Piercing axis-parallel boxes

Let \F be a finite family of axis-parallel boxes in $\R^d$ such that \F contains no $k+1$ pairwise disjoint boxes. We prove that if \F contains a subfamily \M of $k$ pairwise disjoint boxes with the property that for every F\in \F and M\in \M with $F \cap M \neq \emptyset$, either $F$ contains a corner of $M$ or $M$ contains $2^{d-1}$ corners of $F$, then \F can be pierced by $O(k)$ points. One consequence of this result is that if $d=2$ and the ratio between any of the side lengths of any box is bounded by a constant, then \F can be pierced by $O(k)$ points. We further show that if for each two intersecting boxes in \F a corner of one is contained in the other, then \F can be pierced by at most $O(k\log\log(k))$ points, and in the special case where \F contains only cubes this bound improves to $O(k)$

Covering many points with a small-area box

Let $P$ be a set of $n$ points in the plane. We show how to find, for a given integer $k>0$, the smallest-area axis-parallel rectangle that covers $k$ points of $P$ in $O(nk^2 \log n+ n\log^2 n)$ time. We also consider the problem of, given a value $\alpha>0$, covering as many points of $P$ as possible with an axis-parallel rectangle of area at most $\alpha$. For this problem we give a probabilistic $(1-\varepsilon)$-approximation that works in near-linear time: In $O((n/\varepsilon^4)\log^3 n \log (1/\varepsilon))$ time we find an axis-parallel rectangle of area at most $\alpha$ that, with high probability, covers at least $(1-\varepsilon)\mathrm{\kappa^*}$ points, where $\mathrm{\kappa^*}$ is the maximum possible number of points that could be covered

Covering Points by Disjoint Boxes with Outliers

For a set of n points in the plane, we consider the axis--aligned (p,k)-Box Covering problem: Find p axis-aligned, pairwise-disjoint boxes that together contain n-k points. In this paper, we consider the boxes to be either squares or rectangles, and we want to minimize the area of the largest box. For general p we show that the problem is NP-hard for both squares and rectangles. For a small, fixed number p, we give algorithms that find the solution in the following running times: For squares we have O(n+k log k) time for p=1, and O(n log n+k^p log^p k time for p = 2,3. For rectangles we get O(n + k^3) for p = 1 and O(n log n+k^{2+p} log^{p-1} k) time for p = 2,3. In all cases, our algorithms use O(n) space.Comment: updated version: - changed problem from 'cover exactly n-k points' to 'cover at least n-k points' to avoid having non-feasible solutions. Results are unchanged. - added Proof to Lemma 11, clarified some sections - corrected typos and small errors - updated affiliations of two author

Approximation Algorithm for Line Segment Coverage for Wireless Sensor Network

The coverage problem in wireless sensor networks deals with the problem of covering a region or parts of it with sensors. In this paper, we address the problem of covering a set of line segments in sensor networks. A line segment ` is said to be covered if it intersects the sensing regions of at least one sensor distributed in that region. We show that the problem of finding the minimum number of sensors needed to cover each member in a given set of line segments in a rectangular area is NP-hard. Next, we propose a constant factor approximation algorithm for the problem of covering a set of axis-parallel line segments. We also show that a PTAS exists for this problem.Comment: 16 pages, 5 figures

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