Computational complexity and approximability of guarding of proximity graphs? Konstantin Kobylkin Institute of Mathematics and Mechanics, Ural Branch of RAS, Sophya Kovalevskaya str. 16, 620990 Ekaterinburg, Russia, Ural Federal University, Mira str. 19, 620002 Ekaterinburg, Russia, [email protected] http://wwwrus.imm.uran.ru

Abstract. Computational complexity and approximability are studied for a problem of intersecting a set of straight line segments with the smallest cardinality set of disks of fixed radii r ≥ 0 where the set of segments forms a straight line drawing G = (V, E, F ) of a planar graph without edge crossings. This problem arises in network security applications (Agarwal et al., 2013). We claim strong NP-hardness of the problem within the class of (edge sets of) Delaunay, TD-Delaunay triangulations and their subgraphs for r ∈ [dmin , dmax ] and r ∈ [dmax , ηdmax ] for some large constant η as well as within the class of 4-connected TD-Delaunay triangulations for r = 0 where dmax and dmin are Euclidean lengths of the longest and shortest graph edges respectively. Fast O(|E| log |E|)-time O(1)-approximation algorithm is proposed within the class of straight line drawings of planar graphs for which the inequality r ≥ ηdmax holds uniformly for some constant η > 0. Keywords: computational complexity, approximation algorithms, Vertex Cover, Hitting Set, Continuous Disk Cover, Delaunay triangulations, proximity graphs

1

Introduction

Guarding the boundaries of geometric objects and complexes of plane embeddings of planar graphs is widely studied class of problems in computational geometry, see e.g. [7], [11], [21]. Usually in such problems one needs to find the smallest cardinality set C of guards (e.g. points on the plane) having bounded (in some sense) visibility area such that each piece of the boundary or part of the graph complex (e.g. edge or face) is within visibility area of some guard from C. Designing exact and approximate algorithms for these problems finds its applications in security, sensor placement, lighting and robotics. In this paper complexity and approximability of the following problem are studied. Intersecting Plane Graph with Disks (IPGD): given a straight line drawing of an arbitrary simple1 planar graph G = (V, E) without edge crossings and ? 1

This work was supported by Russian Science Foundation, project 14-11-00109. a graph without loops and parallel edges

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a constant r > 0, find the smallest cardinality set C ⊂ Q2 of points (disk centers) such that each edge e ∈ E is within (Euclidean) distance r from some point c = c(e) ∈ C or, equivalently, the disk of radius r centered at c intersects e. In the IPGD problem each guard has circular visibility area which does not depend on the obstacles in contrast to known Art Gallery problem [21] where sensor coverage area is affected by gallery walls. Applications of studying the IPGD problem come from network security [1]. Given a network of physical devices one needs to evaluate its vulnerability to simultaneous technical failures caused by natural (e.g. floods, fire, electromagnetic pulses) and human sources. Here network nodes are modeled by points on the plane while its physical links are given in the form of straight line segments. A catastrophic event (threat) is usually localized in a particular geographical area and modeled by a disk of some fixed radius r > 0. A threat impacts a network link when the corresponding disk and segment intersect. Evaluation of the network vulnerability may be expressed in the form of finding the minimum number of threats along with their positions that cause all network links to be broken. Thus, it brings us to the IPGD problem as usually network links are geographically non-overlapping. Below a triple G = (V, E, F ) is called a plane graph if V ⊂ R2 , the set E consists of nondegenerate straight line segments, crossing only at their endpoints which are from V. Here F denotes the set of all open (in R2 ) regions bounded by segments from E : each f ∈ F does not intersect with any segment from E. The only unbounded set f∞ ∈ F is named as the outer face whereas bounded ones from F are inner faces. A planar (plane) graph is called a planar (plane) triangulation if its face set F consists of triangles except for possibly outer one. In this paper computational complexity and approximability of IPGD are studied for simple plane graphs with either r ∈ [dmin , dmax ] or r = Ω(dmax ) or r = 0 where dmax and dmin are lengths of the longest and shortest edges of G. Our emphasis is on those classes of simple plane graphs that are defined by some distance function, namely, on Delaunay triangulations, some of their connected subgraphs (Gabriel, relative neighbourhood, nearest neighbour graphs, minimum Euclidean spanning trees) and TD-Delaunay triangulations. These graphs are often called proximity graphs. Delaunay and TD-Delaunay triangulations (being geometric spanners) are plane graphs which admit efficient geometric routing algorithms [6], thus, representing convenient network topologies. Gabriel and relative neighbourhood graphs arise in modeling wireless [8] and road networks. Related work. IPGD is related to several well-known combinatorial optimization problems. First, it is the special case of the Hitting Set problem. Hitting Set: given a family N of sets on the plane and a set U ⊆ Q2 , find the smallest cardinality set H ⊆ U such that N ∩ H 6= ∅ for every N ∈ N . IPGD coincides with Hitting Set if we set N := Nr (E) = {Nr (e)}e∈E and U := Q2 where Nr (e) = Br (0) + e = {x + y : x ∈ Br (0), y ∈ e} is Euclidean r-neighbourhood of e (having form of Minkowski sum) and Br (x) is the disk of radius r centered at x ∈ R2 . An aspect ratio of a closed convex set N with int N 6= ∅2 coincides with the ratio of the minimum radius of the disk which 2

int N is the set of interior points of N

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contains N to the maximum radius of the disk which is contained in N. For example, each object Nr (e) has the aspect ratio equal to 1 + d(e) 2r where d(e) is the length of the edge e. APX-hardness of the discrete3 Hitting Set problem is presented for families of (non-equal height and width) axis-parallel rectangles [10] and triangles of bounded aspect ratio [16]. Known Vertex Cover (VC) problem for planar graphs can be obtained from IPGD by setting r = 0. For the case of IPGD where G consists of isolated vertices (i.e. segments from E are all of zero length) we have the Continuous Disk Cover (CDC) problem. Strong NPhardness is well known for CDC [18] and VC problems (see e.g. [11], [19]). Results. Our results report complexity and approximation algorithms for the IPGD problem within several classes of plane graphs under different assumptions on r. Let S be a set of n points in general position on the plane no 4 of which are cocircular. We call a plane graph G = (V, E, F ) a Delaunay triangulation if V = S and [u, v] ∈ E iff there is a disk T such that u, v ∈ bd T 4 and S∩int T = ∅. Consider an oriented equilateral unit sided triangle ∇ whose barycenter is the origin and one of its vertices is on the negative y-axis. Denote ∇(p, λ) = p+λ∇ = {x ∈ R2 : x = p + λa, a ∈ ∇} for some p ∈ R2 and λ > 0. Suppose the set S is such that a straight line through any pair of its points does not make angles 0◦ , 60◦ and 120◦ with x-axis. A plane graph G = (V, E, F ) with V = S is called a TD-Delaunay triangulation when [u, v] ∈ E iff there are p ∈ R2 and λ > 0 such that u, v ∈ bd ∇(p, λ) and S ∩ int ∇(p, λ) = ∅. Finally, a plane graph G = (V, E, F ) with V = S is named as nearest neighbour graph when [u, v] ∈ E iff either u or v is the nearest Euclidean neighbour for v or u respectively. Hardness results. Our first result claims strong NP-hardness of IPGD within the class of Delaunay triangulations and some known classes of their connected subgraphs (Gabriel, relative neighbourhood graphs) for r ∈ [dmin , dmax ] and = O(n) where n is the number of vertices in G. IPGD remains strongly µ = ddmax min NP-hard within the class of nearest neighbour graphs for r = [dmax , ηdmax ] with large constant η and µ ≤ 4. Furthermore, we have the same hardness results under the same restrictions on r and µ even if we are bound to choose points of C close to vertices of G. The upper bound √on µ for Delaunay triangulations is 3 comparable with the lower bound µ = Ω n2 which holds true (with positive probability) for Delaunay triangulations produced by n random independent points on the unit disk [3]. Thus, declared restrictions on r and µ define natural instances of IPGD. An upper bound on µ implies an upper bound on the ratio of the largest and the smallest aspect ratio of objects from Nr (E). The Hitting Set problem is generally easier when sets from N have almost equal aspect ratio bounded from above by some constant. Our result for the class of nearest neighbour graphs gives the problem hardness in the case where objects of Nr (E) have almost equal constant aspect ratio. In distinction to known results for the Hitting Set problem mentioned above our study is mostly for its continuous setting with the structured system Nr (E) formed by an edge set of a specific plane graph; each set from Nr (E) is of the special form of Minkowski sum of 3 4

when U coincides with some prescribed finite set bd T denotes the set of boundary points of T.

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some graph edge and the radius r disk. Our proofs are elaborate complexity reductions from the CDC problem which is intimately related with IPGD. Our second result is about computational complexity of the IPGD problem for r = 0 (which coincides with the VC problem implying C ⊂ V ) within the class of planar triangulations. Namely, strong NP-hardness of VC is reported within the class of 4-connected (Hamiltonian) planar triangulations of bounded vertex degree with triangular outer face. It presents a novel contribution into complexity analysis of the VC problem which is usually taken in the literature within graph classes that are far away from planar and geometric triangulations, see e.g. [11], [19]. It is known that 4-connected planar triangulations with triangular outer face are isomorphic to Delaunay triangulations due to [14] which supports the conjecture that VC is NP-hard within the class of 4-connected Delaunay ones. Applying known result from graph drawing we obtain strong NP-hardness of VC within the class of 4-connected TD-Delaunay triangulations. Positive results. Let R(E) be the smallest radius of the disk that intersects all segments from the edge set E. As opposed to the cases where either r = 0 or r ∈ [dmin , dmax ] or r = [dmax , ηdmax ], IPGD is solvable within the class of simple plane graphs, for which the inequality r ≥ ηR(E) holds uniformly for some fixed l √ m2 η > 0, in time O k 2 |E|2k+1 with k = η2 . Thus, IPGD is parameterized with respect to r. Above inequality implies an upper bound k on its optimum. Taking into account proof of W [1]-hardness of parameterized version of CDC [17] and the reduction used to prove theorem 2 of this paper, it seems unlikely to improve this time bound to O(f (k)|E|c ) for any function f and constant c > 0. Finally, we present an 8p(1+2λ)-approximation O(|E| log |E|)-time algorithm for IPGD when the inequality r ≥ dmax 2λ holds true uniformly within some class of simple plane graphs for a constant λ > 0, where p(x) is the smallest number of unit disks needed to cover any disk of radius x > 1. It corresponds to the case where objects from Nr (E) have their aspect ratio bounded from above by 1 + λ. A similar but more complex Ω(|E|2 )-time constant factor approximation algorithm is given in [9] to approximate the Hitting Set problem for sets of objects whose generalized aspect ratio is bounded from above by some constant.

