Approximation Schemes for Geometric Coverage Problems
Approximation Schemes for Geometric Coverage Problems Steven Chaplick1 , Minati De2 , Alexander Ravsky3 , and Joachim Spoerhase1
arXiv:1607.06665v1 [cs.CG] 22 Jul 2016
1
Lehrstuhl f¨ ur Informatik I, Universit¨at W¨ urzburg, Germany, www1.informatik.uni-wuerzburg.de/en/staff, [email protected]. 2 Department of Computer Science and Automation, Indian Institute of Science, Bangalore, India [email protected]. 3 Pidstryhach Institute for Applied Problems of Mechanics and Mathematics, National Academy of Science of Ukraine, Lviv, Ukraine, [email protected]. July 25, 2016
Abstract In their seminal work, Mustafa and Ray [25] showed that a wide class of geometric set cover (SC) problems admit a PTAS via local search, which appears to be the most powerful approach known for such problems. Their result applies if a naturally defined “exchange graph” for two feasible solutions is planar and is based on subdividing this graph via a planar separator theorem due to Frederickson [16]. Obtaining similar results for the related maximum k-coverage problem (MC) seems challenging due to the hard cardinality constraint. In fact, in a subsequent work, Badanidiyuru, Kleinberg, and Lee [2] study geometric MC. They obtain fixed-parameter approximation schemes for MC instances with bounded VC dimension, but the running times are exponential in k. In this work, we overcome the above-mentioned technical hurdles by proposing a colored version of the planar separator theorem that might be of independent interest. The resulting subdivision approximates locally in each part the global distribution of the colors. This allows us to obtain a PTAS (with running time polynomial in k) for any “planarizable” instance of MC and thus essentially for all cases where the corresponding SC instance can be tackled via the approach of Mustafa and Ray. As a corollary, we resolve an open question by Badanidiyuru, Kleinberg, and Lee regarding real half spaces in dimension three.
1
1
Introduction
The Maximum Coverage (MC) problem is one of the classic combinatorial optimization problems which is well studied due to its wealth of applications. Let U be a set of ground elements, F ⊆ 2U be a family of subsets of U and k be a positive integer. The Coverage S Maximum ′ ′ (MC) problem asks for a k-subset F of F such that the number | F | of ground elements covered by F ′ is maximized. Many real life problems arising from banking [11], social networks, transportation network [23], databases [18], information retrieval, sensor placement, security (and others) can be framed as an instance of MC problem. For example, the problem of placing k sensors so that maximum number of customers are under coverage, or the problem of finding a set of k documents satisfying the information needs of as many users as possible [2], or the problem of placing k security personnel in a terrain so that maximum number of important regions is secured, can be modelled as MC problem. From the result of Cornu´ejols [11], it is well known that greedy algorithm is a 1 − 1/e approximation algorithm for the MC problem. Due to wide applicability of the problem, whether one can achieve an approximation factor better than (1 − 1e ) was subject of research for a long period of time. From the result of Feige [15], it is known that if there exists a polynomial-time algorithm that approximates maximum coverage within a ratio of (1− 1e + ǫ) for some ǫ > 0 then P = NP. Better results can however be obtained for special cases of MC. For example, Ageev and Sviridenko [1] show in their seminal work that their pipage rounding approach gives a factor 1 − (1 − 1/r)r for instances of MC where every element occurs in at most r sets. For constant r this is a strict improvement on 1 − 1/e but this bound is approached if r is unbounded. For example, pipage rounding gives a 3/4-approximation algorithm for Maximum Vertex Cover (MVC), which asks for a k-subset of nodes of a given graph that maximizes the number of edges incident on at least one of the selected nodes. Petrank [26] showed that this special case of MC is APX-hard. In this paper, we study the approximability of MC in geometric settings where elements and sets are represented by geometric objects. Such problems have been considered before and have applications, for example, in information retrieval [2] and in wireless networks [14]. MC is related to the Set Cover problem (SC). For a given set U of ground elements and a family F ⊆ 2U of subsets of U , this problem asks for a minimum cardinality subset of F which covers all the ground elements of U . This problem plays a central role in combinatorial optimization and in particular in the study of approximation algorithms. The best known approximation algorithm has a ratio of ln n, which is essentially the best possible [15] under a plausibly complexity-theoretic assumption. A lot of work has been devoted to beat the logarithmic barrier in the context of geometric set cover problems[7, 28, 8, 24]. Mustafa and Ray [25] introduced a powerful tool which can be used to show that a local search approach provides a PTAS for various geometric SC problems. Their result applies if a naturally defined “exchange graph” (whose nodes are the sets in two feasible solutions) is planar and is based on subdividing this graph via a planar separator theorem due to Frederickson [16]. In the same paper [25], they applied this approach to provide a PTAS for the SC problem when the family F consists of either a set of half spaces in R3 , or a set of disks in R2 . Many results have been obtained using this technique for different problems in geometric setting [9, 12, 17, 21]. Beyond the context of SC, local search has also turned out a very powerful tool also for other geometric problems but the analysis of such algorithms is usually non-trivial and highly tailored to the specific setting. Examples are the Euclidean TSP, Euclidean Steiner tree, facility location, k-median [10]. In very recent break-through results PTASs for k-means problem in finite Euclidean dimension (and more general cases) via local search have been announced [29, 30]. In this paper, we study the effectiveness of local search for geometric MC problems. In
2
the general case, b-swap local search is known to yield a tight approximation ratio of 1/2 [20]. However, for special cases such as geometric MC problems local search is a promising candidate for beating the barrier 1 − 1/e. It seems, however, challenging to obtain such results using the technique of Mustafa and Ray [25]. In their analysis, each part of the subdivided planar exchange graph (see above) corresponds to a feasible candidate swap that replaces some sets of the local optimum with some sets of the global optimum and it is ensured that every element stays covered due to the construction of the exchange graph. It is moreover argued that if the global optimum is sufficiently smaller than the local optimum then one of the considered candidate swaps would actually reduce the size of the solution. It is possible to construct the same exchange graphs also for the case of MC. However, the hard cardinality constraint given by input parameter k poses an obstacle. In particular, when considering a swap corresponding to a part of the subdivision, this swap might be infeasible as it may contain (substantially) more sets from the global optimum than from the local optimum. Another issue is that MC has a different objective function than SC. Namely, the goal is to maximize the number of covered elements rather than minimizing the number of used sets. Finally, while for SC all elements are covered by both solutions, in MC we additionally have elements that are covered by none or only one of the two solutions requiring a more detailed distinction of several types of elements. In fact, subsequent to the work of Mustafa and Ray on SC [25], Badanidiyuru, Kleinberg, and Lee [2] studied geometric MC. They obtained fixed-parameter approximation schemes for MC instances for the very general case where the family F consists of objects with bounded VC dimension, but the running times are exponential in k. In their work, they pose the open question if there is a PTAS for MC where the family F consists of half spaces in R3 . This underlines the observation that geometric MC problems seem more challenging to handle than their SC counterparts as at this point a PTAS for halfspaces in R3 for SC was already known via the approach of Mustafa and Ray.
1.1
Our Contribution
In this paper, we address the above-mentioned technical hurdles. We are able to achieve a PTAS for many geometric MC problems. At a high level also our approach builds on defining a planar (or more generally f -separable) exchange graph and subdividing it into a number of small parts each of them corresponding to a candidate swap. As each part may be (substantially) imbalanced in terms of the number of number of sets of the global optimum and local optimum, respectively, a natural idea seems to swap in only a sufficiently subset of the globally optimal sets. This idea alone is, however, not sufficient. Consider, for example, the case where each part contains either only sets from the local or only sets from the global optimum making it impossible to retrieve any feasible swap from the considering the single parts. To overcome this difficulty, we prove a colored version of the planar separator theorem (Theorem 2) that might be of independent interest. In this theorem, the input is a planar (or more generally f -separable) graph whose nodes are two-colored arbitrarily. The distinctions of our separator theorem from the prior work, are that our separator theorem guarantees that all parts have roughly the same size (rather than simply an upper limit on their size) and that the two colors are represented in each part in roughly the same ratio as in the whole graph. This balancing property allows us to address the issue of the above-mentioned infeasible swaps. Our proofs of the colored separator theorems incorporate solving a vector partitioning problem. The constructions are algorithmic and provide in the case of two colors a linear time algorithm. In the case of more than two colors we provide an iterative rounding algorithm. Using these tools, we are able to prove by a careful analysis (which turns out more intricate than for the SC case) that local search also yields a PTAS for the wide class f -separable MC problems (see Theorem 3). As an immediate consequence, we obtain PTASs for essentially all cases of geometric MC problems where the corresponding SC problem can be tackled via the approach of Mustafa and Ray (Theorem 4). 3
In particular, we resolve the open question by Badanidiyuru, Kleinberg, and Lee [2] regarding halfspaces in R3 . We also obtain immediately PTASs for Maximum Dominating Set and Maximum Vertex Cover on f -separable and minor-closed graph classes, which, to the best of our knowledge, were not known before. We believe that our approach is applicable also to further problems.
2
Tools: Vector Partitioning and Color Balanced Separators
In this section we construct the main tools from which our main result (i.e., Theorem 3) is proven. In Section 2.1, we provide a linear time algorithm to order and partition a given set of two-dimensional vectors in a nearly uniform manner with respect to the size and the coordinate sums of the parts. In Subsection 2.2, we first describe a new specialized separator lemma for f -separable graph classes (see Lemma 2). This builds on the concept of (r, f (r))-divisions (in the sense of Henzinger et al. [19]) of graphs in an f -separable graph class. We then combine this specialized separator lemma with our vector partitioning results to obtain a two-color balanced versions (see Theorems 2) of this new separator theorem. These results generalize to higher dimensions and more than two colors, respectively. These results are described in the appendix. Throughout this section, for a positive integer n, we use [n] to denote the set {1, . . . , n}.
