R. Fowler, M. Paterson and S. Tanimoto, "Optimal packing and covering in the plane are NP complete", Inf. Proc. Letters, Vol. 12, No. 3, pp. 133-137, 1981.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....b, such that every element from N is covered by a disk with radius r centered in one of the points in B. If a point j lies in distance exactly d from some i 2 B, i.e. on a circle with radius r around i, we stipulate that it is covered by a disk centered in i. It is known that DC is NP complete [5], even in the strong sense [7] Lemma 2. For C 12, the problem BSPC is strongly NP hard. Proof. First, we show that BSPC is NP hard by constructing a Turing reduction from DC. Given is the set N of points in the plane, the radius r, and the budget b. We x an arbitrary triangulation T ....

....able to conclude that a constant C, as required in the de nition of BSPC , exists. However, the reduction does not guarantee (R3) Hence, the above reduction does not show that BSPC is hard. Nevertheless, it can be shown, using a modi cation of the reduction from 3SAT to DC given in [5], that the restricted problem is strongly NP hard; hence we conclude: Theorem 4. For D d the problem BSPC admit an FPTAS, unless P = NP. Proof. Sketch. Fix d; D, where D d. Given a 3CNF formula , we rst construct an instance of DC with loops for all variables in as in [5, Theorem ....

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R. J. Fowler, M. S. Paterson, and S. L. Tanimoto. Optimal packing and covering in the plane are NP-complete. Information Processing Letters, 12:133-137, 1981.

....for computing a p center Of n points in R . Therefore, for a fixed value of p, the Euclidean p center (and also the Euclidean discrete p center) problem can be solved in polynomial time in any fixed dimension. However, either of these problems is NP Complete for d 2, if p is part of the input [123, 217]. This has led researchers to develop efficient algorithms for approximate solutions and for small values of p and d. Approximation algorithms. Let r be the minimum value of r for which p disks of radius r cover D. Feder and Green [118] showed that computing a set S of p supply points so that ....

....later improved in [182] See [134, 186] for other approximation algorithms. Another way of seeking an approximation is to find a small number of balls of a fixed radius, say r, that cover all demand points. Computing k , the minimum number of balls of radius r that cover D, is also NP Complete [123]. A greedy algorithm can construct k log n balls of radius r that cover D. Hochbaum and Maass gave a polynomial time algorithm to compute a cover of size (1 )k , for any 0 [152] see also [51, 118, 134] No constant factor approximation algorithm is known for the capacitated covering ....

[Article contains additional citation context not shown here]

R. J. Fowler, M. S. Paterson, and S. L. Tanimoto, Optimal packing and covering in the plane are NP-complete, Inform. Process. Lett., 12 (1981), 133--137.

....the performance of the algorithm in both centralized and distributed settings. Finally Section 6 concludes with some directions for future research. 2 Previous work There is little prior work on this specific mobile clustering problem. The static version of the problem is known to be NP complete [9] and to admit a PTAS (polynomial time approximation scheme) It is equivalent to finding the minimum dominating set in the intersection graph of unit disks. The dominating set problem is defined as follows. Given a graph G = V, E) find a minimum size subset V # of vertices, such that every ....

R. J. Fowler, M. S. Paterson, and S. L. Tanimoto. Optimal packing and covering in the plane are NP-complete. Inform. Process. Lett., 12(3):133--137, 1981.

....alternate definition of the interference graph in [BNR96] The dual of the RPACK problem is to find the minimum cardinality subset of disjoint hyper rectangles with total weight at least w, for some given weight bound w. Since it is NP hard to even find a feasible solution for this dual problem [FPT81], it cannot be approximated to within any factor. Hence, we do not consider this dual problem any further. All our results will generalize to n1 Theta n2 array. Henceforth, when we refer to arbitrary array A we imply one with non negative integers. While our results do extend to the case when ....

....the set of hyper rectangles that is generated is typically much smaller than the set of all possible hyperrectangles. The focus is therefore again on the sparse input case. 1. 3 Summary of Our Results and Related Research Since the RTILE, DRTILE and d RPACK problems are known to be NP Hard [FPT81, KMP98] in two or more dimensions , our goal is to design approximation algorithms with guaranteed performance bounds. Naturally, our focus is to design approximation algorithms with better performance ratios than previously known, but additionally, our focus is to design such approximation algorithms ....