2 2.1

Hardness results Problem preprocessing

Before presenting hardness results for the IPGD problem we are to take some problem preprocessing. It is aimed at reducing a set of points, among which centers of radius r disks are chosen, to a finite set of cardinality which is bounded from above by some polynomial in |E|. As was mentioned in the introduction, the IPGD problem coincides with the Hitting Set problem considered for Euclidean r-neighbourhoods of graph edges which form the system denoted by Nr (E). Their boundaries are composed of four parts: a pair of half-circles and a pair of equal parallel straight line segments. We can assume (roughly) that nonempty intersection of an arbitrary number of objects from Nr (E) contains

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a point from the intersection of boundaries of some pair of objects from Nr (E). Thus, our choice of points, forming a feasible solution to the IPGD problem, can be restricted to the set of intersection points of pairs of boundaries of objects from Nr (E). This preprocessing serves two main purposes: first, it gives that IPGD is in NP; second, it grounds the polynomial time algorithm for IPGD from subsection 3.1. Lemma 1. Let G = (V, E) be a simple plane graph. Each feasible solution C to the IPGD problem for G can be converted in polynomial time and space (in |E|) to a feasible solution D ⊂ Dr (G) of IPGD for G with |D| ≤ |C| where Dr (G) ⊂ Q2 is some set of the cardinality of the order O(|E|2 ) which can be constructed in polynomial time and space. Proof. Below we keep some notations from the introduction. First, let us describe a construction of the set Dr (G). W.l.o.g. it can be assumed that for each N1 ∈ Nr (E) there exists N2 ∈ Nr (E) with N1 ∩N2 6= ∅. Otherwise, an arbitrary vertex of each such edge e ∈ E is added to Dr (G) that has Nr (e) non-intersecting with the other objects from Nr (E). It can be done in polynomial time. Consider an arbitrary maximum (with respect to inclusion) subsystem of objects from Nr (E) whose intersection is nonempty. It is called a maximum feasible subsystem (MFS for short). Let M be the set which coincides with the intersection of all objects from an arbitrary MFS. The set M is bounded, closed and convex. Moreover, M has a boundary which is composed of pieces of boundaries of objects from Nr (E). The set M is contained in the intersection of at least two objects from Nr (E). Consider the degenerate case where two edges e1 and e2 from E are parallel and have their r-neighbourhoods touching in such a way that touching points form a segment. In this case |bd Nr (e1 ) ∩ bd Nr (e2 )| = ∞. The same is also possible if edges e1 and e2 intersect at their common vertex. In both cases the intersection bd Nr (e1 ) ∩ bd Nr (e2 ) has at most two extreme points. In the other (non-degenerate) ones |bd Nr (e1 ) ∩ bd Nr (e2 )| is bounded from above by some constant taking into account the fact that each set bd Nr (e), e ∈ E, is a compound of a pair of parallel straight line segments and a pair of half-circles. Set [ Cr (G) = extr (bd Nr (e1 ) ∩ bd Nr (e2 )) , e1 ,e2 ∈E, e1 6=e2

where extr N means the set of (extreme in two aforementioned degenerate cases) points of one-dimensional set N. The set M has at least one vertex which belongs to Cr (G). Moreover, |Cr (G)| = O(|E|2 ). The system Nr (E) forms an arrangement Ar (E) which can be computed in polynomial time [2]. Therefore for every c ∈ extr (bd Nr (e1 ) ∩ bd Nr (e2 )) a rational point d(c) ∈ Nr (e1 ) ∩ Nr (e2 ) can be found in some cell z of Ar (E) with c ∈ z such that d(c) ∈ z. Set Dr (G) = {d(c) : c ∈ Cr (G)}. Second, let C be a feasible solution to the IPGD problem. For each point c ∈ C a point d(c) ∈ Dr (G) can be found in polynomial time such that {N ∈ Nr (E) : c ∈ N } ⊆ {N ∈ Nr (E) : d(c) ∈ N }.

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Thus, the set D = {d(c) : c ∈ C} is feasible for IPGD and |D| ≤ min{|C|, |Dr (G)|}. The following assertion is an immediate consequence of lemma 1. Corollary 1. The IPGD problem belongs to the class NP. 2.2

Hardness for r ∈ [dmin , dmax ]

We start our complexity analysis for the IPGD problem by considering its setting where r ∈ [dmin , dmax ]. Under this restriction on r IPGD coincides neither with VC nor with CDC. In fact it is equivalent (see the introduction) to the geometric Hitting Set problem for the set Nr (E) of Euclidean r-neighbourhoods of edges of G. For the IPGD problem we claim its hardness even if we restrict the graph G to be either a Delaunay triangulation or some of its known subgraphs or a bounded from above, TD-Delaunay triangulation. We keep the ratio µ = ddmax min thus, imposing an upper bound on the ratio of the largest and the smallest aspect ratio of objects from Nr (E). We show that IPGD remains intractable even in its simple case where r = Θ(dmax ) and µ is bounded by some small constant or, equivalently, when objects of Nr (E) have close constant aspect ratio. Our first hardness result for IPGD is obtained by using a complexity reduction from the CDC problem. Below we describe a class of hard instances of the CDC problem which correspond to hard instances of the IPGD problem for Delaunay triangulations with relatively small upper bound on the parameter µ. Hardness result for the CDC problem. To single out the class of hard instances of the CDC problem we use a reduction from the strongly NP-complete minimum dominating set problem which is formulated as follows: given a simple planar graph G0 = (V0 , E0 ) of degree at most 3, find the smallest cardinality set V00 ⊆ V0 such that for each u ∈ V0 \V00 there is some v = v(u) ∈ V00 which is adjacent to u. Below an integer grid denotes the set of points on the plane with integervalued coordinates within some bounded interval. An orthogonal drawing of the graph G0 on some integer grid is the drawing whose vertices are represented by points on that grid whereas its edges are given in the form of polylines formed by sequences of connected axis-parallel straight line segments of the form [p1 , p2 ], [p2 , p3 ], . . . , [pk−1 , pk ] intersecting only at the edge endpoints, where each point pi again belongs to the grid. In [18] strong NP-completeness of CDC is proved by reduction from the minimum dominating set problem. This reduction involves using plane orthogonal drawing of G0 on some integer grid. More specifically, a set D is build on that grid with V0 ⊂ D. The resulting hard instance of the CDC problem is for the set D and some integer (constant) radius r0 ≥ 1. Let us observe that G0 admits an orthogonal drawing (theorem 1 [22]) on the grid of size O(|V0 |) × O(|V0 |) whereas total length of each edge is O(|V0 |). Proof of the strong NP-completeness of CDC could be conducted taking into account this observation. We can formulate (see combination of theorems 1 and 3 [18]) Theorem 1. (Masuyama et. al., 1981 [18]) The CDC problem is strongly NPcomplete for a constant integer radius r0 and the point set D on the integer

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grid of size O(|D|) × O(|D|). It remains strongly NP-complete even if we restrict centers of radius r0 disks to be at the points of D. Remark 1. For every simple planar graph G0 of degree at most 3 its orthogonal drawing can be constructed such that at least one its edges is a polyline which is composed of two axis-parallel segments. The IPGD problem hardness for Delaunay triangulations. To build a reduction from the CDC problem for the set D constructed in theorem 1 we exploit a simple idea that a radius r disk covers a set of points D0 ⊂ D iff a slightly larger disk covers straight line segments, each of which is close to some point of D0 and has a small length with respect to distances between points of D. Then a proximity graph H is build whose vertex set coincides with the set of endpoints of small segments corresponding to points of D. Since H usually contains these small segments as its edges, this technique gives hardness for the IPGD problem within a variety of classes of proximity graphs. The following technical lemma holds. Lemma 2. Let X ⊂ Z2 , r ≥ 1 be some integer and ρ(u; v, w) be the minimum of two distances from u ∈ X to circles of radius r passing through distinct points v and w from X with |v − w|2 ≤ 2r. Then min

u∈C(v,w),v6 / =w,u,v,w∈X,|v−w|2 ≤2r

ρ(u; v, w) ≥

1 100r4

where C(v, w) is the union of two radius r circles through v and w. The complexity of the following restricted form of IPGD is also studied. Vertex Restricted IPGD (VRIPGD(δ)): given a simple plane graph G = (V, E, F ), a constant δ > 0 and a constant r > 0, find the least cardinality set C ⊂ Q2 such that each S e ∈ E is within (Euclidean) distance r from some point c = c(e) ∈ C and C ⊂ Bδ (v). v∈V

Theorem 2. Both IPGD and VRIPGD(δ) problems are strongly NP-hard for r ∈ [dmin , dmax ], µ = O(n) and δ = Θ(r) within the class of Delaunay triangulations where n is the number of vertices in triangulation. Proof. By corollary 1 IPGD belongs to NP. Using the similar preprocessing algorithm it can be shown that VRIPGD is also in NP: namely, pairwise intersection points are sought for the system of boundaries of objects from Nr (E) augmented with radius δ circles centered at the vertices from V. Let us prove that IPGD is strongly NP-complete. Proof technique for the VRIPGD(δ) problem is analogous. For any hard instance of the CDC problem given in theorem 1 we build the IPGD problem instance with r = r0 + δ as follows where δ = 40012 2r9 . For every u ∈ D points u0 and v0 are found such 0 that |u − u0 |∞ ≤ δ/4 and |u − v0 |∞ ≤ δ/4 where Iu = [u0 , v0 ] has a length at least δ/4. More specifically, let us set ID = {Iu = [u0 , v0 ] : u ∈ D}. Endpoints of

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segments from ID are constructed in sequential manner in polynomial time and space by defining a new segment Iu to provide generality position for the set of endpoints of the set ID0 ∪ {Iu }, D0 ⊂ D, where segments of ID0 are already defined. Here endpoints of Iu are chosen in the rational grid that contains u c1 c1 whose elementary square size is |D| 2 × |D|2 for some small absolute rational constant c1 . Assuming u = (ux , uy ), the point u0 is chosen in the lower part of the grid with y-coordinates less than uy − δ/8 whereas v0 is taken from the upper one for which y-coordinates exceed uy + δ/8. Let S be the set of endpoints of segments from ID . Every disk having Iu as its diameter does not contain any points of S distinct from endpoints of Iu . Let G = (V, E, F ) be a Delaunay triangulation for S which can be computed in polynomial time and space in |D|. Obviously, each segment Iu coincides with some edge from E. We have dmin ≤ r and µ = O(n). It remains to prove that r ≤ dmax . Due to remark 1 and the construction of the set D (see fig. 1 [18]) the set S can be constructed such that the inequality r ≤ dmax holds true for G. Moreover, representation length for vertices of V is polynomial with respect to representation length for points of D. Let k be a positive integer. Obviously, centers of at most k disks of radius r0 containing D in their union give centers of radius r ≥ r0 disks whose union is intersected with each segment from E. Conversely, let T be a disk of radius r0 +δ which intersects a subset ID0 = {Iu : u ∈ D0 } of segments for some D0 ⊆ D. When |D0 | = 1, it is easy to transform T to a disk which contains the segment ID0 . Points of D have integer coordinates. Moreover, squared (Euclidean) distance between each pair of points of the subset D0 does not exceed (2r0 + 4δ)2 = 4r02 + 16r0 δ + 16δ 2 . Therefore points from D0 are located within the distance 2r0 from each other. Let us use Helly theorem. Let R be the minimum radius of the disk T0 containing any triple u1 , u2 and u3 from D0 . W.l.o.g. we suppose that, say, u1 and u2 are on the boundary of T0 . Obviously, R ≤ r0 + 2δ. Let us show that the case R > r0 is void. We slightly shift the center of T0 (along the midperpendicular to [u1 , u2 ]) to have u1 and u2 at the distance r0 from the shifted center O0 . The distance √ from the point u13 to the radius r0 circle centered at O0 does not exceed 2 r0 δ + δ 2 + 2δ < 100r 4 . By lemma 2 we have 0 R ≤ r0 . Thus, D0 is contained in some disk of radius r0 . Given a set of points, the smallest radius disk can be found in polynomial time and space which covers this set. Therefore we can convert any set of at most k disks of radius r whose union is intersected with each segment from E to some set of at most k disks of radius r0 whose union covers D. Using corollary 1 of section √ 4.2 from [3] and theorem 1 from [20] we arrive 3 at the lower bound µ = Ω n2 which holds true with positive probability for Delaunay triangulations produced by n random uniform points on the unit disk. Thus, the order of the parameter µ for the considered class of hard instances of the IPGD problem is comparable with the one for random Delaunay triangulations.

