On Dominating Sets for Pseudo-disks

Transcription

1 On Dominating Sets for Pseudo-disks Boris Aronov 1, Anirudh Donakonda 1, Esther Ezra 2,, and Rom Pinchasi 3 1 Department of Computer Science and Engineering Tandon School of Engineering, New York University Brooklyn, NY 11201, USA boris.aronov@nyu.edu, ad2930@nyu.edu. 2 Department of Mathematics Georgia Institute of Technology Atlanta, Georgia 30332, USA eezra3@math.gatech.edu. 3 Department of Mathematics, Technion - Israel Institute of Technology, Haifa, Israel room@math.technion.ac.il. Abstract. We study the problem of computing the smallest weighted dominating set in a collection of pseudo-disks in the plane, i.e., a minimumsize subset of the given set of pseudo-disks so that every pseudo-disk is either selected or intersected by a selected pseudo-disk. A similar problem has recently been investigated by Gibson and Pirwani (ESA 2010) for disk graphs, whereas the case of pseudo-disks remains open. Using the machinery of Chan et al. (SODA 2012) we present a randomized expected polynomial-time algorithm that achieves a constant approximation factor of the smallest weighted dominating set in a collection of pseudo-disks. 1 Introduction Problem statement. We are given a finite collection P of pseudo-disks in the plane, i.e., a collection of compact regions each bounded by a Jordan curve such that any two of their boundaries cross twice or not at all. We define the intersection graph G of P in the standard manner, that is, the vertex set is P and there is an edge between two pseudo-disks if their intersection is non-empty. Work on this paper by Boris Aronov has been supported by NSA MSP Grant H , by NSF Grants CCF , CCF , CCF , and CCF , and by BSF grant 2014/170. Work on this paper by Anirudh Donakonda has been partially supported by NSF Grant CCF Work on this paper by Esther Ezra has been supported by NSF under grants CAREER CCF , CCF , and CCF Work on this paper by Esther Ezra was initiated when she was at the Courant Institute of Mathematical Sciences, New York University, New York, NY 10012, USA, and continued at the Department of Computer Science and Engineering, Polytechnic (now Tandon) School of Engineering, New York University, Brooklyn, NY 11201, USA.

2 The dominating set problem for G is to find a smallest subset D P, such that each vertex in G is either in D or is adjacent to a vertex in D. In other words, this is a smallest subset of P such that any pseudo-disk in P appears in the subset or is intersected by a pseudo-disk in it. In the weighted dominating set problem, each element of P is assigned a non-negative weight, and the goal is to find a dominating set of overall smallest weight. Related work. For general graphs, solving the dominating set problem is NP-hard [14, 20], and the standard greedy algorithm yields an O(1 + log n)- approximation factor [7, 19], where n is the size of the vertex set, which, up to a constant factor, is the best one can do in polynomial-time time, under reasonable complexity-theoretic assumptions [6, 10]. Many graph problems remain NP-hard in more specialized settings, such as unit disk graphs and growth-bounded graphs [8]. Nevertheless, often one can achieve in polynomial time better approximation factors. Specifically, the dominating set problem admits a polynomial-time approximation scheme (PTAS) for the aforementioned settings [18, 23]; see also [12] for a constant-factor approximation for the weighted dominating set problem on unit disk graphs. Erlebach and van Leeuwen [11] studied the dominating set problem in several natural cases, where they constructed polynomial-time algorithms guaranteeing a constant approximation factor. This includes (i) regular r-gons, where r is an arbitrary constant, (ii) homothetic triangles, and (iii) axis-parallel rectangles of bounded aspect ratio. On the other hand, they showed that in intersection graphs of arbitrary fat convex objects (a fat object is one in which the ratio of the radius of the smallest circumscribing disk to that of the largest inscribed disk is bounded by a constant), of arbitrarily large complexity, the problem is as hard to approximate as in abstract settings, and when the algebraic description complexity of these objects is constant, 4 then the problem is APX-hard. 5 In addition, they showed that for axis-parallel rectangles of arbitrary aspect ratio the problem is APX-hard as well. This result settles a conjecture of Chlebík and Chlebíková [6], who showed APX-hardness for axis-parallel boxes in any dimension d 3. See also the recent paper of Har-Peled and Quanrud [17] studying intractability in geometric settings. Gibson and Pirwani studied the case of disk graphs, that is, the special case of our problem when one replaces P with a set of Euclidean disks [15]. In this situation they demonstrated a PTAS for computing the smallest (unweighted) dominating set via a local search approach. For the weighted problem, they showed that 2 O(log n) -approximation factor is achievable in polynomial time using the quasi-sampling approach of Varadarajan [26]. The latter has been strengthened to O(1) by Chan et al. [5], who improved the sampling scheme of Varadarajan [26] and also presented a simple reduction from set cover to dominating set, considerably simplifying the approach taken in [15]. 4 A planar region is of constant (algebraic description) complexity if it can be described by a Boolean formula constructed from at most c algebraic inequalities in x and y, of total degree at most c, for some constant c. 5 This excludes the case of disks discussed below.

