We give a deterministic polynomial-time method for finding a set cover in a set system (X, ~') of dual VC-dimension d such that the size of our cover is at most a factor of O(d log(dc)) from the optimal size, c. For constant VCdimensional set systems, which are common in computational geometry, our method gives an O(logc) approximation factor. This improves the previous O(logl XI) bound of the greedy method and challenges recent complexity-theoretic lower bounds for set covers (which do not make any assumptions about the VC-dimension). We give several applications of our method to computational geometry, and we show that in some cases, such as those arising in threedimensional polytope approximation and two-dimensional disk covering, we can quickly find O(c)-sized covers. IntroductionA set system (X, ~q~) is a set X along with a collection ~q' of subsets of X, which are sometimes called ranges [25]. Such entities have also been called hypergraphs and range spaces in the computational geometry literature (e.g., see [5], [10]-[16], [20], [24], [25], [34]-[36], and [38]-[41]), and they can be used to model a number of interesting computational geometry problems.

We give a deterministic polynomial time method for finding a set cover in a set system (X, 7?) of VC-dimension d such that the size of our cover is at most a factor of 0 (d log (de)) from the optimal size, c. For constant VC-dimension set systems, which are common in computational geometry, our method gives an O (log c) approximation factor. This improves the previous @(log IX 1) bound of the greedy method and beats recent complexitytheoretic lower bounds for set covers (which don't make any assumptions about VC-dimension). We give several applications of our method to computational geometry, and we show that in some cases, such as those that arise in 3-d polytope approximation and 2-d disc covering, we can quickly find O(c) -sized covers.

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 ⊂ R 2 at the smallest possible cost. We consider cost functions of the form ∑ j f (r j ), where f (r) = r α is the cost of transmission to radius r. Special cases arise for α = 1 (sum of radii) and α = 2 (total area); power consumption models in wireless network design often use an exponent α > 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 ⊂ R 2 ; (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 R 2 and any fixed α > 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.

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