Geometric Set Cover Research Articles

On the geometric priority set cover problem

We study the priority set cover problem for simple geometric set systems in the plane. For pseudo-halfspaces in the plane we obtain a PTAS via local search by showing that the corresponding set system admits a planar support. We show that the problem is APX-hard even for unit disks in the plane and argue that in this case the standard local search algorithm can output a solution that is arbitrarily bad compared to the optimal solution. We then present an LP-relative constant factor approximation algorithm (which also works in the weighted setting) for unit disks via quasi-uniform sampling. As a consequence we obtain a constant factor approximation for the capacitated set cover problem with unit disks. For arbitrary size disks, we show that the problem is at least as hard as the vertex cover problem in general graphs even when the disks have nearly equal sizes. We also present a few simple results for unit squares and orthants in the plane.

Dynamic Geometric Set Cover and Hitting Set

We investigate dynamic versions of geometric set cover and hitting set where points and ranges may be inserted or deleted, and we want to efficiently maintain an (approximately) optimal solution for the current problem instance. While their static versions have been extensively studied in the past, surprisingly little is known about dynamic geometric set cover and hitting set. For instance, even for the most basic case of one-dimensional interval set cover and hitting set, no nontrivial results were known. The main contribution of our article are two frameworks that lead to efficient data structures for dynamically maintaining set covers and hitting sets in ℝ 1 and ℝ 2 . The first framework uses bootstrapping and gives a (1 + ε)-approximate data structure for dynamic interval set cover in ℝ 1 with O ( n α / ε) amortized update time for any constant α > 0; in ℝ 2 , this method gives O (1)-approximate data structures for unit-square set cover and hitting set with O ( n 1/2+α ) amortized update time. The second framework uses local modification and leads to a (1 + ε)-approximate data structure for dynamic interval hitting set in ℝ 1 with Õ(1/ε) amortized update time; in ℝ 2 , it gives O (1)-approximate data structures for unit-square set cover and hitting set in the partially dynamic settings with Õ(1) amortized update time.

Weighted geometric set cover with rectangles of bounded integer side lengths

In this article, we study the gradation of the complexity of the weighted set cover problem with axis-parallel rectangles whose side lengths are bounded integers. We show that the mod-one method of Chan and Hu (2015) for unit squares can be extended to these objects to get a polynomial-time approximation scheme (PTAS). We further show that the problem has a polynomial-time algorithm when all rectangles intersect a given horizontal line. On the contrary, we show that even the unweighted version of the problem is NP-hard when every rectangle intersects at least one of two given horizontal lines at the unit vertical distance.

Quasi-Polynomial Time Approximation Schemes for Packing and Covering Problems in Planar Graphs

We consider two optimization problems in planar graphs. In Maximum Weight Independent Set of Objects we are given a graph G and a family $${\mathcal {D}}$$ of objects, each being a connected subgraph of G with a prescribed weight, and the task is to find a maximum-weight subfamily of $${\mathcal {D}}$$ consisting of pairwise disjoint objects. In Minimum Weight Distance Set Cover we are given a graph G in which the edges might have different lengths, two sets $${\mathcal {D}},{\mathcal {C}}$$ of vertices of G, where vertices of $${\mathcal {D}}$$ have prescribed weights, and a nonnegative radius r. The task is to find a minimum-weight subset of $${\mathcal {D}}$$ such that every vertex of $${\mathcal {C}}$$ is at distance at most r from some selected vertex. Via simple reductions, these two problems generalize a number of geometric optimization tasks, notably Maximum Weight Independent Set for polygons in the plane and Weighted Geometric Set Cover for unit disks and unit squares. We present quasi-polynomial time approximation schemes (QPTASs) for both of the above problems in planar graphs: given an accuracy parameter $$\epsilon >0$$ we can compute a solution whose weight is within multiplicative factor of $$(1+\epsilon )$$ from the optimum in time $$2^{{\mathrm {poly}}(1/\epsilon ,\log |{\mathcal {D}}|)}\cdot n^{{\mathcal {O}}(1)}$$, where n is the number of vertices of the input graph. We note that a QPTAS for Maximum Weight Independent Set of Objects would follow from existing work. However, our main contribution is to provide a unified framework that works for both problems in both a planar and geometric setting and to transfer the techniques used for recursive approximation schemes for geometric problems due to Adamaszek and Wiese (in Proceedings of the FOCS 2013, IEEE, 2013; in Proceedings of the SODA 2014, SIAM, 2014) and Har-Peled and Sariel (in Proceedings of the SOCG 2014, SIAM, 2014) to the setting of planar graphs. In particular, this yields a purely combinatorial viewpoint on these methods as a phenomenon in planar graphs.

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.

Formula omitted]-hardness of geometric set cover and hitting set with rectangles containing a common point

We prove that both set cover and hitting set problems with points and axis-parallel rectangles are NP-hard even when (i) all rectangles share a common point and (ii) there are only two types of rectangles a×b and b×a for some positive integers a and b.

( $$\delta ,{\varepsilon }$$ 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.