3 Our result. We show that, for the case of pseudo-disks, each of which is assigned a non-negative weight, one can achieve a constant approximation factor in expected polynomial time. In our analysis we define an appropriate set system for P, which, roughly speaking, consists of subsets of P which are met by a common pseudodisk (see Section 2 for the precise definition). Our main technical contribution is to show that this set system has VC-dimension of at most 4 (Theorem 3), and, moreover, that it has shallow cell complexity, meaning that the number of sets of size k P is only linear in P and polynomial in k (Definition 1 and Proposition 1). The aforementioned approximation factor is then achieved by using the recent machinery of Chan et al. [5], which allows us to deduce the following main result. Theorem 1. There is a randomized expected polynomial-time algorithm, that given a set P of pseudo-disks in the plane, each with a non-negative weight, computes a dominating set D P of weight O(Opt), where Opt is the smallest total weight of such dominating set. We note that the analysis of Gibson and Pirwani [15], as well as the more recent improvement of Chan et al. [5] strongly rely on the geometry of the disks. This enables them to reduce the dominating set problem for disks to a hitting set or a set cover for other geometric objects. However, neither of the approaches in [5, 15] seems to extend to the case of pseudo-disks, which is the reason we had to resort to another set of tools that largely exploit the combinatorial structure of the setting, in contrast to the more geometric ones used in [5, 15]. 2 Preliminaries Arrangements and levels. Let P be a set of n pseudo-disks in the plane. Without loss of generality, we assume that the pseudo-disks of P are in general position, that is, no point is incident to more than two pseudo-disk boundaries and, whenever two such boundaries meet, they properly cross. Let A(P) denote the arrangement of P (see, e.g., [1]). The level of an (open) face in this arrangement is the number of pseudo-disks containing it in their interior. Well-known results by Kedem et al. [21] and by Clarkson and Shor [9] imply that A(P) has O(n) level-2 faces and O(nk) faces at level at most k. Range spaces of bounded VC-dimension. A range space (or set system) (X, R) is a pair consisting of an underlying universe X of objects (also called the space) and a family R of subsets of X (called ranges). Of particular interest are range spaces of bounded VC-dimension, as, e.g., their restriction to any finite subset X X admits only polynomially many (in X ) ranges (see, e.g., [16, Chapter 5]). They also have other structural properties, which we discuss shortly. We say a subset K X is shattered by R if, for every subset Z of K, Z = K r for some range r R. The VC-dimension of a range space is the size of the largest finite shattered subset; if no such subset exists one says the VC-dimension