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The IPGD problem hardness for other classes of proximity graphs. The same proof technique could be applied for proving the problem hardness within the other classes of proximity graphs. Let us start with some definitions. The following graphs are connected subgraphs of Delaunay triangulations. A plane graph G = (V, E, F ) with V = S is called a Gabriel graph where [u, v] ∈ E iff the disk having [u, v] as its diameter does not contain any other points of S distinct from u and v. A relative neighbourhood graph is the plane graph G with V = S for which [u, v] ∈ E iff there is no any other point w ∈ S such that w 6= u, v with max{|u − w|2 , |v − w|2 } < |u − v|2 . Finally a plane graph is called a minimum Euclidean spanning tree if it is the minimum weight spanning tree of the weighted complete graph K|S| whose vertices are points of S such that its edge weight is given by the Euclidean distance between the edge endpoints. The proof of theorem 2 is extendable for classes of Gabriel, relative neighbourhood and nearest neighbour graphs. It is easy to observe that segments Iu , u ∈ D, form a subset of edges of any minimum Euclidean spanning tree. In contrary, suppose a spanning tree H does not contain segment Iu but contains edges incident to both endpoints u0 and v0 of Iu for some u ∈ D. Let us remove the edge e = [u0 , w0 ] from H and add the edge Iu to it where w0 is the parent of u0 in H. Obviously the resulting tree has the smaller weight, a contradiction. Corollary 2. Both IPGD and VRIPGD(δ) problems are strongly NP-hard for r ∈ [dmin , dmax ], µ = O(n) and δ = Θ(r) within classes of Gabriel, relative neighbourhood graphs, TD-Delaunay triangulations and minimum Euclidean spanning trees and for r ∈ [dmax , ηdmax ] and µ ≤ 4 within the class of nearest neighbour graphs where η is a large constant. Remark 2. In theorem 2 and corollary 2 hardness is given for r = Θ(dmin ) within graph classes except for nearest neighbour ones. 2.3

Hardness for r = 0

We continue our complexity analysis of the IPGD problem by studying its extreme case where r = 0. In this case the IPGD problem coincides with the VC problem. Below the VC problem hardness is given within the class of highly connected TD-Delaunay triangulations. Moreover, some directions are also provided for proving hardness of VC within the class of Delaunay triangulations. A planar triangulation is called special if faces of its plane embedding are all triangles. Our proof method is simple. First, the VC problem hardness is given within the class of highly connected simple special planar triangulations. Second, some result from graph drawing is applied to embed every simple special planar triangulation in the form of TD-Delaunay triangulation in polynomial time which translates obtained complexity result from special planar triangulations to TDDelaunay ones. The VC problem hardness for planar triangulations. We start by formulating an auxiliary result claiming hardness of the following problem.

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Maximum Facial Independent Set (Facial MIS). Given a special plane triangulation G = (V, E, F ) (which may contain parallel edges) find the smallest cardinality set of faces F 0 ⊂ F such that each pair of faces from F 0 does not have an edge from E in common. Here the notion of plane graph is used for plane embeddings of non-simple planar graphs whose edges are of the form of piecewise linear polylines. This problem is the dual form of known maximum independent set problem considered for some simple 2-connected 3-regular plane graph G0 = (V0 , E0 , F0 ). This can be easily seen by applying the classical geometric duality transform [13] for G0 , where vertices of V0 correspond to faces of the dual special triangulation G, whereas faces of F0 correspond to vertices of G; finally edges of E0 are in oneto-one correspondence with edges of G. Theorem 4.1 [19] reports the following Theorem 3. (Mohar, 2001 [19]) Facial MIS problem (in its decision form) is strongly NP-complete within some subclass T0 of special planar triangulations possibly containing parallel edges. Remark 3. In the proof of theorem 4.1 [19] it is also noted that each graph G0 ∈ T0 can be almost uniquely embedded on the plane in polynomial time. Theorem 4. The VC problem is strongly NP-hard within the class of simple special 4-connected (Hamiltonian) planar triangulations of degree at most log n where n is the number of triangulation vertices. Proof. The VC problem is obviously in NP because every subset of triangulation vertices has cardinality of the order O(n). Let T be the class of simple special planar (3-connected) triangulations. We give a reduction to VC from Facial MIS which is strongly NP-complete by theorem 3. For each plane graph G0 = (V0 , E0 , F0 ) ∈ T0 we construct a v1 b

v15 v1

v5

b

v7

b

b

G(f01) v5

v3

b

b

b

b b

b

b

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b b

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v19

G(f0)

G(f02)

v11 b

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Fig. 1. A graph G(f0 ) corresponding to a Fig. 2. Graphs G(f01 ) and G(f02 ) along face f0 of G0 with their connector graph G(e0 )

special plane triangulation G = (V, E, F ) ∈ T as follows. Each face f0 = v1 (f0 )v2 (f0 )v3 (f0 ) ∈ F0 \{f0,∞ } is replaced by a graph G(f0 ) shown on the fig. 1 (symbol f0 is omitted in notations of vertices of G(f0 ) in the sequel) where

Complexity and approximability of guarding of proximity graphs

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f0,∞ is the outer face of G0 . It is easy to see that the minimum vertex cover for the graph G(f0 ) is unique. Moreover, it does not contain v10 and coincides with Uf0 = {v4 , v5 , v6 , v7 , v8 , v9 }. It follows from the fact that each triangle from the triple of triangles ∆v1 v5 v7 , ∆v2 v6 v8 and ∆v3 v4 v9 (see fig. 1) requires two vertices to be covered. Considering all possible cases we get that |Vf00 | ≥ 7 for every vertex cover Vf00 if v10 ∈ Vf00 . Moreover, Wf0 = {v1 , v2 , v3 , v4 , v5 , v6 , v10 } is a vertex cover for G(f0 ). The graph G(f0 ) is placed inside each face f0 such that sides of the generally curvilinear triangle v1 v2 v3 are parallel to sides of the face f0 e.g. by scaling that face using a given rational factor. Graphs G(f01 ) and G(f02 ) corresponding to faces f01 and f02 from F0 \{f0,∞ } are joined using a connector graph G(e0 ) as shown in the fig. 2 where faces f01 and f02 have a common edge e0 ∈ E0 . The graph G(e0 ) has two common edges with each of two graphs G(f01 ) and G(f02 ). The graph G(f0,∞ ) corresponding to the face f0,∞ is isomorphic to the graph G(f0 ) for an arbitrary f0 ∈ F0 \{f0,∞ }. It is shown in the fig. 3 together with respective connector graphs Gi , i = 1, 2, 3. (graphs Gi are depicted schematically without some edges and vertices). Edges of graphs G(e0 ), e0 ∈ E0 , can be

b

v1 b

b

v3 v8

b

b

b

b b

b

b

b

b b

M(v0 )b

b

v6

b

b

b

b

v10

b

b

v7

G1

G3 G b 2 b

v4

v5

b

b

b

v0 b

b

b

b

b

b b

v2 b

v9 b

b b

b

Fig. 3. The graph G(f0,∞ ) constructed for Fig. 4. Graphs G(f0 ) and connectors the outer face of G0 G(e0 ) bounding polygon M (v0 )

embedded in such a way that edges of graphs G(e01 ) and G(e02 ) intersect at most at their common endpoints (if exists) for every e01 and e02 from E0 . For every v0 ∈ V0 each set E0 (v0 ) = {e0 ∈ E0 : v0 ∈ e0 } corresponds to the set Q(v0 ) = {G(e0 ) : e0 ∈ E0 (v0 )} of deg(v0 ) connector graphs where deg(v0 ) = |E0 (v0 )|. The vertex v0 is in the interior of some generally curvilinear simple 2deg(v0 )-gon M (v0 ) bounded by edges of connectors from Q(v0 ) (see fig. 4). Let U (v0 ) = {u1 , . . . , u2deg(v0 ) } be a clockwise ordered sequence of vertices of M (v0 ) where u1 coincides with one of two vertices v14 or v15 of connector graphs from Q(v0 ). Triangulate M (v0 ) as follows (and denote the graph thus obtained by G(v0 )) : add series of edges {u1 , u3 }, {u3 , u5 }, . . . then series of edges {u1 , u5 }, {u5 , u9 }, . . . continuing doing so until M (v0 ) becomes triangulated.

12

Konstantin Kobylkin

Let us set G :=

[ f0 ∈F0

G(f0 ) ∪

[ v0 ∈V0

G(v0 ) ∪

[

G(e0 ).

e0 ∈E0

Since the number of graph connectors is equal to |E0 | and the degree of each vertex from G(v0 ) is of the order O(log |E0 |), the complexity of construction of the graph G for the graph G0 is of the order O(|F0 | + |E0 | log |E0 |). Moreover, the number of vertices, edges and faces of G is of the order O(|F0 | + |E0 |). The length of representation of vertices of G differs by a constant factor from the representation length of vertices of G0 . Since G is a simple special triangulation (we “subdivided” all parallel edges of G0 ), the graph G is 3-connected. The graph G does not have any edges that connect vertices from G(f0 )\G(e0 ) with those from G(e0 )\G(f0 ); the same holds for G(f0 ) and G(v0 ) as well as for G(e0 ) and G(v0 ). Considering all possible cases shows that for every graph G(f0 ), f0 ∈ F0 , and G(e0 ), e0 ∈ E0 , each their 3-cycle coincides with their face. In the process of triangulating of M (v0 ) the only triangles that we have are faces by construction. Therefore each 3-cycle of G is its face. Using lemma 2.3 [12] we get 4-connectedness of G. Let us show that each independent set of faces F00 ⊂ F0 with |F00 | ≥ k can be converted into some vertex cover V 0 for G where |V 0 | ≤ 7|F0 | − k + 4|E0 |. Set Vf00 := Uf0 for every f0 ∈ F00 ; for each f0 ∈ F0 \F00 set Vf00 := Wf0 . To get a vertex cover Ve00 of the graph G(e0 ) that connects graphs G(f01 ) and G(f02 ), f01 , f02 ∈ F0 , vertices v14 and v15 are taken along with one vertex per each pair {v16 , v19 } and {v17 , v18 }. More specifically, we choose a vertex from each such pair depending on whether f01 ∈ F00 and f02 ∈ F00 : namely, if f01 ∈ F00 and f02 ∈ / F00 , then vertices v17 and v19 are chosen; choice of vertices S from0 these S pairs is arbitrary in the case where f01 , f02 ∈ / F00 . Obviously, V 0 = Vf0 ∪ Ve00 f0 ∈F0