Decentralized Message Passing Algorithm for Distributed Minimum Sensor Cover

This paper addresses a decentralized message passing approach for a sensor placement problem in a continuous plane, named minimum sensor cover problem: given pointwise tasks and omnidirectional coverage of sensors deployable by distributed local planners, find the minimum set of sensors that cover all the tasks. Because the solution is required to be derived in the local planners’ network, a decentralized message passing algorithm is proposed with a problem conversion process into a discrete domain for applying the algorithm. By using discretization of the solution space, the minimum sensor cover problem can be converted to a combinatorial optimization (i.e., the geometric set cover), and then it is formulated as a maximum a posteriori state assignment problem. Belief propagation is a decentralized algorithm based on local iterative message passing to solve the maximum a posteriori state assignment problem, but it has convergence and feasibility issues. Therefore, the proposed algorithm is modified from the belief propagation to obtain stable and guaranteed feasible solutions, and its time complexity in a decentralized computation is analyzed. Numerical simulations validate the convergence, feasibility, and preferable solution quality of the proposed algorithm against existing variants of belief propagation and the centralized greedy algorithm for the set cover.

Multivariate Complexity Analysis of Geometric Red Blue Set Cover

We investigate the parameterized complexity of Generalized Red Blue Set Cover (Gen-RBSC), a generalization of the classic Set Cover problem and the more recently studied Red Blue Set Cover problem. Given a universe U containing b blue elements and r red elements, positive integers \(k_\ell \) and \(k_r\), and a family \(\mathcal F \) of \(\ell \) sets over U, the Gen-RBSC problem is to decide whether there is a subfamily \(\mathcal F '\subseteq \mathcal F \) of size at most \(k_\ell \) that covers all blue elements, but at most \(k_r\) of the red elements. This generalizes Set Cover and thus in full generality it is intractable in the parameterized setting. In this paper, we study a geometric version of this problem, called Gen-RBSC-lines, where the elements are points in the plane and sets are defined by lines. We study this problem for an array of parameters, namely, \(k_\ell , k_r, r, b\), and \(\ell \), and all possible combinations of them. For all these cases, we either prove that the problem is W-hard or show that the problem is fixed parameter tractable (FPT). In particular, on the algorithmic side, our study shows that a combination of \(k_\ell \) and \(k_r\) gives rise to a nontrivial algorithm for Gen-RBSC-lines. On the hardness side, we show that the problem is para-NP-hard when parameterized by \(k_r\), and W[1]-hard when parameterized by \(k_\ell \). Finally, for the combination of parameters for which Gen-RBSC-lines admits FPT algorithms, we ask for the existence of polynomial kernels. We are able to provide a complete kernelization dichotomy by either showing that the problem admits a polynomial kernel or that it does not contain a polynomial kernel unless \(\text {co{-}NP}\subseteq \text {NP}/\text{ poly }\).

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.

The Geometry of Scheduling

We consider the following general scheduling problem. The input consists of $n$ jobs, each with an arbitrary release time, size, and monotone function specifying the cost incurred when the job is completed at a particular time. The objective is to find a preemptive schedule of minimum aggregate cost. This problem formulation is general enough to include many natural scheduling objectives, such as total weighted flow time, total weighted tardiness, and sum of flow time squared. We give an $O(\log \log P )$ approximation for this problem, where $P$ is the ratio of the maximum to minimum job size. We also give an $O(1)$ approximation in the special case of identical release times. These results are obtained by reducing the scheduling problem to a geometric capacitated set cover problem in two dimensions.

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.

Exact algorithms and APX-hardness results for geometric packing and covering problems

We study several geometric set cover and set packing problems involving configurations of points and geometric objects in Euclidean space. We show that it is APX-hard to compute a minimum cover of a set of points in the plane by a family of axis-aligned fat rectangles, even when each rectangle is an ϵ-perturbed copy of a single unit square. We extend this result to several other classes of objects including almost-circular ellipses, axis-aligned slabs, downward shadows of line segments, downward shadows of graphs of cubic functions, fat semi-infinite wedges, 3-dimensional unit balls, and axis-aligned cubes, as well as some related hitting set problems. We also prove the APX-hardness of a related family of discrete set packing problems. Our hardness results are all proven by encoding a highly structured minimum vertex cover problem which we believe may be of independent interest.In contrast, we give a polynomial-time dynamic programming algorithm for geometric set cover where the objects are pseudodisks containing the origin or are downward shadows of pairwise 2-intersecting x-monotone curves. Our algorithm extends to the weighted case where a minimum-cost cover is required. We give similar algorithms for several related hitting set and discrete packing problems.