4 P 2 P 3 is infinite, slightly abusing the terminology. If the dimension is bounded by an absolute constant d > 0, we say the range space has bounded VC-dimension, again somewhat abusing the terminology. In this case, the number of ranges in any restriction of (X, R) to a finite collection X X is at most O( X d ) (once again, see [16, Chapter 5]). Hitting sets and dominating sets. Fix any family F of pseudo-disks in the plane. We first define the appropriate range spaces, which correspond to our dominating set problem. Namely, we consider finite range spaces (P, E), where P F is a finite set of pseudo-disks in the plane, and each range r E is precisely the set of pseudo-disks of P intersected by a pseudo-disk of F (which may not belong to P). Consider the intersection graph G of P as defined above. In G, a neighborhood of a pseudo-disk is the set of pseudo-disks intersecting it (therefore, this is a subgraph of G spanned by a star; note that we include the pseudo-disk itself in its neighborhood). The family of all neighborhoods define a range space, arising as a special case of the range space (P, E) defined above, as in this case we have F = P. In what follows and with a slight abuse of notation, we also denote a range space of the latter kind by (P, E). In fact, for our purposes it is sufficient to consider only the case F = P. However, we do not make this assumption in our analysis in Sections 3 and 4. We now observe that a dominating set in G is, in fact, a hitting set for (P, E), where the latter refers to a subset D P, which meets all ranges of E. That is, D meets all objects in P if and only if each neighborhood in the intersection graph (which is a range of E) is hit by an element of D. In particular, the minimum hitting set for (P, E) corresponds to the minimum dominating set of G, and this property holds in the weighted setting as well. See Figure 1 for an example. P 2 P 3 P 6 P 1 P 4 P 1 P 5 (a) P 6 P 5 (b) P 4 Fig. 1. The intersection graph in (b) induced by the pseudo-disks P 1,..., P 6, depicted in (a), is a star centered at P 1. The smallest dominating set is {P 1}. The underlying range space is (P, E), where P = {P 1,..., P 6}, and E = {{P 1,..., P 6}, {P 1, P 2}, {P 1, P 3}, {P 1, P 4}, {P 1, P 5}, {P 1, P 6}}. Finally, {P 1} is the smallest hitting set for this range space.

5 Shallow cell complexity. Following the notation of Chan et al. [5], we represent our set system (P, E) as an incidence matrix A. The columns of A correspond to pseudo-disks of P, and the rows to the sets in E. We have A i,j = 1 if the set e i E contains the pseudo-disk p j P, and 0 otherwise, for i = 1,..., E, j = 1,..., P. In a similar manner we can represent any restriction of (P, E) to a subset P P by considering a submatrix A of A restricted to the corresponding columns. In this manner, we may obtain sets of equivalent rows containing the same 0/1 vector. In the terminology of Chan et al. [5], equivalent classes of rows are referred to as cells, and the depth of a cell is the total number of ones contained in any one of its rows. We next define the notion of shallow cell complexity : Definition 1 (Shallow Cell Complexity; Chan et al. [5]). Let f(n, k) be a non-decreasing function in both n and k. A binary matrix A with N columns has shallow cell complexity (SCC) f if for all 1 k n N and for all submatrices A of A containing exactly n columns, the number of cells of A of depth k or fewer is at most f(n, k). A hitting set instance has SCC f if and only if its element-set incidence matrix does. 6 A main result of Chan et al. [5], which we employ in this study, is that hittingset instances with small SCC admit small approximation factors achievable in expected polynomial time: Theorem 2 (Chan et al. [5]). If I is a class of hitting set instances with SCC f(n, k) = c 1 nk c2, for absolute constants c 1, c 2 0, then there exists a randomized polynomial-time O(1)-approximation algorithm for the weighted hitting set problem for I. We note that the original formulation of Theorem 2 is slightly more general, and covers the possibility that the dependence of f on n is slightly super-linear. However, for our purposes a linear dependence on n is sufficient. In the main result of this study (Proposition 1) we show that the set system (P, E) admits SCC with f(n, k) = O(nk 3 ). Then, Theorem 1 follows by applying Theorem 2, and using our observation that the minimum weight hitting set for (P, E) corresponds to the minimum weight dominating set of G. Note that we need to assume that the intersection predicate for two pseudo-disks can be evaluated in time polynomial in the sum of their sizes, which enables us to produce the incidence matrix A of (P, E) in polynomial time, and therefore eventually apply the algorithm in [5] in overall expected polynomial time. Thus our main goal is to show that (P, E) has small shallow cell complexity. Road map. The rest of the paper is organized as follows. In Section 3 we prove that VC-dimension of (P, E) is at most 4 and that this bound is optimal (Theorem 3). In Section 4 we show that the number of pairs in E (sets containing exactly 6 The original definitions and analysis in [5] are stated using set cover terminology. We will use the dual and equivalent hitting set formulation, as it is more appropriate for our application.