R. Fowler, M. Paterson, and S. Tanimoto. Optimal packing and covering in the plane are np-complete. Information Proc. Letters, 12, 133--137, 1981.

....be achieved in polynomial time for the rectilinear k center problem is 2. Several approximation results (on the radii of the disks) have been obtained in [11, 17, 20, 21] For more results on the k center problem, please refer to [2] Regarding the k piercing set problem in , Fowler et al. [12] proved the NP completeness of nding the minimal value of k for a given set of n disks. Hochbaum and Maas [19] gave an O(l n 2l 1 ) polynomial time algorithm with approximation factor (1 for any xed integer l 1. Thus, for l = 1, their algorithm yields an O(n ) time ....

R. J. Fowler, M. S. Paterson, and S. L. Tanimoto. Optimal packing and covering in the plane are NP-complete. Inform. Process. Lett., 12(3):133-137, 1981.

....be achieved in polynomial time for the rectilinear k center problem is 2. Several approximation results (on the radii of the disks) have been obtained in [11, 17, 20, 21] For more results on the k center problem, please refer to [2] Regarding the k piercing set problem in , Fowler et al. [12] proved the NP completeness of nding the minimum value of k for a given set of n disks. Hochbaum and Maas [19] gave an O(l n 2l 1 polynomial time algorithm for the minimum piercing set problem with approximation factor (1 for any xed integer l 1. Thus, for l = 1, their ....

R. J. Fowler, M. S. Paterson, and S. L. Tanimoto. Optimal packing and covering in the plane are NP-complete. Inform. Process. Lett., 12(3):133-137, 1981.

....at most L points. Previous results. There is a vast literature on clustering problems, see, for example, the books [3, 11, 18] the survey paper [1] and the references there in. Even the simplest clustering problems are known to be NP Hard, including the Euclidean k center problem in the plane [13, 24]. In fact, it is NP Hard to approximate the two dimensional k center problem within a factor of 2 even under the L1 metric [12] The greedy algorithm by Gonzalez [14] gives a 2 approximation algorithm for the k center problem in any metric space. This algorithm requires O(kn) distance ....

R. J. Fowler, M. S. Paterson, and S. L. Tanimoto, Optimal packing and covering in the plane are NP-complete, Inform. Process. Lett., 12 (1981), 133--137.

....pairwise disjoint rectangles in R corresponds to an independent set in G(R) We want to compute a maximum independent set of G(R) Abusing the terminology slightly, we will say that we want to compute a maximum independent set of R. Computing an independent set of rectangles is known to be NP hard [8, 13]. This suggests that one should aim for approximation algorithms. We call an algorithm an approximation algorithm, for 1, if it returns an independent set of size at least fl= where fl is the size of a maximum independent set of R. Although it is known that no ....

R. J. Fowler, M. S. Paterson, and S. L. Tanimoto. Optimal packing and covering in the plane are NP-complete. Inform. Process. Lett., 12(3):133--137, 1981.

....of objects that are not pierced by these cells has a non empty intersection (this is an LP type problem) Thus, we obtain an O(n d(k Gamma1) 1 ) expected time algorithm. Since the problem of finding the minimum integer k so that a set of objects S is k pierceable was shown to be NP complete [Kar72, FPT81], many authors have focused on the problem of approximating k. There exist some polynomial time algorithms for the latter problem with bounded error ratio [Chv79, Hoc82] Bellare et al. BGLR93] show that no polynomial time algorithm can approximate the optimal solution within a factor of ( ....

R. J. Fowler, M. S. Paterson, and S. L. Tanimoto. Optimal packing and covering in the plane are NP-complete. Inform. Process. Lett., 12(3):133--137, 1981.

