Minimum-Cost Coverage of Point Sets by Disks

We consider a class of geometric facility location problems in which the goal is to determine a set X of disks given by their centers (t_j) and radii (r_j) that cover a given set of demand points Y in the plane at the smallest possible cost. We consider cost functions of the form sum_j f(r_j), where f(r)=r^alpha is the cost of transmission to radius r. Special cases arise for alpha=1 (sum of radii) and alpha=2 (total area); power consumption models in wireless network design often use an exponent alpha>2. Different scenarios arise according to possible restrictions on the transmission centers t_j, which may be constrained to belong to a given discrete set or to lie on a line, etc. We obtain several new results, including (a) exact and approximation algorithms for selecting transmission points t_j on a given line in order to cover demand points Y in the plane; (b) approximation algorithms (and an algebraic intractability result) for selecting an optimal line on which to place transmission points to cover Y; (c) a proof of NP-hardness for a discrete set of transmission points in the plane and any fixed alpha>1; and (d) a polynomial-time approximation scheme for the problem of computing a minimum cost covering tour (MCCT), in which the total cost is a linear combination of the transmission cost for the set of disks and the length of a tour/path that connects the centers of the disks.Comment: 10 pages, 4 figures, Latex, to appear in ACM Symposium on Computational Geometry 200

A new proof of the Sylvester-Gallai theorem

problem: Given a finite collection of points in the affine plane, not all lying on a line, show that there exists a line which passes through precisely two of the points. Sylvester’s problem was reposed by Erdős in 1944 [4] and later that year a proof was given by Gallai [6]. Since then, many proofs of the Sylvester-Gallai Theorem have been found. Of these proofs, that given by Kelly (as communicated by Coxeter in [2] and [3]) and that attributed to Melchior (as implied in [7] 1) are particularly elegant. Kelly’s proof uses a simple distance argument while Melchior considers the dual collection of lines and applies Euler’s formula. For more extensive treatments of the Sylvester-Gallai Theorem and its relatives, see [1] and [5]. Given a collection of points, a line passing through just two of the points is commonly referred to as an ordinary line. As in Melchior [7], one can use projective duality to obtain a full

A faster dynamic programming algorithm for facility location

We study the following 1D and 1.5D problems: Given a set of m server locations {t1,...,tm}, lying along the x-axis and specified in increasing order, and n client locations {p1,..., pn}, also specified in increasing order along th

An Improved Bound for the Affine Sylvester Problem

... affine plane, given n lines, not all parallel and not all passing through a common point, there had to be at least n ordinary points. The present paper improves 6 on this result to show that there must be at least 2n−3 7 ordinary points, except for a single arrangement of 6 lines with one ordinary point