Geometric Set Cover Problems Research Articles

On the VC-dimension of half-spaces with respect to convex sets

A family S of convex sets in the plane defines a hypergraph H = (S, E) as follows. Every subfamily S' of S defines a hyperedge of H if and only if there exists a halfspace h that fully contains S' , and no other set of S is fully contained in h. In this case, we say that h realizes S'. We say a set S is shattered, if all its subsets are realized. The VC-dimension of a hypergraph H is the size of the largest shattered set. We show that the VC-dimension for pairwise disjoint convex sets in the plane is bounded by 3, and this is tight. In contrast, we show the VC-dimension of convex sets in the plane (not necessarily disjoint) is unbounded. We provide a quadratic lower bound in the number of pairs of intersecting sets in a shattered family of convex sets in the plane. We also show that the VC-dimension is unbounded for pairwise disjoint convex sets in R^d , for d > 2. We focus on, possibly intersecting, segments in the plane and determine that the VC-dimension is always at most 5. And this is tight, as we construct a set of five segments that can be shattered. We give two exemplary applications. One for a geometric set cover problem and one for a range-query data structure problem, to motivate our findings.

An exact algorithm for a class of geometric set-cover problems

Given a set R of m disjoint finite regions in the 2-dimensional plane, all regions having polygonal boundaries, and given a set D of n discs with fixed centers and radii, we consider the problem of finding a minimum cardinality subset D∗⊆D such that every point in R is covered by at least one disc in D∗. We show that this problem can be solved by using an iterative procedure that alternates between the solution of a traditional set-cover problem and the construction of the Laguerre–Voronoi diagram of a circle set. Computer experiments demonstrate the effectiveness of the proposed algorithm, particularly when the number |D∗| of discs necessary to cover R is low.

A scanning strategy optimized for signal-to-noise ratio for the Geostationary Carbon Cycle Observatory (GeoCarb) instrument

Abstract. The Geostationary Carbon Cycle Observatory (GeoCarb) will make measurements of greenhouse gases over the contiguous North and South American landmasses at daily temporal resolution. The extreme flexibility of observing from geostationary orbit induces an optimization problem, as operators must choose what to observe and when. The proposed scanning strategy for the GeoCarb mission tracks the sun's path from east to west and covers the entire area of interest in five coherent regions in the order of tropical South America east, tropical South America west, temperate South America, tropical North America, and temperate North America. We express this problem in terms of a geometric set cover problem, and use an incremental optimization (IO) algorithm to create a scanning strategy that minimizes expected error as a function of the signal-to-noise ratio (SNR). The IO algorithm used in this studied is a modified greedy algorithm that selects, incrementally at 5 min intervals, the scanning areas with the highest predicted SNR with respect to air mass factor (AF) and solar zenith angle (SZA) while also considering operational constraints to minimize overlapping scans and observations over water. As a proof of concept, two experiments are performed applying the IO algorithm offline to create an SNR-optimized strategy and compare it to the proposed strategy. The first experiment considers all landmasses with equal importance and the second experiment illustrates a temporary campaign mode that gives major urban areas greater importance weighting. Using a simple instrument model, we found that there is a significant performance increase with respect to overall predicted error when comparing the algorithm-selected scanning strategies to the proposed scanning strategy.

( $$\delta ,{\varepsilon }$$ δ , ε )-Ball Approximation of a Shape: Definition and Complexity

Given a set S in \(\mathbb {R}^n\), a \((\delta ,\varepsilon )\)-ball approximation of S is defined as a collection of balls that covers the morphological erosion of S (by a ball of radius \(\varepsilon \)) and remains inside the morphological dilation of S (by a ball of radius \(\delta \)). We study the problem of computing a \((\delta ,\varepsilon )\)-ball approximation when S is itself defined as a finite union of balls. This problem relates to geometric set cover problems but is however different in nature. It offers a new framework for reducing the size of a collection of balls while controlling both the inner and outer distance to the shape. We prove that computing a \((\delta ,\varepsilon )\)-ball approximation of minimum cardinality is NP-complete for \(n = 2\). Along the way, we study the boundary of unions of disks and their erosion, for which we derive a computational description.