2.1
Two-Dimensional Vector Partitioning with Low Discrepancy
In this section, we provide a linear-time algorithm for almost uniformly partitioning a given set of two-dimensional vectors. This lemma provides a key step in our construction of uniform two-color balanced separators in f -separable graph classes (see Theorem 2). Lemma 1. Let c and c′ be positive constants, and A = {(a1 , b1 ), . . . , (an , bn )} ⊆ (Q ∩ [0, ∞))2 be a set of 2-dimensional vectors where ai + bi ∈ [c′ , c] for each i ∈ [n], and α ∈ [0, 1] such that P Pn n ai = α · i=1 bi . There is a permutation π of [n] such that for any 1 ≤ i1 ≤ i2 ≤ n, Pi=1 2 | ii=i a − α · bi | ≤ 2 · c. 1 i As a consequence, for any positive integer q (n is sufficiently larger than q), there is k ≤ n where π can be split into k subsequences I1 , . . . , Ik such that for some q ′ ∈ [q, 2q − 1], and each j ∈ [k], we have: P P (i) |Ij | ∈ {q ′ , q ′ + 1}, i.e., i∈Ij ai + bi ∈ [q ′ · c′ , (q ′ + 1) · c], and (ii) | i∈Ij ai − α · bi | ≤ 2 · c. Moreover, the permutation and partition can be computed in O(n) time. Proof. First, we partition [n] into three sets A>0 , A0 , Aj = Ii1 \ Iij1 . Now, let j ′ be the latest time step when |Vij1 | ≤ |Vij2 | 1 ′
′
(i.e., implying that |Vij1 | < 2r ). We now consider two cases regarding |Ii>j |, either it is precisely 1 ′
′
1, or it is larger than 1. In the former case, we have |Vi1 | = |Vij1 | + |Vi>j | ≤ |Vi2 | + 1 i.e., a contradiction. In the latter case, we note that for each
j ′′
>
j′,
j ′′
r 2
< r, ′′
we have |Vi1 | > |Vij2 |. ′
Thus, we must completely fill Ii2 prior to putting a second index into Ii>j , i.e., |Ii2 | = 4 · cc∗ℓ . In 1 ′
particular, for each index j1 in Ii>j , we can find a unique corresponding index j2 in Ii2 , where 1 ′
|Uj1 | ≤ |Uj2 |. This holds, because |Ii>j | ≤ 4· 1
decreasing cardinality. In particular, |Vi1 | = assumption and proves the claim.
cℓ c∗ = |Ii2 | and ′ ′ |Vij1 | + |Vi>j |≤ 1
because we sorted the sets Uj by 2 · |Vi2 | < r. This contradicts our
Claim 3: If every j ∈ [ℓ] is placed into some Ii , the sets Vi satisfy the conditions of the lemma. First, note that |Ii | is at most 4 · cc∗ℓ , i.e., |N (Vi ) ∩ X | ∈ O(f (r)). Now, from our choice of ∗ t = nr , either there is some index i where |Vi | > r or for every i, |Vi | = r. In the former case, by Claim 2 and since each Uj has at most 2r elements, |Vi | ∈ [ 2r , 3r 2 ] for each i. The latter case is trivial. This proves the claim. We now describe the second step. Namely, when there are unassigned sets Uj . Let Uj ∗ be the largest unassigned set. Note that, for every i with |Ii | < 4 · cc∗ℓ , with |Vi | > r. Moreover, since ℓt ≤ 2 · cc∗ℓ , there must be at least one index i with |Ii | < 4 · cc∗ℓ , i.e., with |Vi | > r. ∗ Hence for every i ∈ [t], |Vi | ≥ r2 by Claim 2. Additionally, since we have t = nr sets which partition at most n∗ elements, there must be some index i′ where |Vi′ | ≤ r and |Ii′ | = 4 · cc∗ℓ , i.e., |Uj ∗ | ≤ r · (4 · cc∗ℓ )−1 . In particular, there are at most ℓ − j ∗ ≤ t · 2 · cc∗ℓ remaining sets, and the union of any 2 · cc∗ℓ of these sets contains at most r2 elements. Thus, we can simply uniformly distribute the remaining indices throughout the sets Ii , and this will add at fewer than 2 · cc∗ℓ indices to each Ii and fewer than 2r elements to each Vi , i.e., for each i, |Ii | ≤ 6 · cc∗ℓ , |Vi | ∈ [ 2r , 2r], and N (Vi ) ∩ X | ≤ 6 · cc∗ℓ · c1 f (⌊ 2r ⌋) ∈ O(f (r)), as needed. We conclude with a brief discussion of the time complexity. First, we generate the (⌊ 2r ⌋, c1 f (⌊ 2r ⌋))division in h(n) time. We then sort the sets |U1 | ≥ . . . ≥ |Uℓ | (this can be done in O(n) time via bucket sort). In the next step we greedily fill the index sets – this takes O(n) time. Finally, we place the remaining “small” sets uniformly throughout the Vi ’s – taking again O(n) time. Thus, we have O(h(n) + n) time in total. We will now use the above lemma and our vector partitioning results from the previous section to prove Theorem 2. In particular, for a given two-colored graph G where G belongs to an f -separable graph class, we first construct a uniform (r, c · f (r))-division (X , V1 , . . . , Vt ) of G as in Lemma 2. From this division we can again carefully partition the Vi ’s into collections Wj where each Wj has roughly the same size and contains roughly the same proportion of each color class as occurring in G. We imagine the regions V1 , . . . , Vt of the uniform (r, c · f (r))-division as a collection of two-dimensional vectors and then simply apply Lemma 1. Theorem 2. Let G be an f -separable graph class and G = (V, E) be a 2-colored n-vertex graph in G with color classes Z1 , Z2 . For any q and r ≪ n where r is suitably large , there is an integer n ) such that V can be partitioned into t + 1 sets X , V1 , . . . , Vt where c1 , c2 are constants t ∈ Θ( q·r independent of our parameters n, r, q and there is an integer q ′ ∈ [q, 2q − 1] all satisfying the following properties. (i) N (Vi ) ∩ Vj = ∅ for each i 6= j, ′
(ii) |Vi | ∈ [ q 2·r , 2 · (q ′ + 1) · r] for each i, 7
(iii) |N (Vi ) ∩ X | ≤ c1 · q · f (r) for each i (i.e., |X | ≤ 1| · |V ∩ Z | (iv) |Vi ∩ Z1 | − |Z ≤2·r i 2 |Z2 |
Pt
i=1 |X
∩ N (Vi )| ≤
c2 ·f (r)·n ). r
Moreover, such a partition can be found in O(h(n) + n) time where h(n) is the amount of time required to produce a uniform (r, c · f (r))-division of G.
3
PTAS for f -Separable Maximum Coverage
In this section we formalize the notion of f -separable instances of the MC problem and prove our main result – see Theorem 3. Definition 1. A class C of instances of MC is called f -separable if for any two disjoint feasible solutions F and F ′ of any instance in C there exists an f -separable graph G with node set F ∪F ′ with the following property. If there is a ground element u ∈ U that is covered both by F and F ′ then there exists an edge (S, S ′ ) in G with S ∈ F and S ′ ∈ F ′ with u ∈ S ∩ S ′ . Theorem 3. Let f ∈ o(n) be non-decreasing sublinear function. Then, any f -separable class of instances of MC that is closed under removing elements and sets admits a PTAS. Proof. Our algorithm is based on local search. We fix a positive constant integer b ≥ 1. Given an f -separable instance of MC, we pick an arbitrary initial solution A. We check if it is possible to replace b sets in A with b sets from F so that the total number of elements covered is increased. We perform such a replacement (swap) as long as there is one. We stop if there is no profitable swap and output the resulting solution. In what follows, we show that for sufficiently large b the above algorithm yields a (1 − 12c1 f (b)/b)-approximate solution and that it runs in polynomial time (for constant b). Here, c1 is the constant from Theorem 2. This will prove the claim of Theorem 3 by letting b sufficiently large. Since each step increases the number of covered elements, the number of iterations of the above algorithm is at most |U |. Each iteration takes O(kb |F|b ) time. Therefore, the total running time of the algorithm is polynomial for constant b. We now analyze the performance guarantee of the algorithm. To this end, let O be an optimum solution to the instance and let A be the (locally optimal) solution output by the algorithm. Let OPT, ALGdenote the number of elements covered by O, A, respectively. Suppose that ALG < 1 − 12c1bf (b) OPT. We want to show that this would imply that there is a profitable swap as this would contradict the local optimality of A and hence complete the proof. We claim that is suffices to consider the case when O, A are disjoint, which is justified as follows. Assume that O ∩ A = 6 ∅. We remove the sets in O ∩ A from F and all the elements covered by these sets from U . Moreover, we decrease k by |O ∩ A| and replace O with O \ A and A with A \ O. Since our class of instances is closed under removing S sets and elements S the resulting instance is still contained in the class. Moreover, | A| < 1 − 12c1bf (b) | O|. Finally, if we are able to show that there exists a feasible and profitable swap in the reduced instance then the same swap is also feasible and profitable in the original instance (with original solutions A and O). Therefore, we assume from now on that A and O are disjoint. Since our instance is f separable, there exists an f -separable graph G with precisely 2k nodes for the two feasible solutions O and A with the properties stated in Definition 1. We now apply our two colored separator theorem (Theorem 2) to G with color classes Z1 = O and Z2 = A and with parameters r = b and q = b. 8
Since |O| = |A| = k, the two color classes in G are perfectly balanced. Let Ai = A ∩ Vi , ¯i = Oi ∪ N O for any part Vi with i ∈ [t] of the resulting Oi = O ∩ Vi , NiO = N (Vi ) ∩ X ∩ O and O i subdivision of G. The idea is to consider for each i ∈ [t] a feasible candidate swap (called candidate swap i) ¯ that replaces in A the sets Ai with some suitably chosen sets from Oi . We will show that if 12c1 f (b) OPT then at least one of the candidate swaps is profitable leading to a ALG < 1 − b contradiction. To accomplish this, we will first show that there exists a profitable swap that replaces Ai ¯i . This swap may be infeasible as |Ai | may be strictly smaller than |O ¯i |. We will, with O however, show that a feasible and profitable swap can be constructed by adding only some of ¯i . the sets in O For technical reasons we are going to define a set Z of elements that we (temporarily) disregard from our calculations because they will remain covered and thus should not impact ¯i we will pick for the feasible swap. More precisely, let our decision which of the sets in O Z = { u ∈ A ∩ B | A ∈ Ai , B ∈ A \ Ai , i ∈ [t] } be the set of elements that are covered by some Ai but that remain covered even if Ai is removed from A. S Let Li = Ai \ S Z be the set of elements that are “lost” when removing the Ai from A. ¯i \ S(A \ Ai ) be the set of elements that are “won” when we add all Moreover, let Wi = O ¯i after removing Ai . the sets of O Pt S We claim that A and that the i=1 |Li | ≤ ALG −|Z|. To this end, note that Z ⊆ family {Li }i∈[t] contains pairwise disjoint sets because all elements that are not exclusively covered Pt by a single Ai are contained in Z and thus removed. On the other hand, we claim that i=1 |Wi | ≥ OPT −|Z|. To see this, note first that every element in Z contributes 0 to the left hand side and 0 or -1 to the right hand side. Every element covered by O but not by A contributes at least 1 to the left hand side and precisely 1 to the right hand side. Finally, consider an element u that is covered both by A and by O but does not lie in Z. This element lies in a set S ∈ Ai for some i ∈ [t]. Because of the definition of the exchange graph G there is some set T ∈ O with u ∈ T and some set S ′ ∈ A with u ∈ S ′ such that S ′ and T are adjacent in G. We have that S ′ ∈ Ai , for, otherwise u ∈ Z. Because of the separator property of X ¯i . Moreover u lies in Wi because it is not (see Property (i) of Theorem 2) we must have T ∈ O ¯ contained in Z but is covered by Oi .PHence u contributes at least 1 to the left hand side and precisely 1 to the right hand side of ti=1 |Wi | ≥ OPT −|Z|, which shows the claim. We have OPT > |Z| and hence Pt |Li | ALG 12c1 f (b) ALG −|Z| |Li | ≤ Pti=1 ≤ 0
Hence, we can pick i ∈ [t] such that
|Li | 12c1 f (b) |Ai | (otherwise we can just add all sets in O ¯i ). Consider the swap Suppose that |O where we replace the |Ai | ≤ b many sets Ai from the local optimum A with at most |Ai | many ¯i . sets {S1 , . . . , S|Ai | } from O We now analyze how this swap affects the ojective function value. By removing the sets in Ai the objective function value drops by (2) 12c1 f (b) · |Wi | |Li | < 1 − b ¯ |A i| [ (4) | 12c1 f (b) |O i Sℓ \ Zi ≤ 1− b |Ai | ℓ=1 |Ai | (3) [ ≤ Sℓ \ Zi . ℓ=1 The right hand side of this inequality is the increase of the objective function due to adding the sets {S1 , . . . , S|Ai | } after removing the sets in Ai . Therefore the above described swap is feasible and also profitable and thus A is not a local optimum leading to a contradiction.