....G(F ) V; E) is planar (see [12] where V and E are defined as follows: V = fx 1 ; x 2 ; x n g E = f(x j ; C i ) j x i or x i appears in C i g : Theorem 2.1 The planar bichromatic partition problem is NP Hard. Proof: Our construction is similar to the one used by Fowler et al. [9], who prove the intractability of certain planar geometric covering problems (without the disjointness condition) see also [4, 8] for similar constructions. We first describe our construction for the bichromatic partition problem. To simplify the proof, our construction allows three or more ....

....1 (iii) Let R be a set of red points satisfying the above two properties. We claim that the set of blue points B can be covered by k disjoint triangles, none of which contains any red point, if and only if the formula F has a truth assignment. Our proof is similar to the one in Fowler et al. [9]; we only sketch the main ideas. The red points act as enforcers, ensuring that only those blue points that are adjacent on the boundary of a P i can be covered by a single triangle. Thus, the minimum number of triangles needed to cover all the points on P i is k i . Further, there are precisely ....

R. Fowler, M. Paterson, and L. Tanimoto, Optimal packing and covering in the plane are NP-complete, Information Processing Letters 35 (1981), 85--92.

....under node motion and analyzes the performance of the algorithm. Finally Section 6 concludes with some directions for future research. 3 2 Previous work There is little prior work on this specific mobile clustering problem. The static version of the problem is known to be NP complete [11] and to admit a PTAS (polynomial time approximation scheme) A variant, the connected dominating set problem, has been studied extensively as well. The static version of the discrete clustering problem is equivalent to finding the minimum dominating set in the intersection graph of unit disks. ....

R. J. Fowler, M. S. Paterson, and S. L. Tanimoto. Optimal packing and covering in the plane are NP-complete. Inform. Process. Lett., 12(3):133--137, 1981.

....the problem is equivalent to nding the Work supported in part by an NSERC Research Grant. E mail: tmchan uwaterloo.ca. 1 maximum independent set in the intersection graph of the objects. The case where the objects are axis aligned unit squares or unit circles in the plane is already NP hard [14, 19], so it is desirable to nd good approximation algorithms. Although the independent set problem in general graphs is hard to approximate, even to within a factor of n 1 [16] the geometric restriction lets us obtain much better approximability results. The problem, particularly in the ....

....geometry literature (see the references in [2] but primarily addressed the case where we are to choose only a constant number p of points. When a large number of chosen points is possible, the problem is NP hard, even if the objects are axis aligned unit squares or unit disks in the plane [14]. Therefore, we again investigate ecient approximation algorithms. The piercing problem is related to the packing problem. Clearly, pierce(C) pack(C) For intervals in one dimension, the two numbers are well known to be equal. For fat objects in any xed dimension, the two numbers are within a ....

R. J. Fowler, M. S. Paterson, and S. L. Tanimoto. Optimal packing and covering in the plane are NP-complete. Inform. Process. Lett., 12:133-137, 1981.

....Compute the corresponding complete label placement. The Label Number Maximization Problem: Find a maximum subset of the features, and for each of these features a label from its set of candidates, such that no two labels overlap. The decision versions of both problems are NP hard in general [5, 4]. In recent years, especially the point labeling problem has achieved some attention in the algorithms community. Formann and Wagner proposed an approximation algorithm that maximizes the size of uniform axis parallel square labels. Their algorithm can be applied to technical maps where labels ....

Robert J. Fowler, Michael S. Paterson, and Steven L. Tanimoto. Optimal packing and covering in the plane are NP-complete. Information Processing Letters, 12(3):133-137, 1981.

....kinetically under node motion and analyzes the performance of the algorithm. Finally Section 6 concludes with some directions for future research. 2. PREVIOUS WORK There is little prior work on this specific mobile clustering problem. The static version of the problem is known to be NP complete [10] and to admit a PTAS (polynomial time approximation scheme) A variant, the connected dominating set problem, has been studied extensively as well. The static version of the discrete clustering problem is equivalent to finding the minimum dominating set in the intersection graph of unit disks. ....

R. J. Fowler, M. S. Paterson, and S. L. Tanimoto. Optimal packing and covering in the plane are NP-complete. Inform. Process. Lett., 12(3):133--137, 1981.