Guest Editorial: Computing and Combinatorics

“Computing and Combinatorics” is an interesting and fundamental research area, which involves several fields, ranging from algorithms, theory of computation, and computational complexity to combinatorics related to computing. This issue consists of six papers, which are briefly discussed as follows. The “Shorthand Universal Cycles for Permutations” paper investigates SP-cycles with maximum and minimum ’weight’. The authors show that periodic min-weight SP-cycles correspond to spanning trees of the (n− 1) permutohedron and provides two constructions by using ’half-hunts’ from bell-ringing and the cool-lex order. In “Zero-Knowledge Argument for Simultaneous Discrete Logarithms,” the authors present an EQDL (the equality of two discrete logarithms) protocol in the ROM (random oracle model) which saves approximately 40 % of the computational cost and approximately 33 % of the prover’s outgoing message size. This improvement benefits a variety of interesting cryptosystems, ranging from signatures and anonymous credential systems, to verifiable secret sharing and threshold cryptosystems. The issue also includes the study of computational complexity. Namely, the paper “Tile-Packing Tomography is NP-hard” shows that for a fixed tile, it is NP-hard to reconstruct its tilings from their projections for all types of tiles. The #NC1 upper bound for the problem of counting accepting paths in any fixed visibly pushdown automaton is considered in “Counting paths in VPA is complete for #NC1”. The paper also shows that the difference between #BWBP and #NC1 is captured exactly by the addition of a visible stack to a nondeterministic finite-state automaton. Due to the NP-hardness of the problems, approximation algorithms are needed to provide a near-optimal solution. In “Exact and Approximation Algorithms for Geometric and General Capacitated Set Cover Problems”, the first known polynomial-

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.

Exact and Approximation Algorithms for Geometric and Capacitated Set Cover Problems

First, we study geometric variants of the standard set cover motivated by assignment of directional antenna and shipping with deadlines, providing the first known polynomial-time exact solutions. Next, we consider the following general (non-necessarily geometric) capacitated set cover problem. There is given a set of elements with real weights and a family of sets of the elements. One can use a set if it is a subset of one of the sets in the family and the sum of the weights of its elements is at most one. The goal is to cover all the elements with the allowed sets. We show that any polynomial-time algorithm that approximates the uncapacitated version of the set cover problem with ratio r can be converted to an approximation algorithm for the capacitated version with ratio r+1.357. In particular, the composition of these two results yields a polynomial-time approximation algorithm for the problem of covering a set of customers represented by a weighted n-point set with a minimum number of antennas of variable angular range and fixed capacity with ratio 2.357. This substantially improves on the best known approximation ratio for the latter antenna problem equal to 3. Furthermore, we provide a PTAS for the dual problem where the number of sets (e.g., antennas) to use is fixed and the task is to minimize the maximum set load, in case the sets correspond to line intervals or arcs. Finally, we discuss the approximability of the generalization of the antenna problem to include several base stations for antennas, and in particular show its APX-hardness already in the uncapacitated case.

Maximising lifetime for fault-tolerant target coverage in sensor networks

Abstract We study the problem of maximising the lifetime of a sensor network for fault-tolerant target coverage in a setting with composite events. Here, a composite event is the simultaneous occurrence of a combination of atomic events, such as the detection of smoke and high temperature. We are given sensor nodes that have an initial battery level and can monitor certain event types, and a set of points at which composite events need to be detected. The points and sensor nodes are located in the Euclidean plane, and all nodes have the same sensing radius. The goal is to compute a longest activity schedule with the property that at any point in time, each event point is monitored by at least two active sensor nodes. We present a (6 + ɛ)-approximation algorithm for this problem by devising an approximation algorithm with the same ratio for the dual problem of minimising the weight of a fault-tolerant sensor cover. The algorithm generalises previous approximation algorithms for geometric set cover with weighted unit disks and is obtained by enumerating properties of the optimal solution that guide a dynamic programming approach.

Improved Approximation Algorithms for Geometric Set Cover

Given a collection S of subsets of some set ${\Bbb U},$ and ${\Bbb M}\subset{\Bbb U},$ the set cover problem is to find the smallest subcollection $C\subset S$ that covers ${\Bbb M},$ that is, ${\Bbb M} \subseteq \bigcup (C),$ where $\bigcup(C)$ denotes $\bigcup_{Y \in C} Y.$ We assume of course that S covers ${\Bbb M}.$ While the general problem is NP-hard to solve, even approximately, here we consider some geometric special cases, where usually ${\Bbb U} = {\Bbb R}^d.$ Combining previously known techniques [4], [5], we show that polynomial-time approximation algorithms with provable performance exist, under a certain general condition: that for a random subset $R\subset S$ and nondecreasing function f(·), there is a decomposition of the complement ${\Bbb U}\backslash\bigcup (R)$ into an expected at most f(|R|) regions, each region of a particular simple form. Under this condition, a cover of size O(f(|C|)) can be found in polynomial time. Using this result, and combinatorial geometry results implying bounding functions f(c) that are nearly linear, we obtain o(log c) approximation algorithms for covering by fat triangles, by pseudo-disks, by a family of fat objects, and others. Similarly, constant-factor approximations follow for similar-sized fat triangles and fat objects, and for fat wedges. With more work, we obtain constant-factor approximation algorithms for covering by unit cubes in ${\Bbb R}^3,$ and for guarding an x-monotone polygonal chain.