e0 ∈E0

covers edges of G and |V 0 | ≤ 7|F0 | − k + 4|E0 |. Conversely, let V 0 be the minimum cardinality vertex cover for G and |V 0 | ≤ 7|F0 | − k + 4|E0 |. Set Vf00 = V 0 ∩V (f0 ) for each f0 ∈ F0 where V (f0 ) is the vertex set of the graph G(f0 ). At least two vertices are required to cover edges that belong to both G(f01 ) and G(e0 ) or to both G(f02 ) and G(e0 ), where e0 is the common edge of faces f01 and f02 . At least 4 vertices are needed to cover edges of triangles over sets of vertices V1 (e0 ) = {v15 , v16 , v19 } and V2 (e0 ) = {v14 , v17 , v18 }. If either |V 0 ∩ V1 (e0 )| = 2 or |V 0 ∩ V2 (e0 )| = 2, then |Vf002 | ≥ 7 for |Vf001 | = 6. In view of minimality of V 0 the case is void when |V 0 ∩ V1 (e0 )| = |V 0 ∩ V2 (e0 )| = 3 and |Vf001 | = |Vf002 | = 6 hold true simultaneously. Therefore the face set F00 = {f ∈ F0 : |Vf00 | = 6} is independent. Since |V 0 ∩ (V1 (e0 ) ∪ V2 (e0 ))| ≥ 4, we have 6|F00 | + 7(|F0 | − |F00 |) + 4|E0 | ≤ |V 0 | so |F00 | ≥ k. The VC problem hardness for geometric triangulations. The following result holds true (corollary 2 [5]). Lemma 3. (Bonichon et. al, 2010 [5]) For every simple special planar triangulation, an isomorphic TD-Delaunay triangulation can be constructed in polynomial time and space.

Complexity and approximability of guarding of proximity graphs

13

Combining results of theorem 4 and lemma 3 we get Corollary 3. The VC problem is strongly NP-hard within the class of 4-connected (Hamiltonian) TD-Delaunay triangulations of degree at most log n where n is the number of triangulation vertices. In spite of the fact that any special 4-connected planar triangulation is isomorphic (in usual graph sense) to some Delaunay triangulation [14] it is still an open (possibly NP-hard) problem of getting combinatorially equivalent Delaunay triangulation for a given special 4-connected planar one. Nevertheless we hope it is possible to do in polynomial time and space for the subclass of special 4-connected planar triangulations constructed in the proof of theorem 4. Conjecture 1. The VC problem is strongly NP-hard within the class of 4-connected Delaunay triangulations.

3 3.1

Positive results Polynomial solvability of IPGD for large r

In distinction to the cases where either r = 0 or r ∈ [dmin , dmax ] or r = Θ(dmax ) the IPGD problem is polynomially solvable for large r = Ω(R(E)) where R(E) is the smallest radius of the disk that intersects straight line segments of the edge set E. It is obvious that IPGD is solvable in time O(|E|) [4] within the class of plane graphs for which the inequality r ≥ R(E) holds uniformly. Let us consider the case of IPGD within the class of plane graphs for which r ≥ ηR(E) for some 0 < η < 1. In this case IPGD is parameterized with respect to r. Since every disk of radius r contains an axis-parallel rectangle l√ m2 l √ m2 √ 2R(E) 2 = whose side is equal to r 2, roughly at most k = k(η) = r η radius r disks are needed to intersect all segments from E. Therefore the bruteforce search algorithm could be applied that just sequentially tries each subset of Dr (G) (see subsection 2.1) of cardinality at most k. This amounts roughly to the O k 2 |E|2k+1 time complexity. Thus, we arrive at the polynomial time algorithm whose complexity depends exponentially on η. This algorithm gives an optimal solution to the IPGD problem taking into account lemma 1. 3.2

Approximation algorithm for IPGD

Let us focus on the case of the IPGD problem where the inequality r ≥ dmax 2λ holds uniformly within some class G of simple plane graphs for a constant λ > 0. It corresponds to the situation where objects from the system Nr (E) have their aspect ratio bounded from above by 1 + λ. It turns out that in this case the problem admits an O(1)-approximation algorithm whose factor depends on λ. Let us consider the following problem: Cover endpoints of segments with disks (CESD): given an arbitrary set SE ⊆ V which covers the set E, find the smallest cardinality set of radius r disks whose union covers SE .

14

Konstantin Kobylkin

Denote by OP TCESD (SE , r) and OP TIP GD (G, r) the optima of CESD and IPGD respectively considered within the graph class G. Let us formulate an Algorithm 1. Compute and output 8-approximate solution to the CESD problem using O(|E| log OP TCESD (SE , r))-time algorithm from [15]. The statement below bounds the ratio of optima for these problems. Statement 5 Suppose the inequality r ≥ dmax 2λ holds uniformly within the class G for some constant λ > 0. The following bound holds true: OP TCESD (SE , r) ≤ p(1 + 2λ) OP TIP GD (G, r) where p(x) is the smallest number of unit disks needed to cover radius x disk. Proof. Let C0 ⊂ Q2 be an optimal solution to the IPGD problem. Set E(c) = {e ∈ E : c ∈ Nr (e)}, c ∈ C0 . For every e ∈ E(c) there is a point de (c) ∈ e with |c − de (c)|2 ≤ r. Each point from the set S(c) of endpoints of segments from E(c) is within Euclidean distance r + dmax from c. Therefore the radius r + dmax disk centered at c covers the set SE ∩ S(c). At most p(1 + 2λ) radius r disks are needed to cover any radius r + dmax disk. Thus, the declared bound for the optima ratio holds true. Corollary 4. The algorithm 1 is 8p(1 + 2λ)-approximate. Remark 4. Approximation factor of the algorithm 1 is in fact lower when G is the subclass of Delaunay triangulations or their subgraphs. Indeed, in this case there is no need to cover the whole radius r + dmax disk with radius r disks.

4

Conclusion

Complexity and approximability are studied for the problem of intersecting a structured set of straight line segments with the smallest number of disks of radii r > 0 where a structural information about segments is given in the form of an edge set of a proximity graph. It is shown that the problem is strongly NPhard within the class of Delaunay, TD-Delaunay triangulations and some of their subgraphs for small and medium values of r while for large r it is polynomially solvable. Fast O(1)-approximation algorithm is given for medium values of r.

References 1. Pankaj K. Agarwal, Alon Efrat, Shashidhara K. Ganjugunte, David Hay, Swaminathan Sankararaman, and Gil Zussman. The resilience of WDM networks to probabilistic geographical failures. IEEE/ACM Trans. Netw., 21(5):1525–1538, 2013. 2. Amato, Nancy M. and Goodrich, Michael T. and Ramos, Edgar A. Computing the arrangement of curve segments: divide-and-conquer algorithms via sampling. Proc. 11th ACM-SIAM Sympos. Discrete Algorithms, 705–706, 2000.

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3. Esther M. Arkin, Antonio Fernndez Anta, Joseph S. B. Mitchell, and Miguel A. Mosteiro. Probabilistic bounds on the length of a longest edge in Delaunay graphs of random points in d dimensions. Comput. Geom., 48(2):134–146, 2015. 4. Bhattacharya B.K., Mukhopadhyay A. Jadhav S., and Robert J.-M. Optimal algorithms for some intersection radius problems. Computing, 52(3):269–279, 1994. 5. Nicolas Bonichon, Cyril Gavoille, Nicolas Hanusse, and David Ilcinkas. Connections between θ-graphs, Delaunay triangulations, and orthogonal surfaces. In Dimitrios M. Thilikos, editor, WG, volume 6410 of Lecture Notes in Computer Science, pages 266–278, 2010. 6. Prosenjit Bose, Jean-Lou De Carufel, Stephane Durocher, and Perouz Taslakian. Competitive online routing on Delaunay triangulations. In R. Ravi and Inge Li Gortz, editors, SWAT, volume 8503 of Lecture Notes in Computer Science, pages 98–109. Springer, 2014. 7. Prosenjit Bose, David G. Kirkpatrick, and Zaiqing Li. Worst-case-optimal algorithms for guarding planar graphs and polyhedral surfaces. Comput. Geom., 26(3):209–219, 2003. 8. Stojmenovic I. Urrutia J. Bose P., Morin P. Routing with guaranteed delivery in ad hoc wireless networks. Wireless Networks, 7(6):609–616, 2001. 9. Timothy M. Chan. Polynomial-time approximation schemes for packing and piercing fat objects. J. Algorithms, 46(2):178–189, 2003. 10. Timothy M. Chan and Elyot Grant. Exact algorithms and APX-hardness results for geometric packing and covering problems. Comput. Geom., 47(2):112–124, 2014. 11. Gautam Das and Michael T. Goodrich. On the complexity of optimization problems for 3-dimensional convex polyhedra and decision trees. Comput. Geom., 8:123–137, 1997. 12. Tamal K. Dey, Michael B. Dillencourt, Subir Kumar Ghosh, and Jason M. Cahill. Triangulating with high connectivity. Comput. Geom., 8:39–56, 1997. 13. R. Diestel. Graph Theory. Springer, 2005. 14. Smith W.D. Dillencourt M.B. Graph-theoretical conditions for inscribability and Delaunay realizability. Discrete Mathematics, 161(1-3):63–77, 1996. 15. Teofilo F. Gonzalez. Covering a set of points in multidimensional space. Inf. Process. Lett., 40(4):181–188, 1991. 16. Quanrud K. Har-Peled S. Approximation algorithms for polynomial-expansion and low-density graphs. In Nikhil Bansal and Irene Finocchi, editors, ESA, volume 9294 of Lecture Notes in Computer Science, pages 717–728. Springer, 2015. 17. Daniel Marx. Efficient approximation schemes for geometric problems? In Gerth Stolting Brodal and Stefano Leonardi, editors, ESA, volume 3669 of Lecture Notes in Computer Science, pages 448–459. Springer, 2005. 18. Hasegawa T. Masuyama S., Ibaraki T. Computational complexity of the m-center problems on the plane. Transactions of the Institute of Electronics and Communication Engineers of Japan. Section E, E64(2):57–64, 1981. 19. Bojan Mohar. Face covers and the genus problem for apex graphs. J. Comb. Theory, Ser. B, 82(1):102–117, 2001. 20. Takuji Onoyama, Masaaki Sibuya, and Hiroshi Tanaka. Limit Distribution of the Minimum Distance between Independent and Identically Distributed d-Dimensional Random Variables, pages 549–562. Springer Netherlands, Dordrecht, 1984. 21. Joseph O’Rourke. Art Gallery Theorems and Algorithms. Oxford University Press, 1987. 22. Tamassia R. and Tollis I.G. Planar grid embedding in linear time. IEEE Transactions on Circuits and Systems, volume 36, pages 1230–1234, 1989.