....for Advanced Networking at Caltech. 1 Application of this theorem to the geometric disk covering problem is in order. Let us try to nd the minimum number of disks of prescribed radius r, which cover a given set of n points in the plane. It is known that this problem is strongly NP complete [6], therefore we are interested in nding approximate solutions that run in polynomial time and have a bounded error ratio. Application of a simple greedy heuristic, originally developed for the set covering problem, leads to a worst case error ratio bounded by (1 ln n) and this bound is tight, as ....

R.J. Fowler, M.S. Paterson and S. L. Tanimoto. \Optimal Packing and Covering in the Plane are NP-Complete". Information Processing Letters, 12-3, pp. 133-137, 1981.

....efficient matching technique of Efrat and Itai [13] Our second application concerns piercing fat objects. A set of points P in R d is a piercing set for a set C of objects, if for each object c 2 C there exists a point in P that lies in c. Finding a minimal piercing set is NP complete for d 2 [16], so it is natural to seek approximate solutions, in which the size of the computed piercing set is not much larger than the optimal size. The problem of finding a minimal piercing set is a special instance of the well known set cover problem, if we regard each cell in the arrangement of C as the ....

R.J. Fowler, M.S. Paterson, and S.L. Tanimoto, Optimal packing and covering in the plane are NP-complete, Information Processing Letters 12 (3) (1981), 133-- 137.

....way that two vertices are joined by an edge iff the corresponding circles intersect. It is assumed that tangent circles intersect. We assume without loss of generality that the radius of each disk is 1. These graphs have been used to model broadcast networks [CCJ90, Ha80, Ra93] image processing [FPT81, HM85], VLSI circuit design [MC80] and optimal facility location [MS84, WK88] Consequently, unit disk graphs have been studied extensively in the literature [CCJ90, FPT81, MHR92, MS84, WK88] As pointed out in [CCJ90] unit disk graphs need not be perfect (any odd cycle of length five or greater is a ....

....that the radius of each disk is 1. These graphs have been used to model broadcast networks [CCJ90, Ha80, Ra93] image processing [FPT81, HM85] VLSI circuit design [MC80] and optimal facility location [MS84, WK88] Consequently, unit disk graphs have been studied extensively in the literature [CCJ90, FPT81, MHR92, MS84, WK88]. As pointed out in [CCJ90] unit disk graphs need not be perfect (any odd cycle of length five or greater is a unit disk graph and is not perfect) Similarly, unit disk graphs need not be planar; in particular, any clique of size five or more is a unit disk graph but is not planar. Thus many of ....

[Article contains additional citation context not shown here]

R.J. Fowler, M.S. Paterson and S.L. Tanimoto, "Optimal Packing and Covering in the Plane are NP-Complete," Inf. Proc. Letters, Vol 12, No.3, June 1981, pp. 133-137.

....and R are geometric ranges. Let H be the set of all d dimensional halfspaces. Here are two examples of geometric set systems: i) R d ; H ) and (ii) H; ffh 2 H j p 2 hg j p 2 R d g. It is known that the set cover problem is NP Hard even in geometric settings. For example, Fowler et al. [78] proved that it is NP Hard to decide whether a given set of n points can be covered by k unit squares. The greedy algorithm can be used for computing an O(log n) approximation. However, one can do slightly better if the VC dimension of the set system is nite; see [95] for the de nition of ....

....higher dimensions. Rectilinear p center. In this problem the metric is the L1 distance, so the decision problem is now to cover the given set D by a set of p axis parallel cubes, each of length 2r. The problem is NP Hard if p is part of the input and d 2, or if d is part of the input and p 3 [78, 143]. Ko et al. 120] showed that computing a solution set S with c(D; S) 2r is also NP Hard. The rectilinear 1 center problem is trivially solved in linear time. See [62, 118, 119, 143] for some earlier results. Sharir and Welzl [172] developed a linear time algorithm for the rectilinear ....

R. J. Fowler, M. S. Paterson, and S. L. Tanimoto, Optimal packing and covering in the plane are NP-complete, Inform. Process. Lett., 12 (1981), 133-137.

