved an upper bound of $O(ncdot (k+1)^{1/3})$. Due to the correspondence between lines in the plane and great-circles on the sphere, the asymptotic bounds carry over to arrangements of great-circles on the sphere, where the $k$-level denotes the vertices at distance at most $k$ to a marked cell, the emph{south pole}. We prove an upper bound of $O((k+1)^2)$ on the expected complexity of the $k$-level in great-circle arrangements if the south pole is chosen uniformly at random among all cells. We also consider arrangements of great $(d-1)$-spheres on the sphere $mathbb{S}^d$ which are orthogonal to a set of random points on $mathbb{S}^d$. In this model, we prove that the expected complexity of the $k$-level is of order $Theta((k+1)^{d-1})$.
The classic Ham-Sandwich theorem states that for any $d$ measurable sets in $mathbb{R}^d$, there is a hyperplane that bisects them simultaneously. An extension by Barany, Hubard, and Jeronimo [DCG 2008] states that if the sets are convex and emph{wel
l-separated}, then for any given $alpha_1, dots, alpha_d in [0, 1]$, there is a unique oriented hyperplane that cuts off a respective fraction $alpha_1, dots, alpha_d$ from each set. Steiger and Zhao [DCG 2010] proved a discrete analogue of this theorem, which we call the emph{$alpha$-Ham-Sandwich theorem}. They gave an algorithm to find the hyperplane in time $O(n (log n)^{d-3})$, where $n$ is the total number of input points. The computational complexity of this search problem in high dimensions is open, quite unlike the complexity of the Ham-Sandwich problem, which is now known to be PPA-complete (Filos-Ratsikas and Goldberg [STOC 2019]). Recently, Fearley, Gordon, Mehta, and Savani [ICALP 2019] introduced a new sub-class of CLS (Continuous Local Search) called emph{Unique End-of-Potential Line} (UEOPL). This class captures problems in CLS that have unique solutions. We show that for the $alpha$-Ham-Sandwich theorem, the search problem of finding the dividing hyperplane lies in UEOPL. This gives the first non-trivial containment of the problem in a complexity class and places it in the company of classic search problems such as finding the fixed point of a contraction map, the unique sink orientation problem and the $P$-matrix linear complementarity problem.
In a emph{fan-planar drawing} of a graph an edge can cross only edges with a common end-vertex. Fan-planar drawings have been recently introduced by Kaufmann and Ueckerdt, who proved that every $n$-vertex fan-planar drawing has at most $5n-10$ edges,
and that this bound is tight for $n geq 20$. We extend their result, both from the combinatorial and the algorithmic point of view. We prove tight bounds on the density of constrain
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