4
Applications
First, we define the following problems which are special instances of MC problem. Problem 1. Let H be a set of ground elements, S ⊆ 2H be a set of ranges and k be a positive integer. A range S ∈ S is hit by a subset H ′ of H if S ∩ H ′ 6= ∅. The Maximum Hitting (MH) problem asks for a k-subset H ′ of H such that the number of ranges hit by H ′ is maximized. Problem 2. Let G = (V, E) be a graph and k be a positive integer. A vertex v ∈ V dominates all the vertices adjacent to v including v. The Maximum Dominating (MD) problem asks for a k-subset V ′ of V such that the number of vertices dominated by V ′ is maximized. Problem 3. Let T be a 1.5D terrain which is an x-monotone polygonal chain in the plane consisting of a set of vertices {v1 , v2 , . . . , vm } sorted in increasing order of their x-coordinate, and vi and vi+1 are connected by an edge for all i ∈ [m − 1]. For any two points x, y ∈ T , we say that y guards x if each point in xy lies above or on the terrain. Given finite sets X, Y ⊆ T and a positive integer k, the Maximum Terrain Guarding (MTG) problem asks for a k sized subset Y ′ of Y such that the number of points of X guarded by Y ′ is maximized. 10
Let r be an even, positive integer. A set of regions in R2 , where each region is bounded by a closed Jordan curve, is called r-admissable if for any two such regions q1 , q2 , the curves bounding them cross s ≤ r times for some even s and q1 \ q2 and q2 \ q1 are connected regions. A set of regions are called pseudo-disks if is 2-admissable. For example a set of disks (of arbitrary size) or a set of squares (of arbitrary size) form a set of pseudo-disks each. We now state the following theorem (whose proof is given in Appendix B). It follows either because it is known from the literature that the corresponding SC problem is planarizable (that √ is, we can define an exchange graph as in Definition 1 that is planar and thus n-separable) or in case of claim (D2 ) and (V ) by construction the exchange graph as a minor of the input graph. Theorem 4. Local search gives a PTAS for the following classes of MC problems: (C1 ) the set of ground elements is a set of points in R3 , and the family of subsets is induced by a set of half spaces in R3 (C2 ) the set of ground elements is a set of points in R2 , and the family of subsets is induced by a set of convex pseudodisks (a set of convex objects where any two objects can have at most two intersections in their boundary). Local search gives a PTAS for the following MH problems: (H1 ) the set of ground elements is a set of points in R2 , and the set of ranges is induced by a set of r-admissible regions (this includes pseudodisks, same-height axis-parallel rectangles, circular disks, translates of convex objects). (H2 ) the set of ground elements is a set of points in R3 , and the set of ranges is induced by a set of half spaces in R3 . Local search gives a PTAS for MD problems in each of the following graph classes: (D1 ) intersection graphs of homothetic copies of convex objects (which includes arbitrary squares, regular k-gons, translated and scaled copies of a convex object). (D2 ) f -separable and minor-closed graph classes. Additionally, the following problems admit a PTAS via local search (V) the MVC problem on f -separable and subgraph-closed graph classes, (T) the MTG problem.
References [1] A. A. Ageev and M. Sviridenko. Pipage rounding: A new method of constructing algorithms with proven performance guarantee. J. Comb. Optim., 8(3):307–328, 2004. Preliminary versions appeared at IPCO’99 and ESA’00. [2] A. Badanidiyuru, R. Kleinberg, and H. Lee. Approximating low-dimensional coverage problems. In Proc. Symposuim on Computational Geometry (SoCG ’12), pages 161–170, 2012. [3] I. B´ar´ any. A vector-sum theorem and its application to improving flow shop guarantees. Mathematics of Operations Research, 6(3):445–452, 1981. [4] I. B´ar´ any and B. Doerr. Balanced partitions of vector sequences. Linear algebra and its applications, 414(2):464–469, 2006. 11
[5] I. B´ar´ any and V. S. Grinberg. On some combinatorial questions in finite-dimensional spaces. Linear Algebra and its Applications, 41:1–9, 1981. [6] J. Beck and T. Fiala. ”Integer-making“ theorems. Discrete Applied Mathematics, 3(1):1 – 8, 1981. [7] H. Br¨onnimann and M. T. Goodrich. Almost optimal set covers in finite vc-dimension. Discrete & Computational Geometry, 14(4):463–479, 1995. [8] T. M. Chan, E. Grant, J. K¨ onemann, and M. Sharpe. Weighted capacitated, priority, and geometric set cover via improved quasi-uniform sampling. In Proceedings of the TwentyThird Annual ACM-SIAM Symposium on Discrete Algorithms, SODA 2012, Kyoto, Japan, January 17-19, 2012, pages 1576–1585, 2012. [9] T. M. Chan and S. Har-Peled. Approximation algorithms for maximum independent set of pseudo-disks. Discrete & Computational Geometry, 48(2):373–392, 2012. [10] V. Cohen-Addad and C. Mathieu. Effectiveness of local search for geometric optimization. In 31st International Symposium on Computational Geometry, SoCG 2015, June 22-25, 2015, Eindhoven, The Netherlands, pages 329–343, 2015. [11] G. Cornu´ejols, G. L. Nemhauser, and L. A. Wolsey. Worst-case and probabilistic analysis of algorithms for a location problem. Operations Research, 28(4):847–858, 1980. [12] M. De and A. Lahiri. Geometric dominating set and set cover via local search. CoRR, abs/1605.02499, 2016. [13] B. Doerr and A. Srivastav. Multicolour discrepancies. Combinatorics, Probability and Computing, 12(04):365–399, 2003. [14] T. Erlebach and E. J. van Leeuwen. Approximating geometric coverage problems. In Proc. Nineteenth Annual ACM-SIAM Symposium on Discrete Algorithms (SODA’08), pages 1267–1276, 2008. [15] U. Feige. A threshold of ln n for approximating set cover. J. ACM, 45(4):634–652, 1998. [16] G. N. Frederickson. Fast algorithms for shortest paths in planar graphs, with applications. SIAM J. Comput., 16(6):1004–1022, 1987. [17] M. Gibson and I. A. Pirwani. Algorithms for dominating set in disk graphs: Breaking the logn barrier - (extended abstract). In Algorithms - ESA 2010, 18th Annual European Symposium, Liverpool, UK, September 6-8, 2010. Proceedings, Part I, pages 243–254, 2010. [18] V. Harinarayan, A. Rajaraman, and J. D. Ullman. Implementing data cubes efficiently. In H. V. Jagadish and I. S. Mumick, editors, Proceedings of the 1996 ACM SIGMOD International Conference on Management of Data, Montreal, Quebec, Canada, June 4-6, 1996., pages 205–216. ACM Press, 1996. [19] M. R. Henzinger, P. N. Klein, S. Rao, and S. Subramanian. Faster shortest-path algorithms for planar graphs. J. Comput. Syst. Sci., 55(1):3–23, 1997. [20] R. B. O. Kerkkamp and K. Aardal. A constructive proof of swap local search worst-case instances for the maximum coverage problem. Oper. Res. Lett., 44(3):329–335, 2016. [21] E. Krohn, M. Gibson, G. Kanade, and K. R. Varadarajan. Guarding terrains via local search. JoCG, 5(1):168–178, 2014.
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[22] R. J. Lipton and R. E. Tarjan. A separator theorem for planar graphs. SIAM Journal on Applied Mathematics, 36(2):177–189, 1979. [23] S. Mecke and D. Wagner. Solving geometric covering problems by data reduction. In Algorithms - ESA 2004, 12th Annual European Symposium, Bergen, Norway, September 14-17, 2004, Proceedings, pages 760–771, 2004. [24] N. H. Mustafa, R. Raman, and S. Ray. Quasi-polynomial time approximation scheme for weighted geometric set cover on pseudodisks and halfspaces. SIAM J. Comput., 44(1):1650– 1669, 2015. [25] N. H. Mustafa and S. Ray. PTAS for geometric hitting set problems via local search. In Proceedings of the 25th ACM Symposium on Computational Geometry (SoCG’09), Aarhus, Denmark, June 8-10, 2009, pages 17–22, 2009. [26] E. Petrank. The hardness of approximation: Gap location. Computational Complexity, 4:133–157, 1994. [27] E. Steinitz. Bedingt konvergente reihen und convexe systeme. J. Reine Angew., 143:128– 175, 1913. [28] K. R. Varadarajan. Weighted geometric set cover via quasi-uniform sampling. In Proceedings of the 42nd ACM Symposium on Theory of Computing (STOC’10), Cambridge, Massachusetts, USA, 5-8 June 2010, pages 641–648, 2010. [29] C. M. Vincent Cohen-Addad, Philip N. Klein. Local search yields approximation schemes for k-means and k-median in Euclidean and minor-free metrics. In Proc. 57th Annual IEEE Symposium on Foundations of Computer Science (FOCS’16). to appear. [30] M. S. Zachary Friggstad, Mohsen Rezapour. Local search yields a PTAS for k-means in doubling metrics. In Proc. 57th Annual IEEE Symposium on Foundations of Computer Science (FOCS’16). to appear.
A
Multicolored Separator Theorem
In this appendix we generalize our vector partitioning and colored separator results, i.e., to dimensions larger than two and more than two color classes. As we expect our tools to have even wider applications than the context in which we use them, we provide quite general versions of them and take some care regarding the computational complexity of the algorithms involved. More specifically, in Subsection A.1 we describe how to compute “low” discrepancy partitions of vectors in the d-dimensional unit cube (see Theorem 6). Here we focus on the L∞ -norm as it is the important case for our application to the multicolored separators. In the second subsection we note that combining this vector partitioning result with our uniform (r, f (r)-division (as given by Lemma 2) provides a notion of a uniform d-color balanced divisions for f -separable graph classes (see Theorem 7).