....for this; the best algorithm we have runs in exponential time. We also do not have a proof that computing the ffi simple cover complexity is NP complete, although the seemingly simpler DISC PACK problem which consists of covering a set of n planar points by a given set of discs is NP hard [7], and a restricted version, in which the discs are identical, is NP complete [7] We leave the determination of the computational complexity as an open problem. 5 Intermezzo: boundary fatness In the definition of fatness given in Section 2, we used the family U(P) of all balls centered inside P ....

.... have a proof that computing the ffi simple cover complexity is NP complete, although the seemingly simpler DISC PACK problem which consists of covering a set of n planar points by a given set of discs is NP hard [7] and a restricted version, in which the discs are identical, is NP complete [7]. We leave the determination of the computational complexity as an open problem. 5 Intermezzo: boundary fatness In the definition of fatness given in Section 2, we used the family U(P) of all balls centered inside P and whose boundary intersects the boundary of P; the fatness of P was then ....

R. J. Fowler, M. S. Paterson, and S. L. Tanimoto. Optimal packing and covering in the plane are NP-complete. Inform. Process. Lett., 12(3):133--137, 1981.

....structured as follows. Section 2 introduces the six models 2 three xed position and three slider that are compared in this paper. We analyze how many more labels can be placed in one model than another, in theory. Point labeling has long been shown to be NP complete for xed position models [10,17,9,15]. However, this does not imply that label placement is also hard for slider models. In Section 3 we show that this is the case; we prove that it is NP complete to decide whether a set of points can be labeled in the four slider model. In Section 4, we show that the slider models allow a simple ....

....model. Marks and Shieber note that their proof also holds for models with an in nite number of positions like that of Hirsch [11] where a label must touch a circle of small radius centered on the point to be labeled. However, their proof cannot be used for our slider model. Fowler et al. [10], and Knuth and Raghunathan [15] have proved the NP hardness of two other labeling models. Recently, Iturriaga and Lubiw have proven that it is NP hard to decide whether a set of points can be labeled with what they call left right sliding labels [14] Their model is similar to our 2 slider model ....

Robert J. Fowler, Michael S. Paterson, and Steven L. Tanimoto. Optimal packing and covering in the plane are NP-complete. Inform. Process. Lett., 12(3):133-137, 1981.

....algorithm for computing a discrete p center. Therefore, for a fixed value of p, the Euclidean p center (and also the Euclidean discrete p center) problem can be solved in polynomial time in any fixed dimension. However, either of these problems is NP complete for d 2, if p is part of the input [104, 187]. This has led researchers to develop efficient algorithms for approximate solutions and for small values of p and d. Approximation algorithms. Let r be the minimum value of r for which p disks of radius r cover D. The greedy algorithm described in Figure 3, originally proposed by Gonzalez [113] ....

....L1 metric, is NP Hard. See [114, 159] for other approximation algorithms. Another way of seeking an approximation is to find a small number of balls of a fixed radius, say r, that cover all demand points. Computing k , the minimum number of balls of radius r that cover D, is also NP complete [104]. A greedy algorithm can construct k log n balls of radius r that cover D. Hochbaum and Maass gave a polynomial time algorithm to compute a cover of size (1 )k , for any 0 [130] see also [45, 101, 114] No constant factor approximation algorithm is known for the capacitated covering ....

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R. J. Fowler, M. S. Paterson, and S. L. Tanimoto, Optimal packing and covering in the plane are NP-complete, Inform. Process. Lett., 12 (1981), 133--137.

....b, such that every element from N is covered by a disk with radius r centered in one of the points in B. If a point j lies in distance exactly d from some i 2 B, i.e. on a circle with radius r around i, we stipulate that it is covered by a disk centered in i. It is known that DC is NP complete [FPT81] even in the strong sense [HM85] Lemma 2. For C 12, the problem BSPC is strongly NP hard. Proof. First, we show that BSPC is NP hard by constructing a Turing reduction from DC. Given is the set N of points in the plane, the radius r, and the budget b. We x an arbitrary triangulation T ....