Quasi-Polynomial Time Approximation Scheme for Weighted Geometric Set Cover on Pseudodisks and Halfspaces

Weighted geometric set-cover problems arise naturally in several geometric and nongeometric settings (e.g., the breakthrough of Bansal and Pruhs [Proceedings of FOCS, 2010, pp. 407--414] reduces a wide class of machine scheduling problems to weighted geometric set cover). More than two decades of research has succeeded in settling the $(1+\epsilon)$-approximability status for most geometric set-cover problems, except for some basic scenarios which are still lacking. One is that of weighted disks in the plane for which, after a series of papers, Varadarajan [Proceedings of STOC'10, 2010, pp. 641--648] presented a clever quasi-sampling technique, which together with improvements by Chan et al. [Proceedings of SODA, 2012, pp. 1576--1585], yielded an $O(1)$-approximation algorithm. Even for the unweighted case, a polynomial time approximation scheme (PTAS) for a fundamental class of objects called pseudodisks (which includes halfspaces, disks, unit-height rectangles, translates of convex sets, etc.) is curren...

Geometric red–blue set cover for unit squares and related problems

We study a geometric version of the Red–Blue Set Cover problem originally proposed by Carr et al. (2000) [1]: given a red point set, a blue point set, and a set of objects, we want to choose a subset of objects to cover all the blue points, while minimizing the number of red points covered. We prove that the problem is NP-hard even when the objects are unit squares in 2D, and we give the first polynomial-time approximation scheme (PTAS) for this case. The technique we use simplifies and unifies previous PTASs for the weighted geometric set cover problem and the unique maximum coverage problem for 2D unit squares.

Approximability issues of guarding a set of segments

In this paper we consider the following problem: given a finite set of straight-line segments S in ℝ2, find minimum in size set V of points on the segments, such that each segment of S contains at least one point in V. We call this problem guarding a set of segments (GSS). GSS is a special case of the set cover problem where the given family of subsets can be taken as a set of intersections of the straight-line segments in S. Requiring that the given subsets can be interpreted geometrically this way is a major restriction on the input, yet it has been shown that the problem is still strongly NP-complete [V.E. Brimkov, A. Leach, M. Mastroianni, and J. Wu, Guarding a set of line segments in the plane, Theor. Comput. Sci. 412 (2011), pp. 1313–1324]. In light of this result, in Brimkov et al. [Experimental studies on approximation algorithms for guarding sets of line segments, in Advances in Visual Computing, G. Bebis, R. Boyle, B. Parvin, D. Koracin, R. Chung, R. Hammoud, M. Hussain, T. Kar-Han, R. Crawfis, D. Thalmann, D. Kao, and L. Avila, eds., ISVC 2010, Part I, Lecture Notes in Computer Science, Vol. 6453, Springer, Berlin, 2010, pp. 592–601; V.E. Brimkov, A. Leach, M. Mastroianni, and J. Wu, Approximation algorithms for a geometric set cover problem, Discrete Appl. Math. 160 (2012), pp. 1039–1052] the GSS approximability was studied both theoretically and experimentally. Here we continue these investigations. In particular, we obtain conditions under which GSS admits good approximation.

Approximation algorithms for a geometric set cover problem

Given a finite set of straight line segments S in R2 and some k∈N, is there a subset V of points on segments in S with |V|≤k such that each segment of S contains at least one point in V? This is a special case of the set covering problem where the family of subsets given can be taken as a set of intersections of the straight line segments in S. Requiring that the given subsets can be interpreted geometrically this way is a major restriction on the input, yet we have shown that the problem is still strongly NP-complete [5]. In light of this result, we studied the accuracy of two polynomial-time approximation algorithms along with a third heuristic based algorithm which return segment coverings. We obtain certain theoretical results, and in particular we show that the performance ratio for each of these algorithms is unbounded, in general. Despite this, our experimental results suggest that the cases where this ratio exceeds 2 are rare for “reasonable” methods of placing segments. This is evidence that these polynomial-time algorithms solve the problem accurately in practice.