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Register a new userFaster Approximation Algorithms for Geometric Set Cover
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Added by
Qizheng He
Publication date
2020
fields
Informatics Engineering
and research's language is
English
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We improve the running times of $O(1)$-approximation algorithms for the set cover problem in geometric settings, specifically, covering points by disks in the plane, or covering points by halfspaces in three dimensions. In the unweighted case, Agarwal and Pan [SoCG 2014] gave a randomized $O(nlog^4 n)$-time, $O(1)$-approximation algorithm, by using variants of the multiplicative weight update (MWU) method combined with geometric data structures. We simplify the data structure requirement in one of their methods and obtain a deterministic $O(nlog^3 nloglog n)$-time algorithm. With further new ideas, we obtain a still faster randomized $O(nlog n(loglog n)^{O(1)})$-time algorithm. For the weighted problem, we also give a randomized $O(nlog^4nloglog n)$-time, $O(1)$-approximation algorithm, by simple modifications to the MWU method and the quasi-uniform sampling technique.
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In this paper, we study two classic optimization problems: minimum geometric dominating set and set cover. Both the problems have been studied for different types of objects for a long time. These problems become APX-hard when the objects are axis-parallel rectangles, ellipses, $alpha$-fat objects of constant description complexity, and convex polygons. On the other hand, PTAS (polynomial time approximation scheme) is known for them when the objects are disks or unit squares. Surprisingly, PTAS was unknown even for arbitrary squares. For homothetic set of convex objects, an $O(k^4)$ approximation algorithm is known for dominating set problem, where $k$ is the number of corners in a convex object. On the other hand, QPTAS (quasi polynomial time approximation scheme) is known very recently for the covering problem when the objects are pseudodisks. For both problems obtaining a PTAS remains open for a large class of objects. For the dominating set problems, we prove that the popular local search algorithm leads to an $(1+varepsilon)$ approximation when objects are homothetic set of convex objects (which includes arbitrary squares, $k$-regular polygons, translated and scaled copies of a convex set etc.) in $n^{O(1/varepsilon^2)}$ time. On the other hand, the same technique leads to a PTAS for geometric covering problem when the objects are convex pseudodisks (which includes disks, unit height rectangles, homothetic convex objects etc.). As a consequence, we obtain an easy to implement approximation algorithm for both problems for a large class of objects, significantly improving the best known approximation guarantees.
Maximum independent set from a given set $D$ of unit disks intersecting a horizontal line can be solved in $O(n^2)$ time and $O(n^2)$ space. As a corollary, we design a factor 2 approximation algorithm for the maximum independent set problem on unit disk graph which takes both time and space of $O(n^2)$. The best known factor 2 approximation algorithm for this problem runs in $O(n^2 log n)$ time and takes $O(n^2)$ space [Jallu and Das 2016, Das et al. 2016].
Given $n$ points in a $d$ dimensional Euclidean space, the Minimum Enclosing Ball (MEB) problem is to find the ball with the smallest radius which contains all $n$ points. We give a $O(ndQcal/sqrt{epsilon})$ approximation algorithm for producing an enclosing ball whose radius is at most $epsilon$ away from the optimum (where $Qcal$ is an upper bound on the norm of the points). This improves existing results using emph{coresets}, which yield a $O(nd/epsilon)$ greedy algorithm. Finding the Minimum Enclosing Convex Polytope (MECP) is a related problem wherein a convex polytope of a fixed shape is given and the aim is to find the smallest magnification of the polytope which encloses the given points. For this problem we present a $O(mndQcal/epsilon)$ approximation algorithm, where $m$ is the number of faces of the polytope. Our algorithms borrow heavily from convex duality and recently developed techniques in non-smooth optimization, and are in contrast with existing methods which rely on geometric arguments. In particular, we specialize the excessive gap framework of citet{Nesterov05a} to obtain our results.
We study geometric set cover problems in dynamic settings, allowing insertions and deletions of points and objects. We present the first dynamic data structure that can maintain an $O(1)$-approximation in sublinear update time for set cover for axis-aligned squares in 2D. More precisely, we obtain randomized update time $O(n^{2/3+delta})$ for an arbitrarily small constant $delta>0$. Previously, a dynamic geometric set cover data structure with sublinear update time was known only for unit squares by Agarwal, Chang, Suri, Xiao, and Xue [SoCG 2020]. If only an approximate size of the solution is needed, then we can also obtain sublinear amortized update time for disks in 2D and halfspaces in 3D. As a byproduct, our techniques for dynamic set cover also yield an optimal randomized $O(nlog n)$-time algorithm for static set cover for 2D disks and 3D halfspaces, improving our earlier $O(nlog n(loglog n)^{O(1)})$ result [SoCG 2020].
We consider the problem of maintaining an approximate maximum independent set of geometric objects under insertions and deletions. We present data structures that maintain a constant-factor approximate maximum independent set for broad classes of fat objects in $d$ dimensions, where $d$ is assumed to be a constant, in sublinear textit{worst-case} update time. This gives the first results for dynamic independent set in a wide variety of geometric settings, such as disks, fat polygons, and their high-dimensional equivalents. For axis-aligned squares and hypercubes, our result improves upon all (recently announced) previous works. We obtain, in particular, a dynamic $(4+epsilon)$-approximation for squares, with $O(log^4 n)$ worst-case update time. Our result is obtained via a two-level approach. First, we develop a dynamic data structure which stores all objects and provides an approximate independent set when queried, with output-sensitive running time. We show that via standard methods such a structure can be used to obtain a dynamic algorithm with textit{amortized} update time bounds. Then, to obtain worst-case update time algorithms, we develop a generic deamortization scheme that with each insertion/deletion keeps (i) the update time bounded and (ii) the number of changes in the independent set constant. We show that such a scheme is applicable to fat objects by showing an appropriate generalization of a separator theorem. Interestingly, we show that our deamortization scheme is also necessary in order to obtain worst-case update bounds: If for a class of objects our scheme is not applicable, then no constant-factor approximation with sublinear worst-case update time is possible. We show that such a lower bound applies even for seemingly simple classes of geometric objects including axis-aligned rectangles in the plane.
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