....the de nition of the general BSPC problem, exists. However, the reduction does not guarantee that D and d are constant. Hence, the above reduction does not show that the restricted BSPC problem is hard. Nevertheless, it can be shown, using a modi cation of the reduction from 3SAT to DC given in [FPT81] that the restricted BSPC problem is strongly NP hard; hence we conclude: Theorem 4. The problem BSPC, restricted to the case that d and D are constant and D d, does not admit an FPTAS, unless P = NP. Proof. Sketch. Fix d; D, where D d. Given a 3 CNF formula , we rst construct an ....

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R. J. Fowler, M. S. Paterson, and S. L. Tanimoto. Optimal packing and covering in the plane are NP-complete. Information Processing Letters, 12:133-137, 1981.

....abstract version of this paper appears in the proceedings APPROX 98 . This work was supported by the German Federal Ministry of Education, Science, Research and Technology (BMBF, F orderkennzeichen 01 IR 411 C7) 1 The following decision problem was shown to be NP complete by Fowler et al. [13]; here and throughout the paper an L square is a rectangle of size L L, and the set of vertices of a polygonal region includes the vertices of all the holes it may have. Pack(k; L) Input: a polygonal region P with n vertices, a parameter k, a parameter L. Question: Can k many L squares be ....

....distance functions for d(v; w) the most natural ones are L 2 distances and L 1 or L1 distances. In the following, we concentrate on rectilinear, i.e. L1 distances; all ideas carry over for other metrics by combining our ideas with the techniques by Hochbaum and Maass [21] and Fowler et al. [13]. We do not include all the details, but sketch the approach at the end of Section 2. The main results of this paper are organized as follows: in Section 2, we show that geometric dispersion with boundaries cannot be approximated arbitrarily well within polynomial time, unless P=NP. In Section 3, ....

[Article contains additional citation context not shown here]

R. J. Fowler, M. S. Paterson, and S. L. Tanimoto. Optimal packing and covering in the plane are NP{complete. Information Processing Letters, 12, 1981, 133-137.

....for A of size at most (1 )c(A) 1 in amortized update time O( log n ) Maintenance of a box cover. Let Q be a set of n points in R d . A cover for Q is a set of (axis parallel) unit hypercubes whose union contains Q. The problem of computing a minimum cover is known to be NP complete [4], and is dual to the following piercing problem. Given a set B of n unit hypercubes in R d , compute a minimum piercing set for B. We present several ecient algorithms for dynamically maintaining a small piercing set for a set of arbitrary (axis parallel) boxes in R d . We obtain an O(c ....

....(axis parallel) hypercubes. Dually, we wish to compute a minimum piercing set for a set Q of n d dimensional unit hypercubes. These problems are referred to in the literature as the BOX COVERING and BOX PIERCING problems. For d 2, these problems were shown to be NPcomplete by Fowler et al. [4]. We consider the BOX PIERCING problem for arbitrary (axis parallel) boxes and in a dynamic setting, where from time to time boxes are inserted and deleted from Q . We begin with a simple observation, and then dynamize an approximation scheme presented in [11] Let S = fB 1 ; B n g be a ....

R. J. Fowler and M. S. Paterson and S. L. Tanimoto \Optimal packing and covering in the plane are NP-complete", Information Processing Letters 12(3) (1981), pp. 133-137.

....more than one rectangle. The implication of this is that more than one matching entry for that cell may exist in the forwarding table. In general, even in one dimension, packing or covering problems involving objects of different sizes are NP complete (the Knapsack problem is one such example) [46]. A number of problems closely related to the address aggregation problem identified above have been shown to be NP complete. In the context of image processing, Fowler, Paterson, and Tanimoto have shown that the planar geometric covering problem using 2x2 squares is NP complete [46] We have ....

....such example) 46] A number of problems closely related to the address aggregation problem identified above have been shown to be NP complete. In the context of image processing, Fowler, Paterson, and Tanimoto have shown that the planar geometric covering problem using 2x2 squares is NP complete [46]. We have proven above that not all possible address aggregations are geographically contiguous. If we define the optimal address aggregation as including also non contiguous cells, then the problem is equivalent to a classical problem of Boolean logic minimization known as the minimum sum problem ....

