Related papers: Separating Two Points with Obstacles in the Plane:…
Recent successes of game-theoretic formulations in ML have caused a resurgence of research interest in differentiable games. Overwhelmingly, that research focuses on methods and upper bounds on their speed of convergence. In this work, we…
Segmenting an image into multiple components is a central task in computer vision. In many practical scenarios, prior knowledge about plausible components is available. Incorporating such prior knowledge into models and algorithms for image…
Efficient trajectory optimization is essential for avoiding collisions in unstructured environments, but it remains challenging to have both speed and quality in the solutions. One reason is that second-order optimality requires calculating…
This paper presents an approximation algorithm for finding a shortest path between two points $s$ and $t$ in a weighted planar subdivision $\PS$. Each face $f$ of $\PS$ is associated with a weight $w_f$, and the cost of travel along a line…
We introduce the parametric matroid one-interdiction problem. Given a matroid, each element of its ground set is associated with a weight that depends linearly on a real parameter from a given parameter interval. The goal is to find, for…
We establish new bounds on the number of tangencies and orthogonal intersections determined by an arrangement of curves. First, given a set of $n$ algebraic plane curves, we show that there are $O(n^{3/2})$ points where two or more curves…
The order in which plane-filling curves visit points in the plane can be exploited to design efficient algorithms. Typically, the curves are useful because they preserve locality: points that are close to each other along the curve tend to…
An obstacle representation of a graph is a mapping of the vertices onto points in the plane and a set of connected regions of the plane (called obstacles) such that the straight-line segment connecting the points corresponding to two…
Parametrized motion planning algorithms have high degrees of universality and flexibility, as they are designed to work under a variety of external conditions, which are viewed as parameters and form part of the input of the underlying…
This work develops feasible path trajectories for a coordinated strike with multiple aircraft in a constrained environment. Using direct orthogonal collocation methods, the two-point boundary value optimal control problem is transcribed…
Let $P$ be a set of $n$ points in the real plane contained in an algebraic curve $C$ of degree $d$. We prove that the number of distinct distances determined by $P$ is at least $c_d n^{4/3}$, unless $C$ contains a line or a circle. We also…
We consider the problem of identifying n points in the plane using disks, i.e., minimizing the number of disks so that each point is contained in a disk and no two points are in exactly the same set of disks. This problem can be seen as an…
This paper considers the collision avoidance problem in a multi-agent multi-obstacle framework. The originality in solving this intensively studied problem resides in the proposed geometrical view combined with differential flatness for…
We consider a variant of the clustering problem for a complete weighted graph. The aim is to partition the nodes into clusters maximizing the sum of the edge weights within the clusters. This problem is known as the clique partitioning…
If A and B are two points in the plane, with B lower and to the right of A, then we may consider the trajectory of an object travelling from A to B under the influence of gravity. The search for the trajectory minimising the time taken by…
This paper improves the algorithms based on supporting halfspaces and quadratic programming for convex set intersection problems in our earlier paper in several directions. First, we give conditions so that much smaller quadratic programs…
The decision to incorporate cross-validation into validation processes of mathematical models raises an immediate question - how should one partition the data into calibration and validation sets? We answer this question systematically: we…
We introduce a new type of distinct distances result: a lower bound on the number of distances between points on a line and points on a two-dimensional strip. This can be seen as a generalization of the well-studied problems of distances…
We introduce an alternative approach for constrained mathematical programming problems. It rests on two main aspects: an efficient way to compute optimal solutions for unconstrained problems, and multipliers regarded as variables for a…
Finding an optimal alignment connecting two end-points in a specified corridor is a complex problem that requires solving three interrelated sub-problems, namely the horizontal alignment, vertical alignment and earthwork optimization…