A.1
Higher Dimensional Vector Partitioning
In this section we consider the problem of partitioning n vectors {a1 , . . . , an } from the rational d-dimensional unit cube (Q ∩ [0, 1])d into k sets I1 , . . ., Ik where the sum P of the vectors in set is “close” (with respect to the L∞ norm) to the expected sum µ = k1 · ni=1 ai of nk vectors, i.e., into sets with “low” discrepancy. Progress on this kind of partitioning problem (usually phrased with respect to an arbitrary normed vector space and its corresponding unit ball) is sometimes referred as a B´ ar´ any-Grinberg type of result due to their initial progress on the case k = 2 [5]. 13
Namely, the discrepancy of such a vector partition is defined as the maximum P deviation of the sum of vectors in a set of the partition from the vector µ, i.e., maxi∈[k] kµ − j∈Ii aj k. Here we will provide a short, self-contained, efficient construction of such a partition in the context of the L∞ -norm (see Theorem 6). This is leveraged in the next subsection to obtain color balanced divisions of f -separable graphs. We first note some context from the literature. For example, one may observe that a result of this kind follows as a corollary of the following algorithmic version [3, Th.1] of Steinitz Lemma [27] (see Lemma 3 below). Namely, this readily provides a partition whose discrepancy is at most 3d + 1 (as described by Lemma 4 below). To our knowledge the strongest B´ar´ anyGrinberg result is given by B´ar´ any and Doerr [4] (included as Theorem 5 below for the sake of completeness and comparison) and is built on a recursive re-partitioning result given by Doerr and Srivastav [13]. The result appears to be constructive, but it is does take quite some effort (going through intermediate results) to establish. As such, it is difficult to estimate the time bounds. Furthermore, the authors did not seem discuss this aspect of their proofs. Note that the bound on the discrepancy here is quite strong (2.005d) and it is provided for all partial sums and is obtained for any norm. Our d-dimensional result (Theorem 6) provides a slightly improved bound (actually 2d), but is given in a more restricted context of the L∞ -norm, and our discrepancy bound is not guaranteed for partial sums, but rather for each entire set in the constructed partition. However, we feel that our proof (involving an iterative rounding process inspired by the Beck-Fiala Theorem [6]) is particularly nice since it is compact, self-contained, and directly provides efficient algorithm to produce the partition. Theorem 5 ([4]). Let d, c ∈ N, k · k be any norm on Rd , and B denote the unit ball with respect to this norm. Any sequence a1 , a2 , . . . of vectors in B can be partitioned into k sub-sequences V1 , . . . , Vk inPa balanced manner with respect to the partial sums, i.e., for all n ∈ N and l ≤ k 1 P we have: k( i≤k,vi ∈Vl vi ) − µk ≤ 2.005d where µ = k · i≤k ai .
Theorem 6. Let A = {a1 , . . . , an } ⊆ (Q ∩ [0, 1])d be a set of d-dimensional vectors, let k ≤ n be a positive integer. Then we can compute in polynomial time a partition I1 , . . . , IP k of [n] into P 1 k sets such that for any j ∈ [k], we have that kµ − i∈Ij ai k∞ ≤ 2d where µ = k · ni=1 ai .
Proof. The idea is to start with a perfect “fractional” assignment of vectors to sets Ij that is iteratively rounded to an integral partition with only a small imbalance. A partition of [n] into sets I1 , . . . , Ik can be described by a matrix x ∈ {0, 1}n×k of binary variables xij with i ∈ [n] and j ∈ [k] such that xij = 1 if and only if vector ai is assigned to partition Ij . The matrix x corresponds to a partition if and only if k X j=1
xij = 1, for every i ∈ [n] .
(5)
Now consider the continuous relaxation, where we allow the variables xij to attain also fractional values in [0, 1]. The intuitive meaning is that a fraction xij of vector ai is assigned to partition Ij . We start with an initial fractional assignment of x = k1 · 1. This assignment achieves perfect balance in the sense that n X xij ai = µ, for every j ∈ [k] . (6) i=1
In what follows, we will iteratively perturb the fractional assignment x while maintaining constraints (5) and (6). Let If ⊆ [n] × [k] be the set of index pairs (i, j) where xij is fractional (i.e. non-integral). Moreover, let Jf ⊆ [k] be the set of indices j where the vector xj := (x1j , . . . , xnj ) ∈ [0, 1]n contains at least 2d + 1 many fractional entries. 14
We now construct a non-trivial perturbation matrix δ ∈ Qk×n that satisfies k X
δij = 0,
for every i ∈ [n] ,
(7)
δij ai = 0,
for every j ∈ Jf .
(8)
(i,j)∈If n X i=1
We further require δij = 0 for every (i, j) ∈ [n] × [k] \ If . The crucial point is that there is a non-trivial solution δ satisfying the above constraints. To see this, we note that there are at least |If | ≥ (2d+1)|Jf | many free variables. On the other hand the number of non-trivial constraints of type (7) is at most |If |/2 since each i ∈ [n] contribute either zero or at least two pairs (i, j) to |If |. Since each constraint of type (8) corresponds to d linear constraints, we have at most |If |/2 + d|Jf | ≤ |If |/2 + d|If |/(2d + 1) < |If | many linear constraints in total. In particular, this means that we have more free variables than constraints and we can find a non-trivial solution δ 6= 0. Now, we determine the smallest factor ǫ > 0 such that x′ := x + ǫδ contains at least one additional integral variable. Such an ǫ exists, because δ is non-trivial and has non-zero entries only for fractional entries of x. Note that for every i ∈ [n], the matrix x′ satisfies the constraint of type (5) because δ satisfies the type (7) constraint. Similarly, for every j ∈ Jf , the matrix x′ satisfies the constraint of type (6) as δ satisfies the constraint of type (8). Since, we only changed fractional entries of x, every integral entry of x is the same in x′ and x′ has at least one additional integral entry. We apply the above rounding step as long as there fractional variables. (Note, that this rounding procedure may still be performed even Jf is empty.) We claim that the final solution x satisfies the claim of the theorem. To see this, note that initially Jf = [k] and that Jf = ∅ at the end of the algorithm. In particular, for each j ∈ [k] there is an iteration where j is removed from Jf for the first time. Since our algorithm does not change entries of x that are already integral, j will not occur Jf at any later iteration. Let x be the vector at the end of this iteration. Note that at this point x satisfies (6) for j and that there are at most 2d many fractional entries xj . Hence, regardless of the assignment x′ at the end of the algorithm, the Pn in ′ expression i=1 xij ai can deviate from µ by at most 2d in every entry.
We conclude this subsection by we showing how the algorithmic version of Steinitz Lemma (see Lemma 3) can be used to obtain a vector partition of d-dimensional vectors whose discrepancy is at most 3d + 1 with respect to the L∞ -norm (see Lemma 4). P Lemma 3. [3, Th.1] For a set {b1 , . . . , bn } ⊆ [−1, 1]d of d-dimensional vectors with ni=1 bi = 0 in O(n2 d3 + nd4 ) steps can be given a permutation π of the set [n] such that for each 1 ≤ l ≤ n
l
X 3
d . bπ(i) ≤
2 i=1
∞
Lemma 4. Let A = {a1 , . . . , an } ⊆ [0, 1]d be a set of d-dimensional vectors, let k ≤ n be a 4 positive integer. Then we can compute in O(n2 d3 + nd I1 , . . . , Ik of [n] into P ) time a partition k sets such that for any j ∈ [k], we have that kµ − i∈Ij ai k∞ < 2 23 d + 1 ≤ 3d + 1, where P µ = k1 · ni=1 ai .
Proof. For each i, let bi = nk · µ − ai . Note that the set {b1 , . . . , bn } satisfies the conditions of Lemma 3. Let π be the permutation provided by this lemma. Let I1′ , . . . , Ik′ be a partition of [n] into k consecutive discrete segments such that sizes of any two elements of the partition differs by at most 1. For each j ∈ [k], let Ij = π(Ij′ ). We now have: 15
X
µ − ai
i∈Ij
∞
X
ai = µ −
i∈π(I ′ ) j
∞
X X
ai ai − = µ +
k≤j, i∈π(I ′ ) k<j, i∈π(I ′ ) k
k
=
∞
X X
kµ kµ
µ − kµ |Ik′ | +
≤ ai − ai − −
n n n
′ ′ k<j, i∈π(Ik ) k≤j, i∈π(Ik ) ∞ k 3 3 3 3 n k kµk∞ − |Ik′ | + d + d < kµk∞ + 2 d ≤ 1 + 2 d . n k 2 2 n 2 2
A.2
Separator Theorem
We will now use the uniform (r, f (r)-division obtained in Lemma 2 and the d-dimensional vector partitioning theorem (Theorem 6) from the previous section to obtain a d-color uniform separator theorem on f -separable graph classes (see Theorem 7 below). In particular, for a given d-colored graph G where G belongs to an f -separable graph class, we first construct a uniform (r, c · f (r))-division (X , V1 , . . . , Vt ) of G as in Lemma 2. From this division we carefully partition the Vi ’s into collections Wj where each Wj has roughly the same size and contains roughly the same proportion of each color class as occurring in G. We imagine the regions V1 , . . . , Vt of the uniform (r, c · f (r))-division as a collection of d-dimensional vectors (whose 1 , and coordinates correspond to the number of vertices of each color), scale these vectors by 2r then apply Theorem 6 with the parameter k = r to obtain the result. Theorem 7. Let G be an f -separable graph class and G = (V, E) be a d-colored n-vertex graph in G with color classes Z1 , . . . Zd . For any r ≪ n where r is suitably large , there is an integer t ∈ Θ( rn2 ) such that V can be partitioned into t + 1 sets X , V1 , . . . , Vt where c1 , c2 are constants independent of n, r, and the following properties are satisfied. (i) N (Vi ) ∩ Vj = ∅ for each i 6= j, (ii) |Vi | ∈ Θ(r 2 ) for each i, (iii) |N (Vi ) ∩ X | ≤ c1 · r · f (r)) for each i (i.e., |X | ≤ (iv) ||Vi ∩ Zq | −
|Zq | t |
≤ 2 · r · d for each q ∈ [d].
Pt
i=1 |X
∩ N (Vi )| ≤
c2 ·f (r)·n ). r
Moreover, such a partition can be found in O(h(n) + p(n)) time where h(n) is the amount of time required to produce a uniform (r, c · f (r))-division of G and p(n) is the polynomial running time of the vector partitioning in Lemma 4.