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Konstantin Kobylkin

A

Proof of Lemma 2

Proof. Let u = (x, y), v = (x1 , y1 ) and w = (x2 , y2 ) be distinct points of X. Consider an arbitrary circle of radius r among a pair of ones through v and w (denote its center by O). We are to bound the distance π(u; v, w) from that circle to the point u ∈ / C(v, w) from below. q Let us denote ∆ = |v − w|2 , λ = ⊥

r2 −

∆2 4 ,

a = (u − v, u − w) and b =

⊥

(u − v, (v − w) ) where (v − w) = ±(y1 − y2 , −x1 + x2 ). The distance π could be written in the form: 2λb ⊥ v + w a + (v − w) ∆ . π = π(u; v, w) = −λ − u − r = q 2 |v − w|2 a + 2λb + r2 + r 2 ∆ W.l.o.g. we can consider the case where u lies inside the disk of radius 2r centered at O. Indeed, otherwise π ≥ r ≥ 1r . Let us bound a denominator of the fraction π taking into account that ∆ ≤ 2r, |u − v|2 ≤ 3r and b/∆ ≤ 3r : r 2λb + r2 + r ≤ 5r. a+ ∆ As points of X have integer coordinates, then a and b are integers and π > 0. 1 When ∆ = 4r2 , we get π ≥ 5r . For ∆2 ≤ 4r2 − 1 let us show that a + 2λb ≥ 1 ∆ 20r3 in each of two cases where 2λb ∆ is either an integer or fractional. For the integer 2λb 1 we arrive at the bound π ≥ 5r . ∆ 2λb Assume that ∆ has a nonzero fractional part. Let q = 2λ and k = 2λ ∆ ∆ where {·} and [·] denote fractional and integer parts of a real number respectively. 2 2 First we consider the case where 4r ∆2 is not an integer. We have 2kq + q ≥ n o

{2kq + q 2 } =

4r 2 ∆2

, whence s

q≥

k2 +

n

4r2 ∆2

−k ≥ 2

4r 2 ∆2

5r

o ≥

1 20r3

1 4r and π ≥ 100r 4 . Now we suppose that ∆2 is an integer. Let us denote by m 2λ 2 2 the smallest integer with m2 < 4λ ≥ 2 < (m + 1) . As q > 0 we have ∆ ∆ √ m2 + 1 . Due to concavity of square root we can write: np o 1 1 1 1 2 m +1 ≥ m+ = ≥ 4λ ≥ 2m + 1 2m + 1 5r ∆ +1

which gives π ≥

1 25r 2 .

Abstract. Computational complexity and approximability are studied for a problem of intersecting a set of straight line segments with the smallest cardinality set of disks of fixed radii r ≥ 0 where the set of segments forms a straight line drawing G = (V, E, F ) of a planar graph without edge crossings. This problem arises in network security applications (Agarwal et al., 2013). We claim strong NP-hardness of the problem within the class of (edge sets of) Delaunay, TD-Delaunay triangulations and their subgraphs for r ∈ [dmin , dmax ] and r ∈ [dmax , ηdmax ] for some large constant η as well as within the class of 4-connected TD-Delaunay triangulations for r = 0 where dmax and dmin are Euclidean lengths of the longest and shortest graph edges respectively. Fast O(|E| log |E|)-time O(1)-approximation algorithm is proposed within the class of straight line drawings of planar graphs for which the inequality r ≥ ηdmax holds uniformly for some constant η > 0. Keywords: computational complexity, approximation algorithms, Vertex Cover, Hitting Set, Continuous Disk Cover, Delaunay triangulations, proximity graphs

1

Introduction

Guarding the boundaries of geometric objects and complexes of plane embeddings of planar graphs is widely studied class of problems in computational geometry, see e.g. [7], [11], [21]. Usually in such problems one needs to find the smallest cardinality set C of guards (e.g. points on the plane) having bounded (in some sense) visibility area such that each piece of the boundary or part of the graph complex (e.g. edge or face) is within visibility area of some guard from C. Designing exact and approximate algorithms for these problems finds its applications in security, sensor placement, lighting and robotics. In this paper complexity and approximability of the following problem are studied. Intersecting Plane Graph with Disks (IPGD): given a straight line drawing of an arbitrary simple1 planar graph G = (V, E) without edge crossings and ? 1

This work was supported by Russian Science Foundation, project 14-11-00109. a graph without loops and parallel edges

2

Konstantin Kobylkin

a constant r > 0, find the smallest cardinality set C ⊂ Q2 of points (disk centers) such that each edge e ∈ E is within (Euclidean) distance r from some point c = c(e) ∈ C or, equivalently, the disk of radius r centered at c intersects e. In the IPGD problem each guard has circular visibility area which does not depend on the obstacles in contrast to known Art Gallery problem [21] where sensor coverage area is affected by gallery walls. Applications of studying the IPGD problem come from network security [1]. Given a network of physical devices one needs to evaluate its vulnerability to simultaneous technical failures caused by natural (e.g. floods, fire, electromagnetic pulses) and human sources. Here network nodes are modeled by points on the plane while its physical links are given in the form of straight line segments. A catastrophic event (threat) is usually localized in a particular geographical area and modeled by a disk of some fixed radius r > 0. A threat impacts a network link when the corresponding disk and segment intersect. Evaluation of the network vulnerability may be expressed in the form of finding the minimum number of threats along with their positions that cause all network links to be broken. Thus, it brings us to the IPGD problem as usually network links are geographically non-overlapping. Below a triple G = (V, E, F ) is called a plane graph if V ⊂ R2 , the set E consists of nondegenerate straight line segments, crossing only at their endpoints which are from V. Here F denotes the set of all open (in R2 ) regions bounded by segments from E : each f ∈ F does not intersect with any segment from E. The only unbounded set f∞ ∈ F is named as the outer face whereas bounded ones from F are inner faces. A planar (plane) graph is called a planar (plane) triangulation if its face set F consists of triangles except for possibly outer one. In this paper computational complexity and approximability of IPGD are studied for simple plane graphs with either r ∈ [dmin , dmax ] or r = Ω(dmax ) or r = 0 where dmax and dmin are lengths of the longest and shortest edges of G. Our emphasis is on those classes of simple plane graphs that are defined by some distance function, namely, on Delaunay triangulations, some of their connected subgraphs (Gabriel, relative neighbourhood, nearest neighbour graphs, minimum Euclidean spanning trees) and TD-Delaunay triangulations. These graphs are often called proximity graphs. Delaunay and TD-Delaunay triangulations (being geometric spanners) are plane graphs which admit efficient geometric routing algorithms [6], thus, representing convenient network topologies. Gabriel and relative neighbourhood graphs arise in modeling wireless [8] and road networks. Related work. IPGD is related to several well-known combinatorial optimization problems. First, it is the special case of the Hitting Set problem. Hitting Set: given a family N of sets on the plane and a set U ⊆ Q2 , find the smallest cardinality set H ⊆ U such that N ∩ H 6= ∅ for every N ∈ N . IPGD coincides with Hitting Set if we set N := Nr (E) = {Nr (e)}e∈E and U := Q2 where Nr (e) = Br (0) + e = {x + y : x ∈ Br (0), y ∈ e} is Euclidean r-neighbourhood of e (having form of Minkowski sum) and Br (x) is the disk of radius r centered at x ∈ R2 . An aspect ratio of a closed convex set N with int N 6= ∅2 coincides with the ratio of the minimum radius of the disk which 2

int N is the set of interior points of N

Complexity and approximability of guarding of proximity graphs

3

contains N to the maximum radius of the disk which is contained in N. For example, each object Nr (e) has the aspect ratio equal to 1 + d(e) 2r where d(e) is the length of the edge e. APX-hardness of the discrete3 Hitting Set problem is presented for families of (non-equal height and width) axis-parallel rectangles [10] and triangles of bounded aspect ratio [16]. Known Vertex Cover (VC) problem for planar graphs can be obtained from IPGD by setting r = 0. For the case of IPGD where G consists of isolated vertices (i.e. segments from E are all of zero length) we have the Continuous Disk Cover (CDC) problem. Strong NPhardness is well known for CDC [18] and VC problems (see e.g. [11], [19]). Results. Our results report complexity and approximation algorithms for the IPGD problem within several classes of plane graphs under different assumptions on r. Let S be a set of n points in general position on the plane no 4 of which are cocircular. We call a plane graph G = (V, E, F ) a Delaunay triangulation if V = S and [u, v] ∈ E iff there is a disk T such that u, v ∈ bd T 4 and S∩int T = ∅. Consider an oriented equilateral unit sided triangle ∇ whose barycenter is the origin and one of its vertices is on the negative y-axis. Denote ∇(p, λ) = p+λ∇ = {x ∈ R2 : x = p + λa, a ∈ ∇} for some p ∈ R2 and λ > 0. Suppose the set S is such that a straight line through any pair of its points does not make angles 0◦ , 60◦ and 120◦ with x-axis. A plane graph G = (V, E, F ) with V = S is called a TD-Delaunay triangulation when [u, v] ∈ E iff there are p ∈ R2 and λ > 0 such that u, v ∈ bd ∇(p, λ) and S ∩ int ∇(p, λ) = ∅. Finally, a plane graph G = (V, E, F ) with V = S is named as nearest neighbour graph when [u, v] ∈ E iff either u or v is the nearest Euclidean neighbour for v or u respectively. Hardness results. Our first result claims strong NP-hardness of IPGD within the class of Delaunay triangulations and some known classes of their connected subgraphs (Gabriel, relative neighbourhood graphs) for r ∈ [dmin , dmax ] and = O(n) where n is the number of vertices in G. IPGD remains strongly µ = ddmax min NP-hard within the class of nearest neighbour graphs for r = [dmax , ηdmax ] with large constant η and µ ≤ 4. Furthermore, we have the same hardness results under the same restrictions on r and µ even if we are bound to choose points of C close to vertices of G. The upper bound √on µ for Delaunay triangulations is 3 comparable with the lower bound µ = Ω n2 which holds true (with positive probability) for Delaunay triangulations produced by n random independent points on the unit disk [3]. Thus, declared restrictions on r and µ define natural instances of IPGD. An upper bound on µ implies an upper bound on the ratio of the largest and the smallest aspect ratio of objects from Nr (E). The Hitting Set problem is generally easier when sets from N have almost equal aspect ratio bounded from above by some constant. Our result for the class of nearest neighbour graphs gives the problem hardness in the case where objects of Nr (E) have almost equal constant aspect ratio. In distinction to known results for the Hitting Set problem mentioned above our study is mostly for its continuous setting with the structured system Nr (E) formed by an edge set of a specific plane graph; each set from Nr (E) is of the special form of Minkowski sum of 3 4

when U coincides with some prescribed finite set bd T denotes the set of boundary points of T.

