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On Guarding the Vertices of Rectilinear Domains

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On Guarding the Vertices of Rectilinear Domains View metadata, citation and similar papers at core.ac.uk brought to you by CORE provided by Elsevier - Publisher Connector Computational Geometry 39 (2008) 219–228 www.elsevier.com/locate/comgeo On guarding the vertices of rectilinear domains Matthew J. Katz ∗,1, Gabriel S. Roisman 2 Department of Computer Science, Ben-Gurion University of the Negev, Beer-Sheva 84105, Israel Received 26 July 2006; received in revised form 31 January 2007; accepted 11 February 2007 Available online 28 February 2007 Communicated by P. Agarwal Abstract We prove that guarding the vertices of a rectilinear polygon P , whether by guards lying at vertices of P , or by guards lying on the boundary of P , or by guards lying anywhere in P , is NP-hard. For the first two proofs (i.e., vertex guards and boundary guards), we construct a reduction from minimum piercing of 2-intervals. The third proof is somewhat simpler; it is obtained by adapting a known reduction from minimum line cover. We also consider the problem of guarding the vertices of a 1.5D rectilinear terrain. We establish an interesting connec- tion between this problem and the problem of computing a minimum clique cover in chordal graphs. This connection yields a 2-approximation algorithm for the guarding problem. © 2007 Elsevier B.V. All rights reserved. Keywords: Geometric optimization; Guarding; NP-hardness; Approximation algorithms 1. Introduction Problems dealing with visibility coverage are often called art-gallery problems. The “classical” art-gallery problem is to place guards in a polygonal region, such that every point in the region is visible to one (or more) of the guards. More formally, given a domain P , one needs to find a set G of points in P , of minimum cardinality, such that every point in P is seen by at least one of the points, called guards,inG. Often there are some restrictions on the location of the guards; e.g., guards may lie only at vertices (in which case they are called vertex guards). The classical art-gallery problem, where guards may lie anywhere in the polygon or only at vertices, is known to be NP-hard, even if the underlying domain is a simple polygon [1,21,26]. Moreover, Eidenbenz et al. [11,12] have shown that these problems are APX-hard. Schuchardt and Hecker [28] proved that these problems remain NP-hard if we restrict our attention to (simple) rectilinear polygons. Their proof is based on a reduction from 3SAT. In this paper we study two art-gallery problems. The first is the problem of guarding the vertices of a rectilinear polygon (GVRP) P . We consider three versions of this problem. In the first version guards may lie anywhere on * Corresponding author. E-mail addresses: [email protected] (M.J. Katz), [email protected] (G.S. Roisman). 1 Partially supported by grant no. 2000160 from the US–Israel Binational Science Foundation. 2 Partially supported by the Lynn and William Frankel Center for Computer Sciences. 0925-7721/$ – see front matter © 2007 Elsevier B.V. All rights reserved. doi:10.1016/j.comgeo.2007.02.002 220 M.J. Katz, G.S. Roisman / Computational Geometry 39 (2008) 219–228 the boundary of P but not in the interior of P , in the second version guards may lie only at vertices of P , and in the third version guards may lie anywhere in P . We prove that despite the weaker requirement (i.e., only the vertices of P must be guarded), the status of the problems does not change and all three versions remain NP-hard. For the first two proofs (i.e., boundary guards and vertex guards), we construct a reduction from minimum piercing of 2-intervals, where a 2-interval is the union of two disjoint line-segments on the real line. For the third proof, we construct a reduction from minimum line cover. (Note that minimum line cover has been used previously in hardness proofs for art-gallery problems by, e.g., Brodén et al. [4] and Joseph Mitchell. However, in order to use it in our setting, one needs sophisticated gadgets.) The second problem that we study is that of guarding the vertices of a 1.5D rectilinear terrain. (A 1.5D rectilinear terrain is defined by an x-monotone chain T of horizontal and vertical line segments; two vertices u, v of T see each other, if the line segment uv does not pass below T .) We establish an interesting connection between this problem and the problem of computing a minimum clique cover in chordal graphs (see below for the definition of chordal graph). This connection