B
Applications
Proof of Theorem 4. In what follows, we refer to several known results for SC where the respective instances are planarizable. This always also implies that the corresponding MC problem is planarizable. By the result of Mustafa and Ray [25], we know that the MC instance is planarizable when the family F is a set of half spaces in R3 , or a set of disks in R2 . Recently, De and Lahiri [12] showed that when the objects are convex pseudodisks, then the corresponding SC (and thus MC) instance is planarizable. Thus, as a consequence of Theorem 3, we have (C1 ) and (C2 ). Note that MH problem is a special instance of MC problem, where the set S of ranges plays the role of U , and the set H plays the role of F, where each set h ∈ H contains all the range 16
S ∈ S such that S ∩h 6= ∅. On the other hand, It follows from the result of Mustafa and Ray [25] that an MH instance is planarizable when the set of ranges are (i) a set of r-admissible regions, (ii) set of half spaces in R3 . Thus, Theorem 3 implies that an MH problem admits PTAS when the ranges are a set of r-admissible region or half spaces in R3 . Thus, we have (H1 ) and (H2 ). Observe that MD is a special instance of MC, where the set V of vertices plays the role of the set U of ground elements , and the family F consists of |V | subsets of V where each subset is corresponding to the set of vertices dominated by each vertex v ∈ V . On the other hand, from the result of De and Lahiri [12], we know that corresponding instance of MD is planarizable when the graph G is an geometric intersection graph induced by homothetic set of convex objects. Thus, as a consequence of Theorem 3, we have (D1 ). To prove (D2 ), we claim that these MC instances are f -separable according to Definition 1. To this end, let D, D ′ be two disjoint feasible solutions. We are going to construct an auxiliary graph H as in Definition 1. We start with the node set D ∪ D ′ and an empty edge set. Let u ∈ V be node that is dominated by D and by D ′ . If u ∈ D then there is a neighbor v ∈ D ′ of u. We add edge uv to H. The case u ∈ D ′ is handled symmetrically. If u ∈ / D and u ∈ / D′ ′ ′ then there are neighbors v ∈ D and v ∈ D of u. In this case we add u and the two edges uv and uv ′ to H. Note that the resulting graph is a subgraph of G. Now, we perform the following operation on H as long as H contains a node that is not in D ∪ D ′ . If such a node u exists it must have precisely two neighbors v ∈ D and v ′ ∈ D ′ by construction of H. We contract the edge uv and identify the resulting node with v (lying in D). As a result we obtain a minor H of G with node set D ∪ D ′ . It is easy to check that this graph fulfills the requirements of Definition 1. Moreover, because G is contained in a minor-closed, f -separable graph class we know that H is f -separable and we obtain that M D problem admits a PTAS on such graph classes. The proof of (V ) is even simpler than the one of (D2 ). Let D, D ′ be two disjoint feasible solutions. We are going to construct an auxiliary graph H as in Definition 1. We start with the node set D ∪ D ′ and an empty edge set. For any edge uu′ that is covered by both D and D ′ we may assume u ∈ D and u′ ∈ D ′ . We add edge uu′ to H. Note that the graph H is a subgraph of G, and it fulfils the requirement of Definition 1. Moreover, because G is contained in a subgraph-closed, f -separable graph class we know that H is f -separable and we obtain that M V C problem admits a PTAS on such graph classes. It is easy to observe that MTG is a special instance of MC, where the set X plays the role of U , and the set Y plays the role of family F of subsets, where each y ∈ Y contains all elements of X which can be guarded by y. On the other hand, we know from the result of Krohn et al. [21, Lem.§2] that MTG is planarizable. Thus, we have (T ).
17
arXiv:1607.06665v1 [cs.CG] 22 Jul 2016
1
Lehrstuhl f¨ ur Informatik I, Universit¨at W¨ urzburg, Germany, www1.informatik.uni-wuerzburg.de/en/staff, [email protected]. 2 Department of Computer Science and Automation, Indian Institute of Science, Bangalore, India [email protected]. 3 Pidstryhach Institute for Applied Problems of Mechanics and Mathematics, National Academy of Science of Ukraine, Lviv, Ukraine, [email protected]. July 25, 2016
Abstract In their seminal work, Mustafa and Ray [25] showed that a wide class of geometric set cover (SC) problems admit a PTAS via local search, which appears to be the most powerful approach known for such problems. Their result applies if a naturally defined “exchange graph” for two feasible solutions is planar and is based on subdividing this graph via a planar separator theorem due to Frederickson [16]. Obtaining similar results for the related maximum k-coverage problem (MC) seems challenging due to the hard cardinality constraint. In fact, in a subsequent work, Badanidiyuru, Kleinberg, and Lee [2] study geometric MC. They obtain fixed-parameter approximation schemes for MC instances with bounded VC dimension, but the running times are exponential in k. In this work, we overcome the above-mentioned technical hurdles by proposing a colored version of the planar separator theorem that might be of independent interest. The resulting subdivision approximates locally in each part the global distribution of the colors. This allows us to obtain a PTAS (with running time polynomial in k) for any “planarizable” instance of MC and thus essentially for all cases where the corresponding SC instance can be tackled via the approach of Mustafa and Ray. As a corollary, we resolve an open question by Badanidiyuru, Kleinberg, and Lee regarding real half spaces in dimension three.
1
1
Introduction
The Maximum Coverage (MC) problem is one of the classic combinatorial optimization problems which is well studied due to its wealth of applications. Let U be a set of ground elements, F ⊆ 2U be a family of subsets of U and k be a positive integer. The Coverage S Maximum ′ ′ (MC) problem asks for a k-subset F of F such that the number | F | of ground elements covered by F ′ is maximized. Many real life problems arising from banking [11], social networks, transportation network [23], databases [18], information retrieval, sensor placement, security (and others) can be framed as an instance of MC problem. For example, the problem of placing k sensors so that maximum number of customers are under coverage, or the problem of finding a set of k documents satisfying the information needs of as many users as possible [2], or the problem of placing k security personnel in a terrain so that maximum number of important regions is secured, can be modelled as MC problem. From the result of Cornu´ejols [11], it is well known that greedy algorithm is a 1 − 1/e approximation algorithm for the MC problem. Due to wide applicability of the problem, whether one can achieve an approximation factor better than (1 − 1e ) was subject of research for a long period of time. From the result of Feige [15], it is known that if there exists a polynomial-time algorithm that approximates maximum coverage within a ratio of (1− 1e + ǫ) for some ǫ > 0 then P = NP. Better results can however be obtained for special cases of MC. For example, Ageev and Sviridenko [1] show in their seminal work that their pipage rounding approach gives a factor 1 − (1 − 1/r)r for instances of MC where every element occurs in at most r sets. For constant r this is a strict improvement on 1 − 1/e but this bound is approached if r is unbounded. For example, pipage rounding gives a 3/4-approximation algorithm for Maximum Vertex Cover (MVC), which asks for a k-subset of nodes of a given graph that maximizes the number of edges incident on at least one of the selected nodes. Petrank [26] showed that this special case of MC is APX-hard. In this paper, we study the approximability of MC in geometric settings where elements and sets are represented by geometric objects. Such problems have been considered before and have applications, for example, in information retrieval [2] and in wireless networks [14]. MC is related to the Set Cover problem (SC). For a given set U of ground elements and a family F ⊆ 2U of subsets of U , this problem asks for a minimum cardinality subset of F which covers all the ground elements of U . This problem plays a central role in combinatorial optimization and in particular in the study of approximation algorithms. The best known approximation algorithm has a ratio of ln n, which is essentially the best possible [15] under a plausibly complexity-theoretic assumption. A lot of work has been devoted to beat the logarithmic barrier in the context of geometric set cover problems[7, 28, 8, 24]. Mustafa and Ray [25] introduced a powerful tool which can be used to show that a local search approach provides a PTAS for various geometric SC problems. Their result applies if a naturally defined “exchange graph” (whose nodes are the sets in two feasible solutions) is planar and is based on subdividing this graph via a planar separator theorem due to Frederickson [16]. In the same paper [25], they applied this approach to provide a PTAS for the SC problem when the family F consists of either a set of half spaces in R3 , or a set of disks in R2 . Many results have been obtained using this technique for different problems in geometric setting [9, 12, 17, 21]. Beyond the context of SC, local search has also turned out a very powerful tool also for other geometric problems but the analysis of such algorithms is usually non-trivial and highly tailored to the specific setting. Examples are the Euclidean TSP, Euclidean Steiner tree, facility location, k-median [10]. In very recent break-through results PTASs for k-means problem in finite Euclidean dimension (and more general cases) via local search have been announced [29, 30]. In this paper, we study the effectiveness of local search for geometric MC problems. In
2
the general case, b-swap local search is known to yield a tight approximation ratio of 1/2 [20]. However, for special cases such as geometric MC problems local search is a promising candidate for beating the barrier 1 − 1/e. It seems, however, challenging to obtain such results using the technique of Mustafa and Ray [25]. In their analysis, each part of the subdivided planar exchange graph (see above) corresponds to a feasible candidate swap that replaces some sets of the local optimum with some sets of the global optimum and it is ensured that every element stays covered due to the construction of the exchange graph. It is moreover argued that if the global optimum is sufficiently smaller than the local optimum then one of the considered candidate swaps would actually reduce the size of the solution. It is possible to construct the same exchange graphs also for the case of MC. However, the hard cardinality constraint given by input parameter k poses an obstacle. In particular, when considering a swap corresponding to a part of the subdivision, this swap might be infeasible as it may contain (substantially) more sets from the global optimum than from the local optimum. Another issue is that MC has a different objective function than SC. Namely, the goal is to maximize the number of covered elements rather than minimizing the number of used sets. Finally, while for SC all elements are covered by both solutions, in MC we additionally have elements that are covered by none or only one of the two solutions requiring a more detailed distinction of several types of elements. In fact, subsequent to the work of Mustafa and Ray on SC [25], Badanidiyuru, Kleinberg, and Lee [2] studied geometric MC. They obtained fixed-parameter approximation schemes for MC instances for the very general case where the family F consists of objects with bounded VC dimension, but the running times are exponential in k. In their work, they pose the open question if there is a PTAS for MC where the family F consists of half spaces in R3 . This underlines the observation that geometric MC problems seem more challenging to handle than their SC counterparts as at this point a PTAS for halfspaces in R3 for SC was already known via the approach of Mustafa and Ray.
