Dolores Lara

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.

Covering moving points with anchored disks

Consider a set of mobile clients represented by n points in the plane moving at constant speed along n different straight lines. We study the problem of covering all mobile clients using a set of k disks centered at k fixed centers. Each disk exists only at one instant and while it does, covers any client within its coverage radius. The task is to select an activation time and a radius for each disk such that every mobile client is covered by at least one disk. In particular, we study the optimization problem of minimizing the maximum coverage radius. First we prove that, although the static version of the problem is polynomial, the kinetic version is NP-hard. Moreover, we show that the problem is not approximable by a constant factor (unless P=NP). We then present a generic framework to solve it for fixed values of k, which in turn allows us to solve more general optimization problems. Our algorithms are efficient for constant values of k.

On crossing families of complete geometric graphs

A crossing family is a collection of pairwise crossing segments, this concept was introduced by Aronov et al. [4]. They proved that any set of n points (in general position) in the plain contains a crossing family of size

On the coarseness of bicolored point sets

{\sqrt{n/12}}

On the connectedness and diameter of a Geometric Johnson Graph

n/12. In this paper we present a generalization of the concept and give several results regarding this generalization.

The Erd{\Ho}s-Faber-Lov\'asz conjecture for geometric graphs

Abstract We study the following combinatorial property of point sets in the plane: For a set S of n points in general position and a point p ∈ S consider the points of S − p in their angular order around p. This gives a star-shaped polygon (or a polygonal path) with p in its kernel. Define c ( p ) as the number of convex angles in this star-shaped polygon around p, and c ( S ) as the sum of all c ( p ) , for p ∈ S . We show that for every point set S, c ( S ) is always at least 1 2 n 3 2 − O ( n ) . This bound is shown to be almost tight. Consequently, every set of n points admits a star-shaped polygonization with at least n 2 − O ( 1 ) convex angles.