Related papers: On the Longest Flip Sequence to Untangle Segments …

For an even set of points in the plane, choose a max-sum matching, that is, a perfect matching maximizing the sum of Euclidean distances of its edges. For each edge of the max-sum matching, consider the ellipse with foci at the edge's…

Given $n$ points in the plane, a \emph{covering path} is a polygonal path that visits all the points. If no three points are collinear, every covering path requires at least $n/2$ segments, and $n-1$ straight line segments obviously suffice…

We provide exact and approximation methods for solving a geometric relaxation of the Traveling Salesman Problem (TSP) that occurs in curve reconstruction: for a given set of vertices in the plane, the problem Minimum Perimeter Polygon (MPP)…

We revisit the classic task of finding the shortest tour of $n$ points in $d$-dimensional Euclidean space, for any fixed constant $d \geq 2$. We determine the optimal dependence on $\varepsilon$ in the running time of an algorithm that…

Given a planar graph $G$, we consider drawings of $G$ in the plane where edges are represented by straight line segments (which possibly intersect). Such a drawing is specified by an injective embedding $\pi$ of the vertex set of $G$ into…

This paper considers the problem of finding a quickest path between two points in the Euclidean plane in the presence of a transportation network. A transportation network consists of a planar network where each road (edge) has an…

We consider the traveling salesman problem when the cities are points in R^d for some fixed d and distances are computed according to geometric distances, determined by some norm. We show that for any polyhedral norm, the problem of finding…

We consider the problem of counting straight-edge triangulations of a given set $P$ of $n$ points in the plane. Until very recently it was not known whether the exact number of triangulations of $P$ can be computed asymptotically faster…

In the longest plane spanning tree problem, we are given a finite planar point set $\mathcal{P}$, and our task is to find a plane (i.e., noncrossing) spanning tree for $\mathcal{P}$ with maximum total Euclidean edge length. Despite more…

We prove that every set of n points in the plane has at most $(16+5/6)^n$ rectangulations. This improves upon a long-standing bound of Ackerman. Our proof is based on the cross-graph charging-scheme technique.

We show a lower bound for the universal traveling salesman heuristic on the plane: for any linear order on the unit square $[0,1]^2$, there are finite subsets $S \subset [0,1]^2$ of arbitrarily large size such that the path visiting each…

Let $k \geq 2$ be a constant. Given any $k$ convex polygons in the plane with a total of $n$ vertices, we present an $O(n\log^{2k-3}n)$ time algorithm that finds a translation of each of the polygons such that the area of intersection of…

In this paper, we study discrepancy questions for spanning subgraphs of $k$-uniform hypergraphs. Our main result is that, for any integers $k \ge 3$ and $r \ge 2$, any $r$-colouring of the edges of a $k$-uniform $n$-vertex hypergraph $G$…

We study the variant of the Euclidean Traveling Salesman problem where instead of a set of points, we are given a set of lines as input, and the goal is to find the shortest tour that visits each line. The best known upper and lower bounds…

We consider systems of "pinned balls," i.e., balls that have fixed positions and pseudo-velocities. Pseudo-velocities change according to the same rules as those for velocities of totally elastic collisions between moving balls. The times…

A star-simple drawing of a graph is a drawing in which adjacent edges do not cross. In contrast, there is no restriction on the number of crossings between two independent edges. When allowing empty lenses (a face in the arrangement induced…

Let $P$ be a set of $n$ points in the plane. A crossing-free structure on $P$ is a plane graph with vertex set $P$. Examples of crossing-free structures include triangulations of $P$, spanning cycles of $P$, also known as polygonalizations…

We describe an exact algorithm for finding the best 2-OPT move which, experimentally, was observed to be much faster than the standard quadratic approach. To analyze its average-case complexity, we introduce a family of heuristic procedures…

We provide an algorithm that verifies the optimal colored Tverberg problem for $10$ points in the plane: Every $10$ points in the plane in color classes of size at most $3$ can be partitioned in $4$ rainbow pieces such that their convex…

Let $L$ be a set of $n$ lines in the plane, not necessarily in general position. We present an efficient algorithm for finding all the vertices of the arrangement $A(L)$ of maximum level, where the level of a vertex $v$ is the number of…