1.1
Our Contribution
In this paper, we address the above-mentioned technical hurdles. We are able to achieve a PTAS for many geometric MC problems. At a high level also our approach builds on defining a planar (or more generally f -separable) exchange graph and subdividing it into a number of small parts each of them corresponding to a candidate swap. As each part may be (substantially) imbalanced in terms of the number of number of sets of the global optimum and local optimum, respectively, a natural idea seems to swap in only a sufficiently subset of the globally optimal sets. This idea alone is, however, not sufficient. Consider, for example, the case where each part contains either only sets from the local or only sets from the global optimum making it impossible to retrieve any feasible swap from the considering the single parts. To overcome this difficulty, we prove a colored version of the planar separator theorem (Theorem 2) that might be of independent interest. In this theorem, the input is a planar (or more generally f -separable) graph whose nodes are two-colored arbitrarily. The distinctions of our separator theorem from the prior work, are that our separator theorem guarantees that all parts have roughly the same size (rather than simply an upper limit on their size) and that the two colors are represented in each part in roughly the same ratio as in the whole graph. This balancing property allows us to address the issue of the above-mentioned infeasible swaps. Our proofs of the colored separator theorems incorporate solving a vector partitioning problem. The constructions are algorithmic and provide in the case of two colors a linear time algorithm. In the case of more than two colors we provide an iterative rounding algorithm. Using these tools, we are able to prove by a careful analysis (which turns out more intricate than for the SC case) that local search also yields a PTAS for the wide class f -separable MC problems (see Theorem 3). As an immediate consequence, we obtain PTASs for essentially all cases of geometric MC problems where the corresponding SC problem can be tackled via the approach of Mustafa and Ray (Theorem 4). 3
In particular, we resolve the open question by Badanidiyuru, Kleinberg, and Lee [2] regarding halfspaces in R3 . We also obtain immediately PTASs for Maximum Dominating Set and Maximum Vertex Cover on f -separable and minor-closed graph classes, which, to the best of our knowledge, were not known before. We believe that our approach is applicable also to further problems.
2
Tools: Vector Partitioning and Color Balanced Separators
In this section we construct the main tools from which our main result (i.e., Theorem 3) is proven. In Section 2.1, we provide a linear time algorithm to order and partition a given set of two-dimensional vectors in a nearly uniform manner with respect to the size and the coordinate sums of the parts. In Subsection 2.2, we first describe a new specialized separator lemma for f -separable graph classes (see Lemma 2). This builds on the concept of (r, f (r))-divisions (in the sense of Henzinger et al. [19]) of graphs in an f -separable graph class. We then combine this specialized separator lemma with our vector partitioning results to obtain a two-color balanced versions (see Theorems 2) of this new separator theorem. These results generalize to higher dimensions and more than two colors, respectively. These results are described in the appendix. Throughout this section, for a positive integer n, we use [n] to denote the set {1, . . . , n}.
2.1
Two-Dimensional Vector Partitioning with Low Discrepancy
In this section, we provide a linear-time algorithm for almost uniformly partitioning a given set of two-dimensional vectors. This lemma provides a key step in our construction of uniform two-color balanced separators in f -separable graph classes (see Theorem 2). Lemma 1. Let c and c′ be positive constants, and A = {(a1 , b1 ), . . . , (an , bn )} ⊆ (Q ∩ [0, ∞))2 be a set of 2-dimensional vectors where ai + bi ∈ [c′ , c] for each i ∈ [n], and α ∈ [0, 1] such that P Pn n ai = α · i=1 bi . There is a permutation π of [n] such that for any 1 ≤ i1 ≤ i2 ≤ n, Pi=1 2 | ii=i a − α · bi | ≤ 2 · c. 1 i As a consequence, for any positive integer q (n is sufficiently larger than q), there is k ≤ n where π can be split into k subsequences I1 , . . . , Ik such that for some q ′ ∈ [q, 2q − 1], and each j ∈ [k], we have: P P (i) |Ij | ∈ {q ′ , q ′ + 1}, i.e., i∈Ij ai + bi ∈ [q ′ · c′ , (q ′ + 1) · c], and (ii) | i∈Ij ai − α · bi | ≤ 2 · c. Moreover, the permutation and partition can be computed in O(n) time. Proof. First, we partition [n] into three sets A>0 , A0 , Aj = Ii1 \ Iij1 . Now, let j ′ be the latest time step when |Vij1 | ≤ |Vij2 | 1 ′
′
(i.e., implying that |Vij1 | < 2r ). We now consider two cases regarding |Ii>j |, either it is precisely 1 ′
′
1, or it is larger than 1. In the former case, we have |Vi1 | = |Vij1 | + |Vi>j | ≤ |Vi2 | + 1 i.e., a contradiction. In the latter case, we note that for each
j ′′
>
j′,
j ′′
r 2
< r, ′′
we have |Vi1 | > |Vij2 |. ′
Thus, we must completely fill Ii2 prior to putting a second index into Ii>j , i.e., |Ii2 | = 4 · cc∗ℓ . In 1 ′
particular, for each index j1 in Ii>j , we can find a unique corresponding index j2 in Ii2 , where 1 ′
|Uj1 | ≤ |Uj2 |. This holds, because |Ii>j | ≤ 4· 1
decreasing cardinality. In particular, |Vi1 | = assumption and proves the claim.
cℓ c∗ = |Ii2 | and ′ ′ |Vij1 | + |Vi>j |≤ 1
because we sorted the sets Uj by 2 · |Vi2 | < r. This contradicts our
Claim 3: If every j ∈ [ℓ] is placed into some Ii , the sets Vi satisfy the conditions of the lemma. First, note that |Ii | is at most 4 · cc∗ℓ , i.e., |N (Vi ) ∩ X | ∈ O(f (r)). Now, from our choice of ∗ t = nr , either there is some index i where |Vi | > r or for every i, |Vi | = r. In the former case, by Claim 2 and since each Uj has at most 2r elements, |Vi | ∈ [ 2r , 3r 2 ] for each i. The latter case is trivial. This proves the claim. We now describe the second step. Namely, when there are unassigned sets Uj . Let Uj ∗ be the largest unassigned set. Note that, for every i with |Ii | < 4 · cc∗ℓ , with |Vi | > r. Moreover, since ℓt ≤ 2 · cc∗ℓ , there must be at least one index i with |Ii | < 4 · cc∗ℓ , i.e., with |Vi | > r. ∗ Hence for every i ∈ [t], |Vi | ≥ r2 by Claim 2. Additionally, since we have t = nr sets which partition at most n∗ elements, there must be some index i′ where |Vi′ | ≤ r and |Ii′ | = 4 · cc∗ℓ , i.e., |Uj ∗ | ≤ r · (4 · cc∗ℓ )−1 . In particular, there are at most ℓ − j ∗ ≤ t · 2 · cc∗ℓ remaining sets, and the union of any 2 · cc∗ℓ of these sets contains at most r2 elements. Thus, we can simply uniformly distribute the remaining indices throughout the sets Ii , and this will add at fewer than 2 · cc∗ℓ indices to each Ii and fewer than 2r elements to each Vi , i.e., for each i, |Ii | ≤ 6 · cc∗ℓ , |Vi | ∈ [ 2r , 2r], and N (Vi ) ∩ X | ≤ 6 · cc∗ℓ · c1 f (⌊ 2r ⌋) ∈ O(f (r)), as needed. We conclude with a brief discussion of the time complexity. First, we generate the (⌊ 2r ⌋, c1 f (⌊ 2r ⌋))division in h(n) time. We then sort the sets |U1 | ≥ . . . ≥ |Uℓ | (this can be done in O(n) time via bucket sort). In the next step we greedily fill the index sets – this takes O(n) time. Finally, we place the remaining “small” sets uniformly throughout the Vi ’s – taking again O(n) time. Thus, we have O(h(n) + n) time in total. We will now use the above lemma and our vector partitioning results from the previous section to prove Theorem 2. In particular, for a given two-colored graph G where G belongs to an f -separable graph class, we first construct a uniform (r, c · f (r))-division (X , V1 , . . . , Vt ) of G as in Lemma 2. From this division we can again carefully partition the Vi ’s into collections Wj where each Wj has roughly the same size and contains roughly the same proportion of each color class as occurring in G. We imagine the regions V1 , . . . , Vt of the uniform (r, c · f (r))-division as a collection of two-dimensional vectors and then simply apply Lemma 1. Theorem 2. Let G be an f -separable graph class and G = (V, E) be a 2-colored n-vertex graph in G with color classes Z1 , Z2 . For any q and r ≪ n where r is suitably large , there is an integer n ) such that V can be partitioned into t + 1 sets X , V1 , . . . , Vt where c1 , c2 are constants t ∈ Θ( q·r independent of our parameters n, r, q and there is an integer q ′ ∈ [q, 2q − 1] all satisfying the following properties. (i) N (Vi ) ∩ Vj = ∅ for each i 6= j, ′
(ii) |Vi | ∈ [ q 2·r , 2 · (q ′ + 1) · r] for each i, 7
(iii) |N (Vi ) ∩ X | ≤ c1 · q · f (r) for each i (i.e., |X | ≤ 1| · |V ∩ Z | (iv) |Vi ∩ Z1 | − |Z ≤2·r i 2 |Z2 |
Pt
i=1 |X
∩ N (Vi )| ≤
c2 ·f (r)·n ). r
Moreover, such a partition can be found in O(h(n) + n) time where h(n) is the amount of time required to produce a uniform (r, c · f (r))-division of G.
