Pablo Pérez-Lantero

Computing optimal islands

Let S be a bicolored set of n points in the plane. A subset I of S is an island if there is a convex set C such that I=C@?S. We give an O(n^3)-time algorithm to compute a monochromatic island of maximum cardinality. Our approach is adapted to optimize similar (decomposable) objective functions. Finally, we use our algorithm to give an O(logn)-approximation for the problem of computing the minimum number of convex polygons that cover a class region.

The Velocity Assignment Problem for Conflict Resolution with Multiple Aerial Vehicles Sharing Airspace

Efficient conflict resolution methods for multiple aerial vehicles sharing airspace are presented. The problem of assigning a velocity profile to each aerial vehicle in real time, such that the separation between them is greater than a given safety distance, is considered and the total deviation from the initial planned trajectory is minimized. The proposed methods involve the use of appropriate airspace discretization. In the paper it is demonstrated that this aerial vehicle velocity assignment problem is NP-hard. Then, the paper presents three different collision detection and resolution methods based on speed planning. The paper also presents simulations and studies for several scenarios.

Maximum-Weight Planar Boxes in O(n 2 ) Time (and Better)

Given a set P of n points in R d , where each point p of P is associated with a weight w(p) (positive or negative), the Maximum-Weight Box problem consists in nding an axis-aligned box B maximizing P p2B\P w(p). We describe algorithms for this problem in two dimensions that run in the worst case inO(n 2 ) time, and much less on more specic classes of instances. In particular, these results imply similar ones for the Maximum Bichromatic Discrepancy Box problem. These improve by a factor of (log

Bichromatic 2-center of pairs of points

We study a class of geometric optimization problems closely related to the 2-center problem: Given a set S of n pairs of points in the plane, for every pair, we want to assign red color to a point of the pair and blue color to the other point in order to optimize the radii of the minimum enclosing ball of the red points and the minimum enclosing ball of the blue points. In particular, we consider the problems of minimizing the maximum and minimizing the sum of the two radii of the minimum enclosing balls. For each case, minmax and minsum, we consider distances measured in the L 2 and in the L ∞ metrics.

New results on stabbing segments with a polygon

We consider a natural variation of the concept of stabbing a set of segments with a simple polygon: a segment s is stabbed by a simple polygon P if at least one endpoint of s is contained in P, and a segment set S is stabbed by P if P stabs every element of S. Given a segment set S, we study the problem of finding a simple polygon P stabbing S in a way that some measure of P (such as area or perimeter) is optimized. We show that if the elements of S are pairwise disjoint, the problem can be solved in polynomial time. In particular, this solves an open problem posed by Loffler and van Kreveld [Algorithmica 56(2), 236-269 (2010)] [16] about finding a maximum perimeter convex hull for a set of imprecise points modeled as line segments. Our algorithm can also be extended to work for a more general problem, in which instead of segments, the set S consists of a collection of point sets with pairwise disjoint convex hulls. We also prove that for general segments our stabbing problem is NP-hard.

The class cover problem with boxes

In this paper we study the following problem: Given sets R and B of r red and b blue points respectively in the plane, find a minimum-cardinality set H of axis-aligned rectangles (boxes) so that every point in B is covered by at least one rectangle of H, and no rectangle of H contains a point of R. We prove the NP-hardness of the stated problem, and give either exact or approximate algorithms depending on the type of rectangles considered. If the covering boxes are vertical or horizontal strips we give an efficient algorithm that runs in O(rlogr+blogb+rb) time. For covering with oriented half-strips an optimal O((r+b)log(min{r,b}))-time algorithm is shown. We prove that the problem remains NP-hard if the covering boxes are half-strips oriented in any of the four orientations, and show that there exists an O(1)-approximation algorithm. We also give an NP-hardness proof if the covering boxes are squares. In this situation, we show that there exists an O(1)-approximation algorithm.

Independent and Hitting Sets of Rectangles Intersecting a Diagonal Line: Algorithms and Complexity

Given a set of

The 1-median and 1-highway problem

n