R. Fowler, M. Paterson, and S. Tanimoto. Optimal Packing and Covering in the Plane are NP-Complete. Information Processing Letters, 12(3):133--37, 1981. 137

.... radius (that is, the minimum radius of a ball containing all the points in the cluster) 1 If such a k way partition exists, then we say that X is (k; b) clusterable (with respect to the diameter or the radius cost) Unfortunately, both decision problems are NP Complete for d 2 (and variable k) [14, 29], and remain hard even when only a certain constant approximation of the cluster size is sought [13] In this work we consider the following relaxation of the above decision problems: For a given approximation parameter fi 0, and distance parameter 0 ffl 1, we would like to determine whether ....

R. J. Fowler, M. S. Paterson, and S. L. Tanimoto. Optimal packing and covering in the plane are NP-complete. IPL, pages 133--137, 1981.

....alternate definition of the interference graph in [BNR96] The dual of the RPACK problem is to find the minimum cardinality subset of disjoint hyper rectangles with total weight at least w, for some given weight bound w. Since it is NP hard to even find a feasible solution for this dual problem [FPT81], it cannot be approximated to within any factor. Hence, we do not consider this dual problem any further. 1.2 Motivating Applications Rectangle tiling and packing problems as defined above are natural combinatorial problems arising in many scenarios. For motivation, we will very briefly review ....

....above, the set of hyper rectangles that is generated is typically much smaller than the set of all possible hyperrectangles. The focus is therefore on the sparse input case. 1. 3 Summary of Our Results and Related Research Since both the tiling and packing problems are known to be NP Hard [FPT81, KMP98] in two or more dimensions 3 , goal is to design approximation algorithms with guaranteed performance bounds. Naturally, our focus is to design approximation algorithms with better performance ratios than previously known, but additionally, our focus is to design such approximation algorithms ....

R. Fowler, M. Paterson, and S. Tanimoto. Optimal packing and covering in the plane are np-complete. Information Proc. Letters, 12, 133--137, 1981.

....and R are geometric ranges. Let H be the set of all d dimensional halfspaces. Here are two examples of geometric set systems: i) R d ; H ) and (ii) H; ffh 2 H j p 2 hg j p 2 R d g. It is known that the set cover problem is NP Hard even in geometric settings. For example, Fowler et al. [82] proved that it is NP Hard to decide whether a given set of n points can be covered by k unit squares. The greedy algorithm can be used for computing an O(log n) approximation. However, one can do slightly better if the VC dimension of the set system Sigma is finite; see [100] for the definition ....

....dimensions. Rectilinear p center. In this problem the metric is the L1 distance, so the decision problem is now to cover the given set D by a set of p axis parallel cubes, each of length 2r. The problem is NP Complete if p is part of the input and d 2, or if d is part of the input and p 3 [82, 149]. Ko et al. 126] showed that computing a solution set S with c(D; S) 2r , where r is the size of an optimal solution, is also NP Complete. The rectilinear 1 center problem is trivially solved in linear time. See [66, 124, 125, 149] for some earlier results. Sharir and Welzl [182] developed a ....

R. J. Fowler, M. S. Paterson, and S. L. Tanimoto, Optimal packing and covering in the plane are NP-complete, Inform. Process. Lett., 12 (1981), 133--137.

....case of the MAX SUM ID metric, it was shown in [18] that the minimum heft cannot be approximated to within a factor of 1:25. We establish similar results for a a different set of metrics that includes SUM SUMSQR DIFF, MAX MAX AVG DIFF, and MAX MAX GEO DIFF. As in [18] and the earlier work in [9], the proof is based on a reduction from the Planar 3SAT problem (shown to be NP complete in [21] though a number of changes are needed to adapt the argument to our types of metrics. Similar results can also be shown for several other metrics, but we restrict ourselves to the most important ones ....

R. Fowler, M. Paterson, and S. Tanimoto. Optimal packing and covering in the plane are np-complete. Information Proc. Letters, 12, 133--137, 1981.

....length five or greater is a unit disk graph. Similarly, unit disk graphs need not be planar; in particular, any clique of size five or more is a unit disk graph. Thus many of the known efficient algorithms for perfect graphs and planar graphs do not apply to unit disk graphs. It has been shown in [CCJ90, FPT81, MS84, WK88] that several standard graph theoretic problems are strongly NP hard even when restricted to unit disk graphs. Given this apparent intractability, we investigate whether these problems have efficient approximation algorithms and approximation schemes. Recall that an approximation algorithm for an ....