3
PTAS for f -Separable Maximum Coverage
In this section we formalize the notion of f -separable instances of the MC problem and prove our main result – see Theorem 3. Definition 1. A class C of instances of MC is called f -separable if for any two disjoint feasible solutions F and F ′ of any instance in C there exists an f -separable graph G with node set F ∪F ′ with the following property. If there is a ground element u ∈ U that is covered both by F and F ′ then there exists an edge (S, S ′ ) in G with S ∈ F and S ′ ∈ F ′ with u ∈ S ∩ S ′ . Theorem 3. Let f ∈ o(n) be non-decreasing sublinear function. Then, any f -separable class of instances of MC that is closed under removing elements and sets admits a PTAS. Proof. Our algorithm is based on local search. We fix a positive constant integer b ≥ 1. Given an f -separable instance of MC, we pick an arbitrary initial solution A. We check if it is possible to replace b sets in A with b sets from F so that the total number of elements covered is increased. We perform such a replacement (swap) as long as there is one. We stop if there is no profitable swap and output the resulting solution. In what follows, we show that for sufficiently large b the above algorithm yields a (1 − 12c1 f (b)/b)-approximate solution and that it runs in polynomial time (for constant b). Here, c1 is the constant from Theorem 2. This will prove the claim of Theorem 3 by letting b sufficiently large. Since each step increases the number of covered elements, the number of iterations of the above algorithm is at most |U |. Each iteration takes O(kb |F|b ) time. Therefore, the total running time of the algorithm is polynomial for constant b. We now analyze the performance guarantee of the algorithm. To this end, let O be an optimum solution to the instance and let A be the (locally optimal) solution output by the algorithm. Let OPT, ALGdenote the number of elements covered by O, A, respectively. Suppose that ALG < 1 − 12c1bf (b) OPT. We want to show that this would imply that there is a profitable swap as this would contradict the local optimality of A and hence complete the proof. We claim that is suffices to consider the case when O, A are disjoint, which is justified as follows. Assume that O ∩ A = 6 ∅. We remove the sets in O ∩ A from F and all the elements covered by these sets from U . Moreover, we decrease k by |O ∩ A| and replace O with O \ A and A with A \ O. Since our class of instances is closed under removing S sets and elements S the resulting instance is still contained in the class. Moreover, | A| < 1 − 12c1bf (b) | O|. Finally, if we are able to show that there exists a feasible and profitable swap in the reduced instance then the same swap is also feasible and profitable in the original instance (with original solutions A and O). Therefore, we assume from now on that A and O are disjoint. Since our instance is f separable, there exists an f -separable graph G with precisely 2k nodes for the two feasible solutions O and A with the properties stated in Definition 1. We now apply our two colored separator theorem (Theorem 2) to G with color classes Z1 = O and Z2 = A and with parameters r = b and q = b. 8
Since |O| = |A| = k, the two color classes in G are perfectly balanced. Let Ai = A ∩ Vi , ¯i = Oi ∪ N O for any part Vi with i ∈ [t] of the resulting Oi = O ∩ Vi , NiO = N (Vi ) ∩ X ∩ O and O i subdivision of G. The idea is to consider for each i ∈ [t] a feasible candidate swap (called candidate swap i) ¯ that replaces in A the sets Ai with some suitably chosen sets from Oi . We will show that if 12c1 f (b) OPT then at least one of the candidate swaps is profitable leading to a ALG < 1 − b contradiction. To accomplish this, we will first show that there exists a profitable swap that replaces Ai ¯i . This swap may be infeasible as |Ai | may be strictly smaller than |O ¯i |. We will, with O however, show that a feasible and profitable swap can be constructed by adding only some of ¯i . the sets in O For technical reasons we are going to define a set Z of elements that we (temporarily) disregard from our calculations because they will remain covered and thus should not impact ¯i we will pick for the feasible swap. More precisely, let our decision which of the sets in O Z = { u ∈ A ∩ B | A ∈ Ai , B ∈ A \ Ai , i ∈ [t] } be the set of elements that are covered by some Ai but that remain covered even if Ai is removed from A. S Let Li = Ai \ S Z be the set of elements that are “lost” when removing the Ai from A. ¯i \ S(A \ Ai ) be the set of elements that are “won” when we add all Moreover, let Wi = O ¯i after removing Ai . the sets of O Pt S We claim that A and that the i=1 |Li | ≤ ALG −|Z|. To this end, note that Z ⊆ family {Li }i∈[t] contains pairwise disjoint sets because all elements that are not exclusively covered Pt by a single Ai are contained in Z and thus removed. On the other hand, we claim that i=1 |Wi | ≥ OPT −|Z|. To see this, note first that every element in Z contributes 0 to the left hand side and 0 or -1 to the right hand side. Every element covered by O but not by A contributes at least 1 to the left hand side and precisely 1 to the right hand side. Finally, consider an element u that is covered both by A and by O but does not lie in Z. This element lies in a set S ∈ Ai for some i ∈ [t]. Because of the definition of the exchange graph G there is some set T ∈ O with u ∈ T and some set S ′ ∈ A with u ∈ S ′ such that S ′ and T are adjacent in G. We have that S ′ ∈ Ai , for, otherwise u ∈ Z. Because of the separator property of X ¯i . Moreover u lies in Wi because it is not (see Property (i) of Theorem 2) we must have T ∈ O ¯ contained in Z but is covered by Oi .PHence u contributes at least 1 to the left hand side and precisely 1 to the right hand side of ti=1 |Wi | ≥ OPT −|Z|, which shows the claim. We have OPT > |Z| and hence Pt |Li | ALG 12c1 f (b) ALG −|Z| |Li | ≤ Pti=1 ≤ 0
Hence, we can pick i ∈ [t] such that
|Li | 12c1 f (b) |Ai | (otherwise we can just add all sets in O ¯i ). Consider the swap Suppose that |O where we replace the |Ai | ≤ b many sets Ai from the local optimum A with at most |Ai | many ¯i . sets {S1 , . . . , S|Ai | } from O We now analyze how this swap affects the ojective function value. By removing the sets in Ai the objective function value drops by (2) 12c1 f (b) · |Wi | |Li | < 1 − b ¯ |A i| [ (4) | 12c1 f (b) |O i Sℓ \ Zi ≤ 1− b |Ai | ℓ=1 |Ai | (3) [ ≤ Sℓ \ Zi . ℓ=1 The right hand side of this inequality is the increase of the objective function due to adding the sets {S1 , . . . , S|Ai | } after removing the sets in Ai . Therefore the above described swap is feasible and also profitable and thus A is not a local optimum leading to a contradiction.
4
Applications
First, we define the following problems which are special instances of MC problem. Problem 1. Let H be a set of ground elements, S ⊆ 2H be a set of ranges and k be a positive integer. A range S ∈ S is hit by a subset H ′ of H if S ∩ H ′ 6= ∅. The Maximum Hitting (MH) problem asks for a k-subset H ′ of H such that the number of ranges hit by H ′ is maximized. Problem 2. Let G = (V, E) be a graph and k be a positive integer. A vertex v ∈ V dominates all the vertices adjacent to v including v. The Maximum Dominating (MD) problem asks for a k-subset V ′ of V such that the number of vertices dominated by V ′ is maximized. Problem 3. Let T be a 1.5D terrain which is an x-monotone polygonal chain in the plane consisting of a set of vertices {v1 , v2 , . . . , vm } sorted in increasing order of their x-coordinate, and vi and vi+1 are connected by an edge for all i ∈ [m − 1]. For any two points x, y ∈ T , we say that y guards x if each point in xy lies above or on the terrain. Given finite sets X, Y ⊆ T and a positive integer k, the Maximum Terrain Guarding (MTG) problem asks for a k sized subset Y ′ of Y such that the number of points of X guarded by Y ′ is maximized. 10
Let r be an even, positive integer. A set of regions in R2 , where each region is bounded by a closed Jordan curve, is called r-admissable if for any two such regions q1 , q2 , the curves bounding them cross s ≤ r times for some even s and q1 \ q2 and q2 \ q1 are connected regions. A set of regions are called pseudo-disks if is 2-admissable. For example a set of disks (of arbitrary size) or a set of squares (of arbitrary size) form a set of pseudo-disks each. We now state the following theorem (whose proof is given in Appendix B). It follows either because it is known from the literature that the corresponding SC problem is planarizable (that √ is, we can define an exchange graph as in Definition 1 that is planar and thus n-separable) or in case of claim (D2 ) and (V ) by construction the exchange graph as a minor of the input graph. Theorem 4. Local search gives a PTAS for the following classes of MC problems: (C1 ) the set of ground elements is a set of points in R3 , and the family of subsets is induced by a set of half spaces in R3 (C2 ) the set of ground elements is a set of points in R2 , and the family of subsets is induced by a set of convex pseudodisks (a set of convex objects where any two objects can have at most two intersections in their boundary). Local search gives a PTAS for the following MH problems: (H1 ) the set of ground elements is a set of points in R2 , and the set of ranges is induced by a set of r-admissible regions (this includes pseudodisks, same-height axis-parallel rectangles, circular disks, translates of convex objects). (H2 ) the set of ground elements is a set of points in R3 , and the set of ranges is induced by a set of half spaces in R3 . Local search gives a PTAS for MD problems in each of the following graph classes: (D1 ) intersection graphs of homothetic copies of convex objects (which includes arbitrary squares, regular k-gons, translated and scaled copies of a convex object). (D2 ) f -separable and minor-closed graph classes. Additionally, the following problems admit a PTAS via local search (V) the MVC problem on f -separable and subgraph-closed graph classes, (T) the MTG problem.
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A
Multicolored Separator Theorem
In this appendix we generalize our vector partitioning and colored separator results, i.e., to dimensions larger than two and more than two color classes. As we expect our tools to have even wider applications than the context in which we use them, we provide quite general versions of them and take some care regarding the computational complexity of the algorithms involved. More specifically, in Subsection A.1 we describe how to compute “low” discrepancy partitions of vectors in the d-dimensional unit cube (see Theorem 6). Here we focus on the L∞ -norm as it is the important case for our application to the multicolored separators. In the second subsection we note that combining this vector partitioning result with our uniform (r, f (r)-division (as given by Lemma 2) provides a notion of a uniform d-color balanced divisions for f -separable graph classes (see Theorem 7).
A.1
Higher Dimensional Vector Partitioning
In this section we consider the problem of partitioning n vectors {a1 , . . . , an } from the rational d-dimensional unit cube (Q ∩ [0, 1])d into k sets I1 , . . ., Ik where the sum P of the vectors in set is “close” (with respect to the L∞ norm) to the expected sum µ = k1 · ni=1 ai of nk vectors, i.e., into sets with “low” discrepancy. Progress on this kind of partitioning problem (usually phrased with respect to an arbitrary normed vector space and its corresponding unit ball) is sometimes referred as a B´ ar´ any-Grinberg type of result due to their initial progress on the case k = 2 [5]. 13
Namely, the discrepancy of such a vector partition is defined as the maximum P deviation of the sum of vectors in a set of the partition from the vector µ, i.e., maxi∈[k] kµ − j∈Ii aj k. Here we will provide a short, self-contained, efficient construction of such a partition in the context of the L∞ -norm (see Theorem 6). This is leveraged in the next subsection to obtain color balanced divisions of f -separable graphs. We first note some context from the literature. For example, one may observe that a result of this kind follows as a corollary of the following algorithmic version [3, Th.1] of Steinitz Lemma [27] (see Lemma 3 below). Namely, this readily provides a partition whose discrepancy is at most 3d + 1 (as described by Lemma 4 below). To our knowledge the strongest B´ar´ anyGrinberg result is given by B´ar´ any and Doerr [4] (included as Theorem 5 below for the sake of completeness and comparison) and is built on a recursive re-partitioning result given by Doerr and Srivastav [13]. The result appears to be constructive, but it is does take quite some effort (going through intermediate results) to establish. As such, it is difficult to estimate the time bounds. Furthermore, the authors did not seem discuss this aspect of their proofs. Note that the bound on the discrepancy here is quite strong (2.005d) and it is provided for all partial sums and is obtained for any norm. Our d-dimensional result (Theorem 6) provides a slightly improved bound (actually 2d), but is given in a more restricted context of the L∞ -norm, and our discrepancy bound is not guaranteed for partial sums, but rather for each entire set in the constructed partition. However, we feel that our proof (involving an iterative rounding process inspired by the Beck-Fiala Theorem [6]) is particularly nice since it is compact, self-contained, and directly provides efficient algorithm to produce the partition. Theorem 5 ([4]). Let d, c ∈ N, k · k be any norm on Rd , and B denote the unit ball with respect to this norm. Any sequence a1 , a2 , . . . of vectors in B can be partitioned into k sub-sequences V1 , . . . , Vk inPa balanced manner with respect to the partial sums, i.e., for all n ∈ N and l ≤ k 1 P we have: k( i≤k,vi ∈Vl vi ) − µk ≤ 2.005d where µ = k · i≤k ai .
Theorem 6. Let A = {a1 , . . . , an } ⊆ (Q ∩ [0, 1])d be a set of d-dimensional vectors, let k ≤ n be a positive integer. Then we can compute in polynomial time a partition I1 , . . . , IP k of [n] into P 1 k sets such that for any j ∈ [k], we have that kµ − i∈Ij ai k∞ ≤ 2d where µ = k · ni=1 ai .