4

Konstantin Kobylkin

some graph edge and the radius r disk. Our proofs are elaborate complexity reductions from the CDC problem which is intimately related with IPGD. Our second result is about computational complexity of the IPGD problem for r = 0 (which coincides with the VC problem implying C ⊂ V ) within the class of planar triangulations. Namely, strong NP-hardness of VC is reported within the class of 4-connected (Hamiltonian) planar triangulations of bounded vertex degree with triangular outer face. It presents a novel contribution into complexity analysis of the VC problem which is usually taken in the literature within graph classes that are far away from planar and geometric triangulations, see e.g. [11], [19]. It is known that 4-connected planar triangulations with triangular outer face are isomorphic to Delaunay triangulations due to [14] which supports the conjecture that VC is NP-hard within the class of 4-connected Delaunay ones. Applying known result from graph drawing we obtain strong NP-hardness of VC within the class of 4-connected TD-Delaunay triangulations. Positive results. Let R(E) be the smallest radius of the disk that intersects all segments from the edge set E. As opposed to the cases where either r = 0 or r ∈ [dmin , dmax ] or r = [dmax , ηdmax ], IPGD is solvable within the class of simple plane graphs, for which the inequality r ≥ ηR(E) holds uniformly for some fixed l √ m2 η > 0, in time O k 2 |E|2k+1 with k = η2 . Thus, IPGD is parameterized with respect to r. Above inequality implies an upper bound k on its optimum. Taking into account proof of W [1]-hardness of parameterized version of CDC [17] and the reduction used to prove theorem 2 of this paper, it seems unlikely to improve this time bound to O(f (k)|E|c ) for any function f and constant c > 0. Finally, we present an 8p(1+2λ)-approximation O(|E| log |E|)-time algorithm for IPGD when the inequality r ≥ dmax 2λ holds true uniformly within some class of simple plane graphs for a constant λ > 0, where p(x) is the smallest number of unit disks needed to cover any disk of radius x > 1. It corresponds to the case where objects from Nr (E) have their aspect ratio bounded from above by 1 + λ. A similar but more complex Ω(|E|2 )-time constant factor approximation algorithm is given in [9] to approximate the Hitting Set problem for sets of objects whose generalized aspect ratio is bounded from above by some constant.

2 2.1

Hardness results Problem preprocessing

Before presenting hardness results for the IPGD problem we are to take some problem preprocessing. It is aimed at reducing a set of points, among which centers of radius r disks are chosen, to a finite set of cardinality which is bounded from above by some polynomial in |E|. As was mentioned in the introduction, the IPGD problem coincides with the Hitting Set problem considered for Euclidean r-neighbourhoods of graph edges which form the system denoted by Nr (E). Their boundaries are composed of four parts: a pair of half-circles and a pair of equal parallel straight line segments. We can assume (roughly) that nonempty intersection of an arbitrary number of objects from Nr (E) contains

Complexity and approximability of guarding of proximity graphs

5

a point from the intersection of boundaries of some pair of objects from Nr (E). Thus, our choice of points, forming a feasible solution to the IPGD problem, can be restricted to the set of intersection points of pairs of boundaries of objects from Nr (E). This preprocessing serves two main purposes: first, it gives that IPGD is in NP; second, it grounds the polynomial time algorithm for IPGD from subsection 3.1. Lemma 1. Let G = (V, E) be a simple plane graph. Each feasible solution C to the IPGD problem for G can be converted in polynomial time and space (in |E|) to a feasible solution D ⊂ Dr (G) of IPGD for G with |D| ≤ |C| where Dr (G) ⊂ Q2 is some set of the cardinality of the order O(|E|2 ) which can be constructed in polynomial time and space. Proof. Below we keep some notations from the introduction. First, let us describe a construction of the set Dr (G). W.l.o.g. it can be assumed that for each N1 ∈ Nr (E) there exists N2 ∈ Nr (E) with N1 ∩N2 6= ∅. Otherwise, an arbitrary vertex of each such edge e ∈ E is added to Dr (G) that has Nr (e) non-intersecting with the other objects from Nr (E). It can be done in polynomial time. Consider an arbitrary maximum (with respect to inclusion) subsystem of objects from Nr (E) whose intersection is nonempty. It is called a maximum feasible subsystem (MFS for short). Let M be the set which coincides with the intersection of all objects from an arbitrary MFS. The set M is bounded, closed and convex. Moreover, M has a boundary which is composed of pieces of boundaries of objects from Nr (E). The set M is contained in the intersection of at least two objects from Nr (E). Consider the degenerate case where two edges e1 and e2 from E are parallel and have their r-neighbourhoods touching in such a way that touching points form a segment. In this case |bd Nr (e1 ) ∩ bd Nr (e2 )| = ∞. The same is also possible if edges e1 and e2 intersect at their common vertex. In both cases the intersection bd Nr (e1 ) ∩ bd Nr (e2 ) has at most two extreme points. In the other (non-degenerate) ones |bd Nr (e1 ) ∩ bd Nr (e2 )| is bounded from above by some constant taking into account the fact that each set bd Nr (e), e ∈ E, is a compound of a pair of parallel straight line segments and a pair of half-circles. Set [ Cr (G) = extr (bd Nr (e1 ) ∩ bd Nr (e2 )) , e1 ,e2 ∈E, e1 6=e2

where extr N means the set of (extreme in two aforementioned degenerate cases) points of one-dimensional set N. The set M has at least one vertex which belongs to Cr (G). Moreover, |Cr (G)| = O(|E|2 ). The system Nr (E) forms an arrangement Ar (E) which can be computed in polynomial time [2]. Therefore for every c ∈ extr (bd Nr (e1 ) ∩ bd Nr (e2 )) a rational point d(c) ∈ Nr (e1 ) ∩ Nr (e2 ) can be found in some cell z of Ar (E) with c ∈ z such that d(c) ∈ z. Set Dr (G) = {d(c) : c ∈ Cr (G)}. Second, let C be a feasible solution to the IPGD problem. For each point c ∈ C a point d(c) ∈ Dr (G) can be found in polynomial time such that {N ∈ Nr (E) : c ∈ N } ⊆ {N ∈ Nr (E) : d(c) ∈ N }.

6

Konstantin Kobylkin

Thus, the set D = {d(c) : c ∈ C} is feasible for IPGD and |D| ≤ min{|C|, |Dr (G)|}. The following assertion is an immediate consequence of lemma 1. Corollary 1. The IPGD problem belongs to the class NP. 2.2

Hardness for r ∈ [dmin , dmax ]

We start our complexity analysis for the IPGD problem by considering its setting where r ∈ [dmin , dmax ]. Under this restriction on r IPGD coincides neither with VC nor with CDC. In fact it is equivalent (see the introduction) to the geometric Hitting Set problem for the set Nr (E) of Euclidean r-neighbourhoods of edges of G. For the IPGD problem we claim its hardness even if we restrict the graph G to be either a Delaunay triangulation or some of its known subgraphs or a bounded from above, TD-Delaunay triangulation. We keep the ratio µ = ddmax min thus, imposing an upper bound on the ratio of the largest and the smallest aspect ratio of objects from Nr (E). We show that IPGD remains intractable even in its simple case where r = Θ(dmax ) and µ is bounded by some small constant or, equivalently, when objects of Nr (E) have close constant aspect ratio. Our first hardness result for IPGD is obtained by using a complexity reduction from the CDC problem. Below we describe a class of hard instances of the CDC problem which correspond to hard instances of the IPGD problem for Delaunay triangulations with relatively small upper bound on the parameter µ. Hardness result for the CDC problem. To single out the class of hard instances of the CDC problem we use a reduction from the strongly NP-complete minimum dominating set problem which is formulated as follows: given a simple planar graph G0 = (V0 , E0 ) of degree at most 3, find the smallest cardinality set V00 ⊆ V0 such that for each u ∈ V0 \V00 there is some v = v(u) ∈ V00 which is adjacent to u. Below an integer grid denotes the set of points on the plane with integervalued coordinates within some bounded interval. An orthogonal drawing of the graph G0 on some integer grid is the drawing whose vertices are represented by points on that grid whereas its edges are given in the form of polylines formed by sequences of connected axis-parallel straight line segments of the form [p1 , p2 ], [p2 , p3 ], . . . , [pk−1 , pk ] intersecting only at the edge endpoints, where each point pi again belongs to the grid. In [18] strong NP-completeness of CDC is proved by reduction from the minimum dominating set problem. This reduction involves using plane orthogonal drawing of G0 on some integer grid. More specifically, a set D is build on that grid with V0 ⊂ D. The resulting hard instance of the CDC problem is for the set D and some integer (constant) radius r0 ≥ 1. Let us observe that G0 admits an orthogonal drawing (theorem 1 [22]) on the grid of size O(|V0 |) × O(|V0 |) whereas total length of each edge is O(|V0 |). Proof of the strong NP-completeness of CDC could be conducted taking into account this observation. We can formulate (see combination of theorems 1 and 3 [18]) Theorem 1. (Masuyama et. al., 1981 [18]) The CDC problem is strongly NPcomplete for a constant integer radius r0 and the point set D on the integer

Complexity and approximability of guarding of proximity graphs

7

grid of size O(|D|) × O(|D|). It remains strongly NP-complete even if we restrict centers of radius r0 disks to be at the points of D. Remark 1. For every simple planar graph G0 of degree at most 3 its orthogonal drawing can be constructed such that at least one its edges is a polyline which is composed of two axis-parallel segments. The IPGD problem hardness for Delaunay triangulations. To build a reduction from the CDC problem for the set D constructed in theorem 1 we exploit a simple idea that a radius r disk covers a set of points D0 ⊂ D iff a slightly larger disk covers straight line segments, each of which is close to some point of D0 and has a small length with respect to distances between points of D. Then a proximity graph H is build whose vertex set coincides with the set of endpoints of small segments corresponding to points of D. Since H usually contains these small segments as its edges, this technique gives hardness for the IPGD problem within a variety of classes of proximity graphs. The following technical lemma holds. Lemma 2. Let X ⊂ Z2 , r ≥ 1 be some integer and ρ(u; v, w) be the minimum of two distances from u ∈ X to circles of radius r passing through distinct points v and w from X with |v − w|2 ≤ 2r. Then min

u∈C(v,w),v6 / =w,u,v,w∈X,|v−w|2 ≤2r

ρ(u; v, w) ≥

1 100r4

where C(v, w) is the union of two radius r circles through v and w. The complexity of the following restricted form of IPGD is also studied. Vertex Restricted IPGD (VRIPGD(δ)): given a simple plane graph G = (V, E, F ), a constant δ > 0 and a constant r > 0, find the least cardinality set C ⊂ Q2 such that each S e ∈ E is within (Euclidean) distance r from some point c = c(e) ∈ C and C ⊂ Bδ (v). v∈V

Theorem 2. Both IPGD and VRIPGD(δ) problems are strongly NP-hard for r ∈ [dmin , dmax ], µ = O(n) and δ = Θ(r) within the class of Delaunay triangulations where n is the number of vertices in triangulation. Proof. By corollary 1 IPGD belongs to NP. Using the similar preprocessing algorithm it can be shown that VRIPGD is also in NP: namely, pairwise intersection points are sought for the system of boundaries of objects from Nr (E) augmented with radius δ circles centered at the vertices from V. Let us prove that IPGD is strongly NP-complete. Proof technique for the VRIPGD(δ) problem is analogous. For any hard instance of the CDC problem given in theorem 1 we build the IPGD problem instance with r = r0 + δ as follows where δ = 40012 2r9 . For every u ∈ D points u0 and v0 are found such 0 that |u − u0 |∞ ≤ δ/4 and |u − v0 |∞ ≤ δ/4 where Iu = [u0 , v0 ] has a length at least δ/4. More specifically, let us set ID = {Iu = [u0 , v0 ] : u ∈ D}. Endpoints of