6 two elements of P) is only linear in P, which is a consequence of the analysis concerning the VC-dimension. Then we show in Section 4 that the number of triples and quadruples in E is linear in P as well. Our proof technique exploits several ideas from the work of Buzaglo et al. [4], who studied the corresponding hitting-set problem for points and pseudo-disks in the plane. Our analysis is a variant of that of Buzaglo et al. [4], but our setting is more general. Specifically, Buzaglo et al. [4] studied topological hypergraphs defined by points and pseudo-disks enclosing them, whereas we consider a hypergraph of pseudo-disks, and subsets of them intersected by pseudo-disks. By viewing a point to be a sufficiently small pseudo-disk one observes that our result is a non-trivial generalization of that in [4]. Finally, we comment that it is crucial to consider pseudo-disks rather than pseudo-circles (that is, full regions rather than their boundaries). Indeed, range spaces of pseudo-circles and subsets of them met by a pseudo-circle do not satisfy Theorem 4: Consider n pairwise intersecting circles in general position and for each of the ( n 2) pairs of circles, place a tiny circle at one of their intersection points. This yields a collection of quadratically many circle pairs, rather than a linear number asserted in Theorem 4. 3 The VC-Dimension is Bounded In this section we prove: Theorem 3. A range space (P, E) as defined above has VC-dimension at most 4. This bound is the best possible. We start by stating the following technical lemma from [4] (see figure below): γ D γ D Lemma 1 (Buzaglo et al. [4]). Let γ and γ be arbitrary curves contained in pseudo-disks D and D, respectively. If the endpoints of γ lie outside of D and the endpoints of γ lie outside of D, then γ and γ must cross an even number of times. We say that a set K of pseudo-disks is well behaved if every pseudo-disk in K has a point not covered by the union of other pseudo-disks in K. We begin with an auxiliary construction. Let K be a finite well-behaved set of pseudo-disks. We construct a graph H = H(K) whose vertices correspond to pseudo-disks of K and whose edges correspond to pseudo-disks of F that meet precisely two sets of K. More specifically, we draw H as follows:

7 Vertices of H: For each pseudo-disk D K, we fix a point v(d) (which need not lie on the boundary of D), not contained in any other pseudo-disk of K; it exists since K is well behaved. The points {v(d) : D K} form the vertex set of H. Edges of H: Let D 1, D 2 K, v 1 = v(d 1 ) and v 2 = v(d 2 ). Suppose there exists P F that intersects D 1 and D 2 but not any other disk in K; fix one such P. Notice that it is possible that P K. We will add an edge v 1 v 2 to H, drawn as described below. We call a connected portion of the edge inside P a red arc and such a portion outside P a blue arc. The edge v 1 v 2 consists of at most one red arc and at most two blue arcs. In figures below, we also use the convention of drawing pseudo-disks of K in blue and the connecting pseudo-disk(s) in red. P contains both v 1 and v 2 : Draw a red arc in P from v 1 to v 2. This forms the edge v 1 v 2 H. See figure below. v 2 D 1 γ D 2 v 1 P P v 2 D 1 v D 2 1 γ P contains v 1 but does not contain v 2 : Draw a red arc in P that starts at v 1 and ends at the boundary of P inside D 2. Now draw a blue arc in D 2 that starts at this point, ends at v 2 and lies totally outside P. The concatenation of these two arcs forms the edge v 1 v 2 of H. See figure below. P D 1 D 2 D 2 D 1 v γ γ 1 v v 2 2 v 1 P D 1 =P v 1 D 2 γ v 2 P contains neither v 1 nor v 2 : Draw a blue arc in D 1 that starts at v 1, ends at the boundary of P inside D 1, and otherwise stays outside of P. From its endpoint, draw a red arc in P to a point of the boundary of P inside D 2 and from there, draw the final blue arc outside P in D 2 to the vertex v 2. The concatenation of these three arcs constitutes the edge v 1 v 2. See figure below. D 1 D 2 P γ v 2 v 1 v 1 D 1 P γ D 2 v 2