Proof. The idea is to start with a perfect “fractional” assignment of vectors to sets Ij that is iteratively rounded to an integral partition with only a small imbalance. A partition of [n] into sets I1 , . . . , Ik can be described by a matrix x ∈ {0, 1}n×k of binary variables xij with i ∈ [n] and j ∈ [k] such that xij = 1 if and only if vector ai is assigned to partition Ij . The matrix x corresponds to a partition if and only if k X j=1
xij = 1, for every i ∈ [n] .
(5)
Now consider the continuous relaxation, where we allow the variables xij to attain also fractional values in [0, 1]. The intuitive meaning is that a fraction xij of vector ai is assigned to partition Ij . We start with an initial fractional assignment of x = k1 · 1. This assignment achieves perfect balance in the sense that n X xij ai = µ, for every j ∈ [k] . (6) i=1
In what follows, we will iteratively perturb the fractional assignment x while maintaining constraints (5) and (6). Let If ⊆ [n] × [k] be the set of index pairs (i, j) where xij is fractional (i.e. non-integral). Moreover, let Jf ⊆ [k] be the set of indices j where the vector xj := (x1j , . . . , xnj ) ∈ [0, 1]n contains at least 2d + 1 many fractional entries. 14
We now construct a non-trivial perturbation matrix δ ∈ Qk×n that satisfies k X
δij = 0,
for every i ∈ [n] ,
(7)
δij ai = 0,
for every j ∈ Jf .
(8)
(i,j)∈If n X i=1
We further require δij = 0 for every (i, j) ∈ [n] × [k] \ If . The crucial point is that there is a non-trivial solution δ satisfying the above constraints. To see this, we note that there are at least |If | ≥ (2d+1)|Jf | many free variables. On the other hand the number of non-trivial constraints of type (7) is at most |If |/2 since each i ∈ [n] contribute either zero or at least two pairs (i, j) to |If |. Since each constraint of type (8) corresponds to d linear constraints, we have at most |If |/2 + d|Jf | ≤ |If |/2 + d|If |/(2d + 1) < |If | many linear constraints in total. In particular, this means that we have more free variables than constraints and we can find a non-trivial solution δ 6= 0. Now, we determine the smallest factor ǫ > 0 such that x′ := x + ǫδ contains at least one additional integral variable. Such an ǫ exists, because δ is non-trivial and has non-zero entries only for fractional entries of x. Note that for every i ∈ [n], the matrix x′ satisfies the constraint of type (5) because δ satisfies the type (7) constraint. Similarly, for every j ∈ Jf , the matrix x′ satisfies the constraint of type (6) as δ satisfies the constraint of type (8). Since, we only changed fractional entries of x, every integral entry of x is the same in x′ and x′ has at least one additional integral entry. We apply the above rounding step as long as there fractional variables. (Note, that this rounding procedure may still be performed even Jf is empty.) We claim that the final solution x satisfies the claim of the theorem. To see this, note that initially Jf = [k] and that Jf = ∅ at the end of the algorithm. In particular, for each j ∈ [k] there is an iteration where j is removed from Jf for the first time. Since our algorithm does not change entries of x that are already integral, j will not occur Jf at any later iteration. Let x be the vector at the end of this iteration. Note that at this point x satisfies (6) for j and that there are at most 2d many fractional entries xj . Hence, regardless of the assignment x′ at the end of the algorithm, the Pn in ′ expression i=1 xij ai can deviate from µ by at most 2d in every entry.
We conclude this subsection by we showing how the algorithmic version of Steinitz Lemma (see Lemma 3) can be used to obtain a vector partition of d-dimensional vectors whose discrepancy is at most 3d + 1 with respect to the L∞ -norm (see Lemma 4). P Lemma 3. [3, Th.1] For a set {b1 , . . . , bn } ⊆ [−1, 1]d of d-dimensional vectors with ni=1 bi = 0 in O(n2 d3 + nd4 ) steps can be given a permutation π of the set [n] such that for each 1 ≤ l ≤ n
l
X 3
d . bπ(i) ≤
2 i=1
∞
Lemma 4. Let A = {a1 , . . . , an } ⊆ [0, 1]d be a set of d-dimensional vectors, let k ≤ n be a 4 positive integer. Then we can compute in O(n2 d3 + nd I1 , . . . , Ik of [n] into P ) time a partition k sets such that for any j ∈ [k], we have that kµ − i∈Ij ai k∞ < 2 23 d + 1 ≤ 3d + 1, where P µ = k1 · ni=1 ai .
Proof. For each i, let bi = nk · µ − ai . Note that the set {b1 , . . . , bn } satisfies the conditions of Lemma 3. Let π be the permutation provided by this lemma. Let I1′ , . . . , Ik′ be a partition of [n] into k consecutive discrete segments such that sizes of any two elements of the partition differs by at most 1. For each j ∈ [k], let Ij = π(Ij′ ). We now have: 15
X
µ − ai
i∈Ij
∞
X
ai = µ −
i∈π(I ′ ) j
∞
X X
ai ai − = µ +
k≤j, i∈π(I ′ ) k<j, i∈π(I ′ ) k
k
=
∞
X X
kµ kµ
µ − kµ |Ik′ | +
≤ ai − ai − −
n n n
′ ′ k<j, i∈π(Ik ) k≤j, i∈π(Ik ) ∞ k 3 3 3 3 n k kµk∞ − |Ik′ | + d + d < kµk∞ + 2 d ≤ 1 + 2 d . n k 2 2 n 2 2
A.2
Separator Theorem
We will now use the uniform (r, f (r)-division obtained in Lemma 2 and the d-dimensional vector partitioning theorem (Theorem 6) from the previous section to obtain a d-color uniform separator theorem on f -separable graph classes (see Theorem 7 below). In particular, for a given d-colored graph G where G belongs to an f -separable graph class, we first construct a uniform (r, c · f (r))-division (X , V1 , . . . , Vt ) of G as in Lemma 2. From this division we carefully partition the Vi ’s into collections Wj where each Wj has roughly the same size and contains roughly the same proportion of each color class as occurring in G. We imagine the regions V1 , . . . , Vt of the uniform (r, c · f (r))-division as a collection of d-dimensional vectors (whose 1 , and coordinates correspond to the number of vertices of each color), scale these vectors by 2r then apply Theorem 6 with the parameter k = r to obtain the result. Theorem 7. Let G be an f -separable graph class and G = (V, E) be a d-colored n-vertex graph in G with color classes Z1 , . . . Zd . For any r ≪ n where r is suitably large , there is an integer t ∈ Θ( rn2 ) such that V can be partitioned into t + 1 sets X , V1 , . . . , Vt where c1 , c2 are constants independent of n, r, and the following properties are satisfied. (i) N (Vi ) ∩ Vj = ∅ for each i 6= j, (ii) |Vi | ∈ Θ(r 2 ) for each i, (iii) |N (Vi ) ∩ X | ≤ c1 · r · f (r)) for each i (i.e., |X | ≤ (iv) ||Vi ∩ Zq | −
|Zq | t |
≤ 2 · r · d for each q ∈ [d].
Pt
i=1 |X
∩ N (Vi )| ≤
c2 ·f (r)·n ). r
Moreover, such a partition can be found in O(h(n) + p(n)) time where h(n) is the amount of time required to produce a uniform (r, c · f (r))-division of G and p(n) is the polynomial running time of the vector partitioning in Lemma 4.
B
Applications
Proof of Theorem 4. In what follows, we refer to several known results for SC where the respective instances are planarizable. This always also implies that the corresponding MC problem is planarizable. By the result of Mustafa and Ray [25], we know that the MC instance is planarizable when the family F is a set of half spaces in R3 , or a set of disks in R2 . Recently, De and Lahiri [12] showed that when the objects are convex pseudodisks, then the corresponding SC (and thus MC) instance is planarizable. Thus, as a consequence of Theorem 3, we have (C1 ) and (C2 ). Note that MH problem is a special instance of MC problem, where the set S of ranges plays the role of U , and the set H plays the role of F, where each set h ∈ H contains all the range 16
S ∈ S such that S ∩h 6= ∅. On the other hand, It follows from the result of Mustafa and Ray [25] that an MH instance is planarizable when the set of ranges are (i) a set of r-admissible regions, (ii) set of half spaces in R3 . Thus, Theorem 3 implies that an MH problem admits PTAS when the ranges are a set of r-admissible region or half spaces in R3 . Thus, we have (H1 ) and (H2 ). Observe that MD is a special instance of MC, where the set V of vertices plays the role of the set U of ground elements , and the family F consists of |V | subsets of V where each subset is corresponding to the set of vertices dominated by each vertex v ∈ V . On the other hand, from the result of De and Lahiri [12], we know that corresponding instance of MD is planarizable when the graph G is an geometric intersection graph induced by homothetic set of convex objects. Thus, as a consequence of Theorem 3, we have (D1 ). To prove (D2 ), we claim that these MC instances are f -separable according to Definition 1. To this end, let D, D ′ be two disjoint feasible solutions. We are going to construct an auxiliary graph H as in Definition 1. We start with the node set D ∪ D ′ and an empty edge set. Let u ∈ V be node that is dominated by D and by D ′ . If u ∈ D then there is a neighbor v ∈ D ′ of u. We add edge uv to H. The case u ∈ D ′ is handled symmetrically. If u ∈ / D and u ∈ / D′ ′ ′ then there are neighbors v ∈ D and v ∈ D of u. In this case we add u and the two edges uv and uv ′ to H. Note that the resulting graph is a subgraph of G. Now, we perform the following operation on H as long as H contains a node that is not in D ∪ D ′ . If such a node u exists it must have precisely two neighbors v ∈ D and v ′ ∈ D ′ by construction of H. We contract the edge uv and identify the resulting node with v (lying in D). As a result we obtain a minor H of G with node set D ∪ D ′ . It is easy to check that this graph fulfills the requirements of Definition 1. Moreover, because G is contained in a minor-closed, f -separable graph class we know that H is f -separable and we obtain that M D problem admits a PTAS on such graph classes. The proof of (V ) is even simpler than the one of (D2 ). Let D, D ′ be two disjoint feasible solutions. We are going to construct an auxiliary graph H as in Definition 1. We start with the node set D ∪ D ′ and an empty edge set. For any edge uu′ that is covered by both D and D ′ we may assume u ∈ D and u′ ∈ D ′ . We add edge uu′ to H. Note that the graph H is a subgraph of G, and it fulfils the requirement of Definition 1. Moreover, because G is contained in a subgraph-closed, f -separable graph class we know that H is f -separable and we obtain that M V C problem admits a PTAS on such graph classes. It is easy to observe that MTG is a special instance of MC, where the set X plays the role of U , and the set Y plays the role of family F of subsets, where each y ∈ Y contains all elements of X which can be guarded by y. On the other hand, we know from the result of Krohn et al. [21, Lem.§2] that MTG is planarizable. Thus, we have (T ).
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