Related papers: Sweeping $x$-monotone pseudolines
Let $P$ be a simple polytope with $n-d = 2$, where $d$ is the dimension and $n$ is the number of facets. The graph of such a polytope is also called a grid. It is known that the directed random walk along the edges of $P$ terminates after…
We study the classic sliding cube model for programmable matter under parallel reconfiguration in three dimensions, providing novel algorithmic and surprising complexity results in addition to generalizing the best known bounds from two to…
In many macroscopic dynamic wetting problems, it is assumed that the macroscopic interface is quasistatic, and the dissipation appears only in the region close to the contact line. When approaching the moving contact line, a microscopic…
Polytopes are the basic finite data structures for convex sets: they appear as feasible regions in linear optimization, as geometric summaries in algorithms, and as random objects in stochastic geometry. A natural geometric question is…
The simplex algorithm is one of the most popular algorithms to solve linear programs (LPs). Starting at an extreme point solution of an LP, it performs a sequence of basis exchanges (called pivots) that allows one to move to a better…
Let $d \geq 3$ be an integer. It is known that the number of edges of the edge polytope of the complete graph with $d$ vertices is $d(d-1)(d-2)/2$. In this paper, we study the maximum possible number $\mu_d$ of edges of the edge polytope…
We show that the problem of covering a set of points in the plane with a minimum number of guillotine cuts is NP-complete. To that end, first we present a new NP-completeness proof for the problem of covering points with disjoint line…
A (multi)set of segments in the plane may form a TSP tour, a matching, a tree, or any multigraph. If two segments cross, then we can reduce the total length with the following flip operation. We remove a pair of crossing segments, and…
Given a set of coins arranged in a line, we remove heads-up coins one at a time and flip any adjacent coins after each removal. The coin-removal problem is to determine for which arrangements of coins it is possible to remove all of the…
The ropelength of a knot or link is the minimal number of inches of 1-inch-thick rope that it takes to tie it. The relationship of this measurement to knot and link invariants has been studied by various authors. We give the first results…
Let $ex(n, P)$ be the maximum possible number of ones in any 0-1 matrix of dimensions $n \times n$ that avoids $P$. Matrix $P$ is called minimally non-linear if $ex(n, P) = \omega(n)$ but $ex(n, P') = O(n)$ for every strict subpattern $P'$…
Given a set of points in the plane, a covering path is a polygonal path that visits all the points. In this paper we consider covering paths of the vertices of an n x m grid. We show that the minimal number of segments of such a path is…
The paper presents an algorithm for minimum vertex cover problem, which is an NP-Complete problem. The algorithm computes a minimum vertex cover of each input simple graph. Tested by the attached MATLAB programs, Stage 1 of the algorithm is…
Let $P\subset\mathbb{R}^{2}$ be a set of $n$ points. In this paper we show two new algorithms, one to compute the number of triangulations of $P$, and one to compute the number of pseudo-triangulations of $P$. We show that our algorithms…
Chains of minimal degree rational curves have been used as an important tool in the study of Fano manifolds. Their own geometric properties, however, have not been studied much. The goal of the paper is to introduce an infinitesimal method…
Let $P$ be a polygonal curve in $\mathbb{R}^d$ of length $n$, and $S$ be a point-set of size $k$. The Curve/Point Set Matching problem consists of finding a polygonal curve $Q$ on $S$ such that the Fr\'echet distance from $P$ is less than a…
We consider the polyhedral properties of two spanning tree problems with additional constraints. In the first problem, it is required to find a tree with a minimum sum of edge weights among all spanning trees with the number of leaves less…
Many algorithms for clipping a line by a rectangular area or a convex polygon in E2 or by a non-convex or convex polyhedron in E3 have been published. The line segment clipping by the rectangular window in E2 is often restricted to the use…
It has been known for more than 300 years that the shape of an inelastic hanging cable, chain, or rope of uniform linear mass density is the graph of the hyperbolic cosine, up to scaling and shifting coordinates. But given two points at…
We construct ancient solutions to Curve Shortening in the plane whose total curvature is uniformly bounded by gluing together an arbitrary chain of given Grim Reapers along their common asymptotes.