8 By construction, for each arc of the constructed edge, either red or blue, there is a pseudo-disk that completely contains it. Lemma 2. The graph H = H(K) is planar. Proof. We will prove H is planar using the strong Hanani-Tutte theorem [25]. Consider two edges e, e that connect v 1 = v(d 1 ) to v 2 = v(d 2 ), and v 3 = v(d 3 ) to v 4 = v(d 4 ) in H(K), respectively, and do not share a vertex so that D 1, D 2, D 3, D 4 K are pairwise distinct. We will prove that e and e intersect an even number of times, by considering their red and blue portions separately. Let P 12 F be the pseudo-disk intersecting only D 1 and D 2 that was used to draw e and let P 34 F be the corresponding pseudo-disk intersecting only D 3 and D 4. Red-Blue Intersections: Consider the red portion of e. This red arc is contained in P 12 and therefore does not meet any pseudo-disk of K other than D 1, D 2. As the blue portions of e lie inside D 3, D 4, this implies that the red arc of e does not meet the blue portions of e. Symmetrically the red portion of e cannot intersect the blue portions of e. Red-Red Intersections: The red arc α along e lies entirely in P 12 and has one endpoint in D 1 and the other in D 2. Similarly, the red arc α along e lies entirely in P 34 and has one endpoint in D 3 and the other in D 4. As P 12 does not intersect D 3 and D 4 and P 34 does not intersect D 1 and D 2, the endpoints of α do not lie in P 34 and the endpoints of α do not lie in P 12. By Lemma 1, α and α intersect an even number of times. Blue-Blue Intersections: Consider blue arcs β e and β e. The blue arc β starts, say, at vertex v 1 of pseudo-disk D 1 and ends at p in D 1 on the boundary of pseudo-disk P 12, and β starts, say, at vertex v 3 of pseudo-disk D 3 and ends at p in D 3 on the boundary of pseudo-disk P 34. By the construction of the vertices of H, v 1 / D 3 and v 3 / D 1. Now, p cannot lie in D 3 because P 12 meets only D 1 and D 2 and similarly p cannot lie in D 1. Hence, by Lemma 1 we deduce once again that β and β intersect an even number of times. There is a possibility that some edges of H self-intersect, but such intersections can be removed using standard methods: see for example [24], and Figure 2 in Appendix A. Thus, any two edges of H that do not share an endpoint cross an even number of times, and therefore H is planar by the strong Hanani-Tutte theorem [25]. Proof (of Theorem 3). Let K P be a set shattered by E. Since K is shattered, for every disk P K there is a disk F F that intersects P but not other disk in K. Therefore, K is well behaved. For a well-behaved set K, H(K) is planar, by Lemma 2, and therefore has at most 3 K 6 edges. However, K is shattered, so H(K) is a complete graph and has ( ) ( K 2 edges. Therefore, K ) 2 3 K 6, implying K 4. This means that the VC-dimension of (P, E) is at most 4. Refer to Figure 3 in Appendix B that shows that this bound is tight.

9 4 Bounding the Number of Shallow Sets Given a family F of pseudo-disks in the plane and a subfamily P F, recall that we denote by (P, E) the hypergraph (or set system) whose vertices are the pseudo-disks in P and whose edges (or sets) are the subsets of pseudo-disks in P intersected by a pseudo-disk in F. For the family E and an integer k > 0, let E k denote the family of subsets of E of size exactly k, and let E k be the family of subsets of E of size at most k. In this section we bound the number E k of sets of size at most k in the set system E, where k n is an integer parameter. We show that this number is only linear in n and polynomial in k. Moreover, this property is hereditary, that is, it holds for any restriction of E to a subset of P. Then, using the machinery of Chan et al. [5], we conclude that a constant approximation factor for the weighted dominating set problem can be achieved in expected polynomial time. We begin with the case k = 2 and proceed to the cases k = 3, k = 4, and k > 4. In the latter three cases we use a variant of the analysis and some of the observations made by Buzaglo et al. [4], rephrasing them in the context of dominating sets, rather than that of hitting sets, as they were originally formulated. Using the analysis in Section 3, we first show that the number of pairs is linear: Theorem 4. In any range space (P, E), P F, as above, E 2 = O( P ). Proof. Let U be the set of pseudo-disks of P, each of which has at least one point not covered by other pseudo-disks of P, and put C = P \ U. Equivalently, each pseudo-disk of U contains at least one level-1 face of the arrangement A = A(P), while those of C contain only faces of level-2 or higher, as we have assumed general position. By definition, the family U is well behaved. Ranges of E 2 (or pairs) can be classified into those that only contain pseudodisks of U and those that contain at least one pseudo-disk of C. As U is well behaved, the number of pairs of the former kind is O( U ) = O( P ), by Lemma 2. Now consider a pair {D, D } E 2 with D C. It must correspond to some pseudo-disk P F meeting D and D and no other pseudo-disks of P. By the assumption of general position P D must contain a point p in the interior of face f of A contained in D. Since D C, f must be at level 2 (if f is at level above 2, p is contained in more than two pseudo-disks of P, contrary to our assumption). Thus it must be contained in another pseudo-disk of P, necessarily D. Hence fixing f A determines the pair {D, D } E 2 uniquely and the number of such faces f is at most O( P ), as follows by the discussion about arrangements and levels in Section 2. This concludes the proof of the theorem. 4.1 Bounding the number of triples and quadruples We next show:

