Locating a Communication Path in a Competitive Scenario M. Abellanas ∗ J.M. D´ ıaz-B´a˜ nez † P. P´ erez-Lantero † I. Ventura † December 29, 2010 Abstract Consider a set of receptors belonging to two competitive telecommunication firms, the blue firm and the red firm. The receptors are represented as points in the plane, b are blue and belong to the blue firm and r are red and belong to the red firm. The blue firm has an emitting device represented as a point that moves along a path sending information to blue receptors as follows: At any time, the device sends information to all blue receptors covered by the largest disk centered at it and containing no red receptor. In this scenario, we study two optimization problems. The first problem is to compute a path P , such that the number of blue receptors served by a moving device is maximized. In particular, we give efficient algorithms when P is a straight line, an anchored half-line, and an axis-parallel double ray. As a second task, we study the problem of removing the minimum number of red receptors in such a way there exists a straight line path P so that if the device moves along P all blue receptors are served. We prove geometrical properties of an optimal straight line and propose efficient algorithms depending on the degrees of freedom of the line. Keywords: Extensive Facility Location, Geometric Optimization, Algorithms, Competitive Loca- tion. ∗ Dep. Matem´atica Aplicada, Universidad Polit´ ecnica de Madrid, mabellanas@fi.upm.es, partially supported by grant MTM2008-05043. † Dep. Matem´atica Aplicada II,Universidad de Sevilla, {dbanez,pplantero,iventura}@us.es, partially supported by grant MTM2009-08625. phone: 34-95-4482157; fax: 34-95-4486165. 1