yields a 2-approximation algorithm for the guarding problem. Ben-Moshe et al. [2] presented a constant-factor approximation algorithm for computing a set of guards for a 1.5D terrain that is defined by a strictly x-monotone polygonal chain. Their algorithm, however, cannot be applied (at least not immediately) to a 1.5D rectilinear terrain, since strict x-monotonicity is necessary at several places in their work. Moreover, the constant of approximation of their algorithm, as well as of the subsequent, purely theoretical, algorithm of Clarkson and Varadarajan [8], is big. Very recently King [19] gave a 4-approximation algorithm for minimum guarding of a 1.5D terrain. Again, strict x-monotonicity of the terrain is assumed. We also note that the idea of using perfect graph theory in the context of guarding is not new; see, e.g., [23]. More related work. Combinatorial art-gallery problems have been studied for three decades; see, e.g., [18,25,27,29] for surveys. The classical combinatorial result, the “art gallery theorem”, states that n/3 guards are sufficient and sometimes necessary to guard an n-vertex simple polygon [7]. Combinatorial results on the number of guards needed for various forms of guarding on terrains are given in [3]. Researches have mostly concentrated on obtaining good approximations. Ghosh [15] gave an O(log n)-approxima- tion for optimal guarding of a polygon by vertex guards, based on standard set cover results. Recent work [10,16] has focused on methods that efficiently apply the Brönnimann–Goodrich technique [5]. Efrat and Har-Peled [10] obtain an O(log k∗)-approximation algorithm for simple polygon guarding with vertex guards, using time O(n(k∗)2 log4 n), where k∗ is the optimal number of vertex guards. Their technique can be applied to non-vertex guards, lying at points of a dense grid, adding a factor polylogarithmic in the grid density to the time bound. (No approximation algorithm is known if the guards are completely unrestricted and every point in the polygon must be guarded.) Their results apply also to polygons with holes and to 2.5D terrains, still with polylogarithmic approximation factors. Very recently, Nilsson [24] presented a constant-factor approximation algorithm for guarding a monotone polygon. Using this algorithm, he also obtains an O((c∗)2)-algorithm for guarding a rectilinear polygon, where c∗ is the size of an optimal guarding set. Finally, for 1.5D terrains (i.e., for an x-monotone polygonal chain), Chen et al. [6] claim that by modifying the hardness proof of [26] one can show that the problem is NP-hard; details are omitted and are still to be verified. 2. GVRP is NP-hard In this section we show that all three versions of GVRP are NP-hard. We begin with the version where guards may lie only on the boundary of the polygon. 2.1. Guards may lie only on the boundary We show that if the guards are restricted to lie on the boundary of the polygon, then GVRP is NP-hard. We construct a reduction from minimum piercing of 2-intervals. 2.1.1. The 2-interval piercing problem A 2-interval o is the union of two line-segments ta and tb on the x-axis, that can be separated by a vertical slab of constant width c0. The minimum 2-interval piercing problem is defined as follows. Let O be a set of n 2-intervals. Find a set P of points on the x-axis, such that (i) for each 2-interval o ∈ O there exists a point p ∈ P that pierces o M.J. Katz, G.S. Roisman / Computational Geometry 39 (2008) 219–228 221 Fig. 1. Proof of Lemma 2.1: A reduction from vertex cover. Fig. 2. A d-gadget. (i.e., that lies in o), and (ii) P is as small as possible. Let D2IP denote the corresponding decision problem, that is, given O and an integer k>0, decide whether there exists a piercing set for O of cardinality k. For completeness we show that D2IP is NP-hard, although we suspect that it is well known. Lemma 2.1. D2IP is NP-Hard. Proof. We construct a reduction from the decision version of vertex cover; see Fig. 1. Given a graph G = (V, E), construct for each edge e = (u, v) ∈ E a 2-interval oe, such that its two line segments represent the two vertices u and v, respectively. Let O denote the set of 2-intervals that is obtained. It is clear that there exists a vertex cover of size k if and only if there exists a piercing set for O of size k. 2 2.1.2. Reduction from D2IP We first present the gadget that we shall use. We call it d-gadget (short for double gadget), see Fig. 2. Any guard below the line l is local. Some of the vertices of a d-gadget can only be guarded by a local guard (e.g., vertices x, y, and z).
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