10 Theorem 5. In any set system (P, E), P F as above, we have (i) E 3 = O( P ), and (ii) E 4 = O( P ). In order to prove Theorem 5, we first need the following key lemma: Lemma 3. Let k 2 be a fixed integer. Let F be a family of pseudo-disks in the plane. Let H be a subfamily of m pseudo-disks from F. We call a pair of pseudo-disks {D 1, D 2 } from H k-good if there exists a pseudo-disk in F that intersects D 1, D 2, and at most k 2 additional pseudo-disks from H, for a total of most k pseudo-disks from H. Then the number of k-good pairs in H is at most c k m, where c k is an absolute constant depending only on k. Proof. We prove the lemma by induction on k. Case k = 2 is precisely Theorem 4. Suppose k 3. We choose each pseudo-disk in H independently with probability p = 1/2 (but we keep F intact). We denote the resulting sample of pseudo-disks by H. We say that a k-good pair {D 1, D 2 } from H survives if D 1, D 2 H and there is a pseudo-disk in F that intersects D 1, D 2, and a total of at most k 1 pseudo-disks in H. In other words, after sampling {D 1, D 2 } becomes (k 1)-good. We observe that a k-good pair {D 1, D 2 } in H survives with probability of at least 1/8. Indeed, because {D 1, D 2 } is a k-good pair, there exists F F such that F intersects D 1 and D 2 and a total of l k pseudo-disks in H. If l k 1, then {D 1, D 2 } is (k 1)-good as soon as both D 1 and D 2 are in H ; this happens with probability 1/4. If l = k, let S H be a pseudo-disk other than D 1 and D 2 intersected by F. If D 1 and D 2 are in H and S is not in H, then {D 1, D 2 } becomes (k 1)-good. This happens with probability 1/8; there may be other ways for {D 1, D 2 } to become (k 1)-good. Therefore the expected number of (k 1)-good pairs in H is at least 1 8 of the number of k-good pairs in H. By the inductive hypothesis on H, there are at most c k 1 H (k 1)-good pairs of disks in H. Therefore, the expected number of (k 1)-good pairs of disks in H is at most c k 1 m/2. Combining the two estimates, the number of k-good pairs in H is at most 4c k 1 m, as claimed. Theorem 5 is then proved using the following result from [4] (see Appendix C): Lemma 4 (Buzaglo et al. [4]). Let H be a graph on m vertices, with the property that, in any subgraph induced by a subset V of vertices, the number of edges is at most c V, where c > 0 is an absolute constant. Then, for any k 2, the number of copies of K k (the complete graph on k vertices) in H is at most d k m, where d k = (2c)k 1 k!. 4.2 Wrapping up Next we would like to bound the number of sets in E of cardinality at most k, for k 5. We show below that the number of such sets is O( P k 3 ). We could easily generalize Theorem 5 to every k and get a linear bound in P for the number of such sets at the cost of a multiplicative constant that grows extremely fast

