# Questions tagged [computational-geometry]

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### “Fast” approximation algorithm for geometric hitting set of same-height rectangles

In the Geometric Hitting Set problem, we are given a set of $m$ geometric objects and a set of $n$ points in $\mathbb{R}^2$, and we wish to find a small subset of the points that hits all the objects. ...
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### Are generalized map involutions captured by GNN?

In coming across a post related to this question's topic, posting in Theoretical Computer Science might be the right place. In combinatorial maps (1,2,3), are generalized map involutions captured by ...
28 views

### Complexity of best folding of a 2D set (or how to optimize a sandwich)

Motivation: I was making lunch for my son, part of which is making a sandwich from two halves of a slice of bread. In order to minimize the parts of bread that have cheese on them, and are not covered ...
88 views

### How many maximal planar graphs are there?

We denote by a triangulation a (simple) maximal planar graph. How many triangulations on $n$ vertices are there? How many triangulations are there if we cannot distinguish the vertices, i.e. ...
147 views

### How to calculate complexity in a high dimensional space?

Edit: 'Fitness landscape analysis' was mentioned as a relevant measure. If you're going to downvote the post, at least leave a comment what is wrong. For a specific f(), I'm defining a term '...
410 views

### In an $m$ by $n$ Boolean matrix, can you find a square block whose four corners are ones in $O(m \cdot n)$ time?

Decision Problem Input: An $m$ by $n$ Boolean matrix $M$. Decision Question: Does there exist a square block within $M$ such that upper-left corner entry == upper-right corner entry == lower-left ...
33 views

### What is the best way to find circles that contain a given point (in 2D)?

Given $n$ circles all with radius $r$ and one point on a 2D plane, what is the best algorithm to find all circles that contain the given point. The circles and the point can change their positions. ...
49 views

### Find a boundary from set of 3d line segments

I have a set of n 3d line segments ...
82 views

### Complexity of computing the union of H-polytopes in three dimensions

Consider a set $P_1,\ldots,P_k$ of polytopes in $\mathbb{R}^3$, each given as an intersection of halfspaces with rational normals (in particular, they are all convex). We are also given a target ...
39 views

### Practical worst-case polylogarithmic dynamic orthogonal range queries?

There are a number of data structures in the literature that solve the dynamic orthogonal range search problem in polylogarithmic time (say, range trees). My understanding is that these structures ...
85 views

### Optimal point placement on integer lattice

What is known about the following point placement problem? For positive integers $N$, $n<N^2$, and $N\times N$ grid $\mathcal{G}$, compute \begin{eqnarray*} \mu_1(N,n)\triangleq\min_{\mathcal{P}\...
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### Example of Delaunay Triangulation where it does not minimize the maximum angle

I know that that the Delaunay triangulation maximizes the minimum angle of triangulation. And it does not minimize the maximum angle. If we consider the set of points in general position(no four ...
72 views

### Computing 3D viewpoint of a set of non-intersecting segments

Consider the following problem: we are given a finite set of bounded line-segments in ${\mathbb R}^3$, and we want to decide whether there exists a point $p\in {\mathbb R}^3$ from which no two ...
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### How to choose good diagonals when partitioning an orthogonal polygon into rectangles?

Following this answer on MathOverflow and section 3 of the linked paper by David Eppstein on how to split an orthogonal polygon into rectangles I came to a point where I just fail to understand how to ...
160 views

### Convex polygons inclusion relation

I have the following problem which came as a subproblem in some work I was doing and I am completely stuck. Note that I am interested in it only in terms of worst case time complexity (not heuristics ...
106 views

### Data structures for embedded simplicial complexes

I am looking for a data structure to encode an $n$-dimensional simplicial complex with an embedding in $\mathbb{R}^{n+1}$. I am aware of combinatorial maps, which generalize rotation systems of planar ...
151 views

### The decision procedure of theory of closed real field is in NP-hard?

The decision procedure of theory of closed real field refers to https://en.wikipedia.org/wiki/Decidability_of_first-order_theories_of_the_real_numbers
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### Minimal number of hyperplanes needed to separate sets of points from one other set

Let $\mathbb{R}^d$ be our space. We have a single good set of points $g$, and a collection of bad sets of points $B$. We assume that for all $b \in B$ the convex hulls of $g$ and $b$ are disjoint. ...
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### Volume computation of special polytopes

I'm interested in computing the volume of a special class of $\mathcal{H}$-polytopes and the complexity of doing so. I know that in general it is #P-hard to compute the volume of $\mathcal{H}$ -...
51 views

### $\ell_\infty$ partially enclosing ball problem

Suppose I have $n$ points in $\mathbb{R}^d$ endowed with the $\ell_\infty$ metric, and I wish to find a minimum-diameter ball that contains some $k$ of these points. What is known about this problem? ...
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### Select circle with given radius that contains most points

Given some points on a coordinate system and some radius r, I need to place a circle with radius r somewhere on the coordinate system such that that circle includes the most points. I tried solving ...
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### Points of a finite set wihtin a ball

I am looking for data-structures to store efficiently a set of points $E$ in an euclidean space of dimension $d$. In particular, I would like to be able to solve the problem of finding all the point ...
141 views

### A least sized partition of a set under a distance metric

What is the worst case complexity of an algorithm to find a least partition of a set under a distance metric, described as follows: Input: A set $S=\{s_1,\ldots,s_n\}$, where the elements $s_i$ are ...
54 views

### Inapproximability Results for APX-hard Geometric Optimization Problems

A lot of Geometric Optimization problems are NP-hard and APX-hard to approximate (No PTAS unless P=NP). One example of the geometric set cover problem can be found here. However, it is not easy to ...
592 views

### Complexity of testing if two sets of $m$ points in $\mathbb{R}^n$ differ only by rotation?

Imagine we have two size $m$ sets of points $X,Y\subset \mathbb{R}^n$. What is (time) complexity of testing if they differ only by rotation?: there exists rotation matrix $OO^T=O^TO=I$ such that $X=OY$...
52 views

### Inclusion probability of irregularly shaped polygon

I have two shapes, one is a circle, let's call it Circle A, and another irregularly shaped Polygon B. A will always have a greater area than B. Both of these polygons exist inside an area S. My task ...
54 views

### Congruency check for set of points in 3D using inertia tensor

You're given two set of points $A, B\subset \mathbb R^3:|A|=|B|=n$. You have to check if those sets are congruent, i.e. there exist some mapping $\sigma : A \to B$ and combination of translation and ...
49 views

### Given an algebraic variaties of n multivarieties polynomial equations, is there any algorithm to decide whether there is n-cube inscribing to it?

Given an algebraic variaties of n multivarieties polynomial equations, is there any algorithm to decide whether there is n-cube inscribing to it? And if there is, what is the computational ...
It is well-known that Euclidean space of dimension $d$ has a $(1+\epsilon)$-spanner of weight at most $\epsilon^{-O(d)}\cdot w(MST)$ (see Chapter 14 of Geometric Spanner Network book by Narashimhan ...