8

Konstantin Kobylkin

segments from ID are constructed in sequential manner in polynomial time and space by defining a new segment Iu to provide generality position for the set of endpoints of the set ID0 ∪ {Iu }, D0 ⊂ D, where segments of ID0 are already defined. Here endpoints of Iu are chosen in the rational grid that contains u c1 c1 whose elementary square size is |D| 2 × |D|2 for some small absolute rational constant c1 . Assuming u = (ux , uy ), the point u0 is chosen in the lower part of the grid with y-coordinates less than uy − δ/8 whereas v0 is taken from the upper one for which y-coordinates exceed uy + δ/8. Let S be the set of endpoints of segments from ID . Every disk having Iu as its diameter does not contain any points of S distinct from endpoints of Iu . Let G = (V, E, F ) be a Delaunay triangulation for S which can be computed in polynomial time and space in |D|. Obviously, each segment Iu coincides with some edge from E. We have dmin ≤ r and µ = O(n). It remains to prove that r ≤ dmax . Due to remark 1 and the construction of the set D (see fig. 1 [18]) the set S can be constructed such that the inequality r ≤ dmax holds true for G. Moreover, representation length for vertices of V is polynomial with respect to representation length for points of D. Let k be a positive integer. Obviously, centers of at most k disks of radius r0 containing D in their union give centers of radius r ≥ r0 disks whose union is intersected with each segment from E. Conversely, let T be a disk of radius r0 +δ which intersects a subset ID0 = {Iu : u ∈ D0 } of segments for some D0 ⊆ D. When |D0 | = 1, it is easy to transform T to a disk which contains the segment ID0 . Points of D have integer coordinates. Moreover, squared (Euclidean) distance between each pair of points of the subset D0 does not exceed (2r0 + 4δ)2 = 4r02 + 16r0 δ + 16δ 2 . Therefore points from D0 are located within the distance 2r0 from each other. Let us use Helly theorem. Let R be the minimum radius of the disk T0 containing any triple u1 , u2 and u3 from D0 . W.l.o.g. we suppose that, say, u1 and u2 are on the boundary of T0 . Obviously, R ≤ r0 + 2δ. Let us show that the case R > r0 is void. We slightly shift the center of T0 (along the midperpendicular to [u1 , u2 ]) to have u1 and u2 at the distance r0 from the shifted center O0 . The distance √ from the point u13 to the radius r0 circle centered at O0 does not exceed 2 r0 δ + δ 2 + 2δ < 100r 4 . By lemma 2 we have 0 R ≤ r0 . Thus, D0 is contained in some disk of radius r0 . Given a set of points, the smallest radius disk can be found in polynomial time and space which covers this set. Therefore we can convert any set of at most k disks of radius r whose union is intersected with each segment from E to some set of at most k disks of radius r0 whose union covers D. Using corollary 1 of section √ 4.2 from [3] and theorem 1 from [20] we arrive 3 at the lower bound µ = Ω n2 which holds true with positive probability for Delaunay triangulations produced by n random uniform points on the unit disk. Thus, the order of the parameter µ for the considered class of hard instances of the IPGD problem is comparable with the one for random Delaunay triangulations.

Complexity and approximability of guarding of proximity graphs

9

The IPGD problem hardness for other classes of proximity graphs. The same proof technique could be applied for proving the problem hardness within the other classes of proximity graphs. Let us start with some definitions. The following graphs are connected subgraphs of Delaunay triangulations. A plane graph G = (V, E, F ) with V = S is called a Gabriel graph where [u, v] ∈ E iff the disk having [u, v] as its diameter does not contain any other points of S distinct from u and v. A relative neighbourhood graph is the plane graph G with V = S for which [u, v] ∈ E iff there is no any other point w ∈ S such that w 6= u, v with max{|u − w|2 , |v − w|2 } < |u − v|2 . Finally a plane graph is called a minimum Euclidean spanning tree if it is the minimum weight spanning tree of the weighted complete graph K|S| whose vertices are points of S such that its edge weight is given by the Euclidean distance between the edge endpoints. The proof of theorem 2 is extendable for classes of Gabriel, relative neighbourhood and nearest neighbour graphs. It is easy to observe that segments Iu , u ∈ D, form a subset of edges of any minimum Euclidean spanning tree. In contrary, suppose a spanning tree H does not contain segment Iu but contains edges incident to both endpoints u0 and v0 of Iu for some u ∈ D. Let us remove the edge e = [u0 , w0 ] from H and add the edge Iu to it where w0 is the parent of u0 in H. Obviously the resulting tree has the smaller weight, a contradiction. Corollary 2. Both IPGD and VRIPGD(δ) problems are strongly NP-hard for r ∈ [dmin , dmax ], µ = O(n) and δ = Θ(r) within classes of Gabriel, relative neighbourhood graphs, TD-Delaunay triangulations and minimum Euclidean spanning trees and for r ∈ [dmax , ηdmax ] and µ ≤ 4 within the class of nearest neighbour graphs where η is a large constant. Remark 2. In theorem 2 and corollary 2 hardness is given for r = Θ(dmin ) within graph classes except for nearest neighbour ones. 2.3

Hardness for r = 0

We continue our complexity analysis of the IPGD problem by studying its extreme case where r = 0. In this case the IPGD problem coincides with the VC problem. Below the VC problem hardness is given within the class of highly connected TD-Delaunay triangulations. Moreover, some directions are also provided for proving hardness of VC within the class of Delaunay triangulations. A planar triangulation is called special if faces of its plane embedding are all triangles. Our proof method is simple. First, the VC problem hardness is given within the class of highly connected simple special planar triangulations. Second, some result from graph drawing is applied to embed every simple special planar triangulation in the form of TD-Delaunay triangulation in polynomial time which translates obtained complexity result from special planar triangulations to TDDelaunay ones. The VC problem hardness for planar triangulations. We start by formulating an auxiliary result claiming hardness of the following problem.

10

Konstantin Kobylkin

Maximum Facial Independent Set (Facial MIS). Given a special plane triangulation G = (V, E, F ) (which may contain parallel edges) find the smallest cardinality set of faces F 0 ⊂ F such that each pair of faces from F 0 does not have an edge from E in common. Here the notion of plane graph is used for plane embeddings of non-simple planar graphs whose edges are of the form of piecewise linear polylines. This problem is the dual form of known maximum independent set problem considered for some simple 2-connected 3-regular plane graph G0 = (V0 , E0 , F0 ). This can be easily seen by applying the classical geometric duality transform [13] for G0 , where vertices of V0 correspond to faces of the dual special triangulation G, whereas faces of F0 correspond to vertices of G; finally edges of E0 are in oneto-one correspondence with edges of G. Theorem 4.1 [19] reports the following Theorem 3. (Mohar, 2001 [19]) Facial MIS problem (in its decision form) is strongly NP-complete within some subclass T0 of special planar triangulations possibly containing parallel edges. Remark 3. In the proof of theorem 4.1 [19] it is also noted that each graph G0 ∈ T0 can be almost uniquely embedded on the plane in polynomial time. Theorem 4. The VC problem is strongly NP-hard within the class of simple special 4-connected (Hamiltonian) planar triangulations of degree at most log n where n is the number of triangulation vertices. Proof. The VC problem is obviously in NP because every subset of triangulation vertices has cardinality of the order O(n). Let T be the class of simple special planar (3-connected) triangulations. We give a reduction to VC from Facial MIS which is strongly NP-complete by theorem 3. For each plane graph G0 = (V0 , E0 , F0 ) ∈ T0 we construct a v1 b

v15 v1

v5

b

v7

b

b

G(f01) v5

v3

b

b

b

b b

b

b

v16

v9

b

b

v12

v6 b

b

v3

b

b

b

b

v4

b b

v18 b

v17

v8 b

v2

b

b

b

v10

v8 b

v4

b

v7

b

b

b

b

v6

v10 v9

b

b

b

v19

G(f0)

G(f02)

v11 b

v2

v13

v14

Fig. 1. A graph G(f0 ) corresponding to a Fig. 2. Graphs G(f01 ) and G(f02 ) along face f0 of G0 with their connector graph G(e0 )

special plane triangulation G = (V, E, F ) ∈ T as follows. Each face f0 = v1 (f0 )v2 (f0 )v3 (f0 ) ∈ F0 \{f0,∞ } is replaced by a graph G(f0 ) shown on the fig. 1 (symbol f0 is omitted in notations of vertices of G(f0 ) in the sequel) where

Complexity and approximability of guarding of proximity graphs

11

f0,∞ is the outer face of G0 . It is easy to see that the minimum vertex cover for the graph G(f0 ) is unique. Moreover, it does not contain v10 and coincides with Uf0 = {v4 , v5 , v6 , v7 , v8 , v9 }. It follows from the fact that each triangle from the triple of triangles ∆v1 v5 v7 , ∆v2 v6 v8 and ∆v3 v4 v9 (see fig. 1) requires two vertices to be covered. Considering all possible cases we get that |Vf00 | ≥ 7 for every vertex cover Vf00 if v10 ∈ Vf00 . Moreover, Wf0 = {v1 , v2 , v3 , v4 , v5 , v6 , v10 } is a vertex cover for G(f0 ). The graph G(f0 ) is placed inside each face f0 such that sides of the generally curvilinear triangle v1 v2 v3 are parallel to sides of the face f0 e.g. by scaling that face using a given rational factor. Graphs G(f01 ) and G(f02 ) corresponding to faces f01 and f02 from F0 \{f0,∞ } are joined using a connector graph G(e0 ) as shown in the fig. 2 where faces f01 and f02 have a common edge e0 ∈ E0 . The graph G(e0 ) has two common edges with each of two graphs G(f01 ) and G(f02 ). The graph G(f0,∞ ) corresponding to the face f0,∞ is isomorphic to the graph G(f0 ) for an arbitrary f0 ∈ F0 \{f0,∞ }. It is shown in the fig. 3 together with respective connector graphs Gi , i = 1, 2, 3. (graphs Gi are depicted schematically without some edges and vertices). Edges of graphs G(e0 ), e0 ∈ E0 , can be

b

v1 b

b

v3 v8

b

b

b

b b

b

b

b

b b

M(v0 )b

b

v6

b

b

b

b

v10

b

b

v7

G1

G3 G b 2 b

v4

v5

b

b

b

v0 b

b

b

b

b

b b

v2 b

v9 b

b b

b

Fig. 3. The graph G(f0,∞ ) constructed for Fig. 4. Graphs G(f0 ) and connectors the outer face of G0 G(e0 ) bounding polygon M (v0 )

embedded in such a way that edges of graphs G(e01 ) and G(e02 ) intersect at most at their common endpoints (if exists) for every e01 and e02 from E0 . For every v0 ∈ V0 each set E0 (v0 ) = {e0 ∈ E0 : v0 ∈ e0 } corresponds to the set Q(v0 ) = {G(e0 ) : e0 ∈ E0 (v0 )} of deg(v0 ) connector graphs where deg(v0 ) = |E0 (v0 )|. The vertex v0 is in the interior of some generally curvilinear simple 2deg(v0 )-gon M (v0 ) bounded by edges of connectors from Q(v0 ) (see fig. 4). Let U (v0 ) = {u1 , . . . , u2deg(v0 ) } be a clockwise ordered sequence of vertices of M (v0 ) where u1 coincides with one of two vertices v14 or v15 of connector graphs from Q(v0 ). Triangulate M (v0 ) as follows (and denote the graph thus obtained by G(v0 )) : add series of edges {u1 , u3 }, {u3 , u5 }, . . . then series of edges {u1 , u5 }, {u5 , u9 }, . . . continuing doing so until M (v0 ) becomes triangulated.