11 (super exponentially) in k. In order to overcome this problem and improve the dependence on k, we use Theorem 5 and a fundamental property shown in [4], namely, that in a set system of bounded VC-dimension every set has a unique small signature. Specifically: Theorem 6 (Buzaglo et al. [4]). Let S = {S 1,..., S m } be a set family with VC-dimension d. Then it is possible to assign to each set S S a subset S S (its signature), of cardinality at most d, such that different sets S, S S are assigned distinct signatures. Given this machinery we conclude (see Appendix C for the proof): Proposition 1. Let F be a family of pseudo-disks in the plane and let P F be a finite subfamily. Consider the set system (P, E), as above. Then the number of sets in E of cardinality at most k is O(nk 3 ), for any k 2, where n = P. Remark: From the considerations made by our analysis (see also Appendix C), it is easy to verify that the bound in Proposition 1 is hereditary, that is, for any subcollection P P of m n pseudo-disks, the number of sets in the restriction of E to P, of cardinality at most k, is O(mk 3 ). 5 Concluding Remarks and Open Problems In this paper we have presented an expected polynomial-time algorithm, which produces the smallest weight dominating set in a collection of pseudo-disks in the plane, up to a constant factor. Note that our construction of the dominating set does not exploit the geometry of the input objects. In fact, one only needs a predicate that tests whether a pair of pseudo-disks intersect, in order to produce the incidence matrix A. The main open problem raised by our study is whether one can obtain a PTAS, such as the one shown by Gibson and Pirwani [15] for the case of Euclidean disks. References 1. P. K. Agarwal and M. Sharir. Arrangements and their applications. Handbook of Computational Geometry. (J. Sack and J. Urrutia, eds.), Elsevier, Amsterdam, 2000, B. Aronov, E. Ezra, and M. Sharir, Small-size ε-nets for axis-parallel rectangles and boxes. SIAM J. Comput., 39 (2010), H. Brönnimann and M. T. Goodrich. Almost optimal set covers in finite VC dimensions. Discrete Comput. Geom., 14 (1995), S. Buzaglo, R. Pinchasi, and G. Rote. Topological Hyper-graphs. In J. Pach (ed.), Thirty Essays on Geometric Graph Theory, pp , T. Chan, E. Grant, J. Koenemann, and M. Sharpe. Weighted capacitated, priority, and geometric set cover via improved quasi-uniform sampling. In Proc. Symp. Discrete Algorithms (SODA 2012), pp , 2012.

12 6. M. Chlebík and J. Chlebíková. Approximation hardness of dominating set problems. In Proc. 12th Annu. Europ. Sympos. Algorithms (ESA 2004), pp , V. Chvátal. A greedy heuristic for the set-covering problem. Mathematics of Operations Research, 4(3): , B. N. Clark, C. J. Colbourn, and D. S. Johnson. Unit disk graphs. Discrete Mathematics, 86(1-3): , K. L. Clarkson and P. W. Shor. Applications of random sampling in computational geometry, II. Discrete Comput. Geom., 4 (1989), I. Dinur and D. Steurer. Analytical approach to parallel repetition. In Proc. 46th Annu. ACM Symp. Theory Comput., pp , T. Erlebach and E. J. van Leeuwen. Domination in geometric intersection graphs. In LATIN, pages , T. Erlebach and M. Mihalák. A (4 + ε)-approximation for the minimum-weight dominating set problem in unit disk graphs. In Proc. 7th Workshop Approx. and Online Algorithms (WAOA 2009), Revised Papers. LNCS 5893, Springer-Verlag, pp , G. Even, D. Rawitz, and S. Shahar. Hitting sets when the VC-dimension is small. Inform. Process. Letts., 95 (2005), M. R. Garey and D. S. Johnson. Computers and Intractability: A Guide to the Theory of NP-Completeness. W. H. Freeman, New York, NY, M. Gibson and I. A. Pirwani. Approximation algorithms for dominating set in disk graphs: Breaking the log n barrier. Proc. 18th Annu. Europ. Symp. Algorithms, 2010, S. Har-Peled. Geometric Approximation Algorithms. Mathematical Surveys and Monographs, Vol. 173, S. Har-Peled, and K. Quanrud. Approximation algorithms for polynomial-expansion and low-density graphs. In Proc. 23rd Annual European Symposium, pp , H. B. Hunt III, M. V. Marathe, V. Radhakrishnan, S. S. Ravi, D. J. Rosenkrantz, and R. E. Stearns. NC-approximation schemes for NP- and PSPACE-hard problems for geometric graphs. J. Algorithms, 26(2): , L. Lovász. On the ratio of optimal integral and fractional covers. Discrete Mathematics, 13: , R. M. Karp. Reducibility among combinatorial problems. In R. E. Miller, J. W. Thatcher, Eds., Complexity of Computer Computations, pages , Plenum Press, New York, K. Kedem, R. Livne, J. Pach, and M. Sharir. On the union of Jordan regions and collision-free translational motion amidst polygonal obstacles. Discrete Comput. Geom., 1 (1986), J. Matoušek. Approximations and optimal geometric divide-and-conquer. J. Comput Sys. Sci., 50 (1995), pp T. Nieberg, and J. L. Hurink. A PTAS for the minimum dominating set problem in unit disk graphs. Proc. Third Workshop Approx. and Online Algorithms (WAOA 2005), Revised Papers, LNCS 3879, Springer-Verlag, pp , M.J. Pelsmajer, M. Schaefer, D. Štefankovič. Removing even crossings. Journal of Combinatorial Theory, Ser. B, 97(4): , W. T. Tutte. Toward a theory of crossing numbers. J. Combinat. Theory, 8:45 53, K. Varadarajan. Weighted geometric set cover via quasi-uniform sampling. In Proc. 42th Annu. ACM Symp. Theory Comput., pp , 2010.

