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

Minimum Perimeter-Sum Partitions in the Plane

Let P be a set of n points in the plane. We consider the problem of partitioning P into two subsets P_1 and P_2 such that the sum of the perimeters of CH(P_1) and CH(P_2) is minimized, where CH(P_i) denotes the convex hull of P_i. The problem was first studied by Mitchell and Wynters in 1991 who gave an O(n^2) time algorithm. Despite considerable progress on related problems, no subquadratic time algorithm for this problem was found so far. We present an exact algorithm solving the problem in O(n log^4 n) time and a (1+e)-approximation algorithm running in O(n + 1/e^2 log^4(1/e)) time

Clustering with Few Disks to Minimize the Sum of Radii

Given a set of $n$ points in the Euclidean plane, the $k$-MinSumRadius problem asks to cover this point set using $k$ disks with the objective of minimizing the sum of the radii of the disks. After a long line of research on related problems, it was finally discovered that this problem admits a polynomial time algorithm [GKKPV~'12]; however, the running time of this algorithm is $O(n^{881})$, and its relevance is thereby mostly of theoretical nature. A practically and structurally interesting special case of the $k$-MinSumRadius problem is that of small $k$. For the $2$-MinSumRadius problem, a near-quadratic time algorithm with expected running time $O(n^2 \log^2 n \log^2 \log n)$ was given over 30 years ago [Eppstein~'92]. We present the first improvement of this result, namely, a near-linear time algorithm to compute the $2$-MinSumRadius that runs in expected $O(n \log^2 n \log^2 \log n)$ time. We generalize this result to any constant dimension $d$, for which we give an $O(n^{2-1/(\lceil d/2\rceil + 1) + \varepsilon})$ time algorithm. Additionally, we give a near-quadratic time algorithm for $3$-MinSumRadius in the plane that runs in expected $O(n^2 \log^2 n \log^2 \log n)$ time. All of these algorithms rely on insights that uncover a surprisingly simple structure of optimal solutions: we can specify a linear number of lines out of which one separates one of the clusters from the remaining clusters in an optimal solution