12

Konstantin Kobylkin

Let us set G :=

[ f0 ∈F0

G(f0 ) ∪

[ v0 ∈V0

G(v0 ) ∪

[

G(e0 ).

e0 ∈E0

Since the number of graph connectors is equal to |E0 | and the degree of each vertex from G(v0 ) is of the order O(log |E0 |), the complexity of construction of the graph G for the graph G0 is of the order O(|F0 | + |E0 | log |E0 |). Moreover, the number of vertices, edges and faces of G is of the order O(|F0 | + |E0 |). The length of representation of vertices of G differs by a constant factor from the representation length of vertices of G0 . Since G is a simple special triangulation (we “subdivided” all parallel edges of G0 ), the graph G is 3-connected. The graph G does not have any edges that connect vertices from G(f0 )\G(e0 ) with those from G(e0 )\G(f0 ); the same holds for G(f0 ) and G(v0 ) as well as for G(e0 ) and G(v0 ). Considering all possible cases shows that for every graph G(f0 ), f0 ∈ F0 , and G(e0 ), e0 ∈ E0 , each their 3-cycle coincides with their face. In the process of triangulating of M (v0 ) the only triangles that we have are faces by construction. Therefore each 3-cycle of G is its face. Using lemma 2.3 [12] we get 4-connectedness of G. Let us show that each independent set of faces F00 ⊂ F0 with |F00 | ≥ k can be converted into some vertex cover V 0 for G where |V 0 | ≤ 7|F0 | − k + 4|E0 |. Set Vf00 := Uf0 for every f0 ∈ F00 ; for each f0 ∈ F0 \F00 set Vf00 := Wf0 . To get a vertex cover Ve00 of the graph G(e0 ) that connects graphs G(f01 ) and G(f02 ), f01 , f02 ∈ F0 , vertices v14 and v15 are taken along with one vertex per each pair {v16 , v19 } and {v17 , v18 }. More specifically, we choose a vertex from each such pair depending on whether f01 ∈ F00 and f02 ∈ F00 : namely, if f01 ∈ F00 and f02 ∈ / F00 , then vertices v17 and v19 are chosen; choice of vertices S from0 these S pairs is arbitrary in the case where f01 , f02 ∈ / F00 . Obviously, V 0 = Vf0 ∪ Ve00 f0 ∈F0

e0 ∈E0

covers edges of G and |V 0 | ≤ 7|F0 | − k + 4|E0 |. Conversely, let V 0 be the minimum cardinality vertex cover for G and |V 0 | ≤ 7|F0 | − k + 4|E0 |. Set Vf00 = V 0 ∩V (f0 ) for each f0 ∈ F0 where V (f0 ) is the vertex set of the graph G(f0 ). At least two vertices are required to cover edges that belong to both G(f01 ) and G(e0 ) or to both G(f02 ) and G(e0 ), where e0 is the common edge of faces f01 and f02 . At least 4 vertices are needed to cover edges of triangles over sets of vertices V1 (e0 ) = {v15 , v16 , v19 } and V2 (e0 ) = {v14 , v17 , v18 }. If either |V 0 ∩ V1 (e0 )| = 2 or |V 0 ∩ V2 (e0 )| = 2, then |Vf002 | ≥ 7 for |Vf001 | = 6. In view of minimality of V 0 the case is void when |V 0 ∩ V1 (e0 )| = |V 0 ∩ V2 (e0 )| = 3 and |Vf001 | = |Vf002 | = 6 hold true simultaneously. Therefore the face set F00 = {f ∈ F0 : |Vf00 | = 6} is independent. Since |V 0 ∩ (V1 (e0 ) ∪ V2 (e0 ))| ≥ 4, we have 6|F00 | + 7(|F0 | − |F00 |) + 4|E0 | ≤ |V 0 | so |F00 | ≥ k. The VC problem hardness for geometric triangulations. The following result holds true (corollary 2 [5]). Lemma 3. (Bonichon et. al, 2010 [5]) For every simple special planar triangulation, an isomorphic TD-Delaunay triangulation can be constructed in polynomial time and space.

Complexity and approximability of guarding of proximity graphs

13

Combining results of theorem 4 and lemma 3 we get Corollary 3. The VC problem is strongly NP-hard within the class of 4-connected (Hamiltonian) TD-Delaunay triangulations of degree at most log n where n is the number of triangulation vertices. In spite of the fact that any special 4-connected planar triangulation is isomorphic (in usual graph sense) to some Delaunay triangulation [14] it is still an open (possibly NP-hard) problem of getting combinatorially equivalent Delaunay triangulation for a given special 4-connected planar one. Nevertheless we hope it is possible to do in polynomial time and space for the subclass of special 4-connected planar triangulations constructed in the proof of theorem 4. Conjecture 1. The VC problem is strongly NP-hard within the class of 4-connected Delaunay triangulations.

3 3.1

Positive results Polynomial solvability of IPGD for large r

In distinction to the cases where either r = 0 or r ∈ [dmin , dmax ] or r = Θ(dmax ) the IPGD problem is polynomially solvable for large r = Ω(R(E)) where R(E) is the smallest radius of the disk that intersects straight line segments of the edge set E. It is obvious that IPGD is solvable in time O(|E|) [4] within the class of plane graphs for which the inequality r ≥ R(E) holds uniformly. Let us consider the case of IPGD within the class of plane graphs for which r ≥ ηR(E) for some 0 < η < 1. In this case IPGD is parameterized with respect to r. Since every disk of radius r contains an axis-parallel rectangle l√ m2 l √ m2 √ 2R(E) 2 = whose side is equal to r 2, roughly at most k = k(η) = r η radius r disks are needed to intersect all segments from E. Therefore the bruteforce search algorithm could be applied that just sequentially tries each subset of Dr (G) (see subsection 2.1) of cardinality at most k. This amounts roughly to the O k 2 |E|2k+1 time complexity. Thus, we arrive at the polynomial time algorithm whose complexity depends exponentially on η. This algorithm gives an optimal solution to the IPGD problem taking into account lemma 1. 3.2

Approximation algorithm for IPGD

Let us focus on the case of the IPGD problem where the inequality r ≥ dmax 2λ holds uniformly within some class G of simple plane graphs for a constant λ > 0. It corresponds to the situation where objects from the system Nr (E) have their aspect ratio bounded from above by 1 + λ. It turns out that in this case the problem admits an O(1)-approximation algorithm whose factor depends on λ. Let us consider the following problem: Cover endpoints of segments with disks (CESD): given an arbitrary set SE ⊆ V which covers the set E, find the smallest cardinality set of radius r disks whose union covers SE .

14

Konstantin Kobylkin

Denote by OP TCESD (SE , r) and OP TIP GD (G, r) the optima of CESD and IPGD respectively considered within the graph class G. Let us formulate an Algorithm 1. Compute and output 8-approximate solution to the CESD problem using O(|E| log OP TCESD (SE , r))-time algorithm from [15]. The statement below bounds the ratio of optima for these problems. Statement 5 Suppose the inequality r ≥ dmax 2λ holds uniformly within the class G for some constant λ > 0. The following bound holds true: OP TCESD (SE , r) ≤ p(1 + 2λ) OP TIP GD (G, r) where p(x) is the smallest number of unit disks needed to cover radius x disk. Proof. Let C0 ⊂ Q2 be an optimal solution to the IPGD problem. Set E(c) = {e ∈ E : c ∈ Nr (e)}, c ∈ C0 . For every e ∈ E(c) there is a point de (c) ∈ e with |c − de (c)|2 ≤ r. Each point from the set S(c) of endpoints of segments from E(c) is within Euclidean distance r + dmax from c. Therefore the radius r + dmax disk centered at c covers the set SE ∩ S(c). At most p(1 + 2λ) radius r disks are needed to cover any radius r + dmax disk. Thus, the declared bound for the optima ratio holds true. Corollary 4. The algorithm 1 is 8p(1 + 2λ)-approximate. Remark 4. Approximation factor of the algorithm 1 is in fact lower when G is the subclass of Delaunay triangulations or their subgraphs. Indeed, in this case there is no need to cover the whole radius r + dmax disk with radius r disks.

4

Conclusion

Complexity and approximability are studied for the problem of intersecting a structured set of straight line segments with the smallest number of disks of radii r > 0 where a structural information about segments is given in the form of an edge set of a proximity graph. It is shown that the problem is strongly NPhard within the class of Delaunay, TD-Delaunay triangulations and some of their subgraphs for small and medium values of r while for large r it is polynomially solvable. Fast O(1)-approximation algorithm is given for medium values of r.

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Complexity and approximability of guarding of proximity graphs

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16

Konstantin Kobylkin

A

Proof of Lemma 2

Proof. Let u = (x, y), v = (x1 , y1 ) and w = (x2 , y2 ) be distinct points of X. Consider an arbitrary circle of radius r among a pair of ones through v and w (denote its center by O). We are to bound the distance π(u; v, w) from that circle to the point u ∈ / C(v, w) from below. q Let us denote ∆ = |v − w|2 , λ = ⊥

r2 −

∆2 4 ,

a = (u − v, u − w) and b =

⊥

(u − v, (v − w) ) where (v − w) = ±(y1 − y2 , −x1 + x2 ). The distance π could be written in the form: 2λb ⊥ v + w a + (v − w) ∆ . π = π(u; v, w) = −λ − u − r = q 2 |v − w|2 a + 2λb + r2 + r 2 ∆ W.l.o.g. we can consider the case where u lies inside the disk of radius 2r centered at O. Indeed, otherwise π ≥ r ≥ 1r . Let us bound a denominator of the fraction π taking into account that ∆ ≤ 2r, |u − v|2 ≤ 3r and b/∆ ≤ 3r : r 2λb + r2 + r ≤ 5r. a+ ∆ As points of X have integer coordinates, then a and b are integers and π > 0. 1 When ∆ = 4r2 , we get π ≥ 5r . For ∆2 ≤ 4r2 − 1 let us show that a + 2λb ≥ 1 ∆ 20r3 in each of two cases where 2λb ∆ is either an integer or fractional. For the integer 2λb 1 we arrive at the bound π ≥ 5r . ∆ 2λb Assume that ∆ has a nonzero fractional part. Let q = 2λ and k = 2λ ∆ ∆ where {·} and [·] denote fractional and integer parts of a real number respectively. 2 2 First we consider the case where 4r ∆2 is not an integer. We have 2kq + q ≥ n o

{2kq + q 2 } =

4r 2 ∆2

, whence s

q≥

k2 +

n

4r2 ∆2

−k ≥ 2

4r 2 ∆2

5r

o ≥

1 20r3

1 4r and π ≥ 100r 4 . Now we suppose that ∆2 is an integer. Let us denote by m 2λ 2 2 the smallest integer with m2 < 4λ ≥ 2 < (m + 1) . As q > 0 we have ∆ ∆ √ m2 + 1 . Due to concavity of square root we can write: np o 1 1 1 1 2 m +1 ≥ m+ = ≥ 4λ ≥ 2m + 1 2m + 1 5r ∆ +1

which gives π ≥

1 25r 2 .