13 Appendix A Undoing Self-intersections Fig. 2. How to undo self-intersections B Shattering Four Pseudo-disks Fig. 3. How to shatter a set of four shaded pseudo-disk objects: the pseudo-disks meeting all or none of four objects are not shown. The pseudo-disks meeting exactly one object are the objects themselves. C Omitted Proofs from Section 4 Proof of Theorem 5 (i): We follow the approach in [4]. We define a graph G whose vertices are the pseudo-disks in P. Two pseudo-disks in P form an edge in G if they belong to some set in E of cardinality 3.

14 We claim that, if G is an induced subgraph of G, then the number of edges in G is O( V (G ) ), where V (G ) is the set of vertices of G. To see this we only need to apply Lemma 3 with k = 3 and H = V (G ). We can now use Lemma 4 and conclude that the number of triangles in G is at most linear in V (G). This is sufficient to prove the assertion in Theorem 5, since every triple {D 1, D 2, D 3 } E gives rise to a unique triangle in G. Proof of Theorem 5 (ii): We repeat the proof of part (i) almost verbatim. We define a graph G whose vertices are the pseudo-disks in P. Two pseudo-disks in P form an edge in G if they belong to some set in E of cardinality 4. We claim that, if G is an induced subgraph of G, then the number of edges in G is O( V (G ) ). This follows from Lemma 3 with k = 4 and H = V (G ). We can now apply Lemma 4 and conclude that the number of copies of K 4 in G is at most linear in V (G). This completes the proof of the theorem, as is easily verified. Proof of Proposition 1: We follow almost verbatim the random sampling approach in [4]. Using Theorem 3, the VC-dimension of (P, E) is at most 4. Applying Theorem 6, we assign to each S E a unique subset B S S of cardinality at most 4. Let 0 < q < 1/2 be a parameter to be fixed shortly. We now select each pseudo-disk in P independently with probability q. Let P be the resulting sample, and consider the set system E restricted to P, denote it by (P, E ). We say that S E survives if all the disks in B S are in P but none of the remaining disks in S \ B S are in P. It is easy to verify that, if a set S of E has cardinality at most k, then S survives with probability q B S (1 q) S B S q B S (1 q) k B S q 4 (1 q) k 4, where the first inequality follows from the assumption S k, and the second from the fact that q < 1/2. By Theorems 4 and 5, the number of sets in E of cardinality 2, 3, and 4 is O( P ). Clearly, the number of sets in E of cardinality 1 is at most P. It thus follows that the number of surviving sets S from (P, E) is O( P ). Taking expectations, we see that the expected number of surviving sets S from E is O( P ) = O(q P ). On the other hand, the expected number of surviving sets S from E with cardinality at most k is greater than or equal to q 4 (1 q) k 4 E k (recall that E k is the number of sets in E with cardinality at most k). Therefore, q 4 (1 q) k 4 E k = O(q P ). By setting q = 1/k, we obtain E k = O( P k 3 ), as asserted.