....sufficient syntactic restrictions on BOW specifications, which allow for rapid processing of the designs and also include many realistic designs. 2 Related Work The complexity of finding exact solutions to graph problems, when restricted to unit disk graphs, has been studied extensively in [CCJ90, FPT81, MS84, WK88]. In [MB 95] we showed that several natural graph problems such as MAXIMUM INDEPENDENT SET, MINIMUM VERTEX COVER and MINIMUM DOMINATING SET can be approximated to within a constant factor of the optimum for unit disk graphs specified using a graph theoretic representation. Other researchers have ....

R. J. Fowler, M. S. Paterson and S. L. Tanimoto, "Optimal Packing and Covering in the Plane are NP-Complete," Information Processing Letters, Vol. 12, No. 3, June 1981, pp. 133--137.

.... [BOW83] or [LW92] Most of the PSPACE hardness results hold for unit disk graphs specified using either the specification language of Lengauer et al. LW92] or that of Bentley et al. BOW83] Here we show that for all the problems (except the Steiner tree problem) which were shown to be NP hard in [CCJ90, FPT81, WK88] for non hierarchically specified graphs, the corresponding problems for hierarchically specified graphs are PSPACE hard. The polynomial time approximation algorithms given in this paper along with our results in [MHR93, MHSR94] are among the first ever polynomial time approximation algorithms for ....

....is illustrated by the following example. Example: Consider the independent set (IS) problem for unit disk graphs. When the input instance (in this case, the input graph) is specified using one of the standard representations, the problem can be shown to be NP hard by a local reduction from 3SAT [FPT81, WK88]. Roughly speaking, the phrase local reduction refers to a reduction where each clause and each variable is replaced by a fixed size subgraph or gadget. Problems such as independent set, dominating set, and clique cover have been proved to be NP hard for unit disk graphs using such local ....

[Article contains additional citation context not shown here]

R. J. Fowler, M. S. Paterson and S. L. Tanimoto, "Optimal Packing and Covering in the Plane are NP-Complete," Information Processing Letters, Vol. 12, No. 3, June 1981, pp. 133-137.

....where we have to pack identical rectangles into a larger rectangle; it is still unclear whether this problem belongs to the class NP, since there may not be an optimal solution that can be described in polynomial time. The following decision problem was shown to be NP complete by Fowler et al. [15]; here and throughout the paper an L square is a rectangle of size L Theta L, and the number of vertices of a polygonal region includes the vertices of all the holes it may have. Pack(k; L) Input: a polygonal region P with n vertices, a parameter k, a parameter L. Question: Can k many ....

....be considered for d(v; w) the most natural ones are L 2 distances and L 1 or L1 distances. In the following, we concentrate on rectilinear, i.e. L1 distances; most of the ideas carry over for L 2 distances by combining our ideas with the techniques by Hochbaum and Maass [23] and Fowler et al. [15]. Details are omitted from this abstract. We concentrate on the most interesting case of dispersion with boundaries, and only summarize the results for pure dispersion and dispersional packing; it is not hard to see that these variants are related via shrinking or expanding the region P in an ....

[Article contains additional citation context not shown here]

R. J. Fowler, M. S. Paterson, and S. L. Tanimoto. Optimal packing and covering in the plane are NP--complete. Information Processing Letters, 12, 1981, 133--137.

....with weighted edges, 2 and define the distance between two points as the weight of the corresponding edge. In this case, we will call the problem clustering on graph. This is the approach taken by the applications we describe in the next section. Most of the clustering problems are NP complete [13, 33], even when an approximate solution is sought [11, 17, 26] However, approximation algorithms exist for many of the geometric clustering problems, and for some clustering on graph problems, when the edge weights satisfy the triangle inequality. Unfortunately, the majority of applications we review ....

R. J. Fowler, M. S. Paterson, and S. L. Tanimoto, Optimal packing and covering in the plane are NP-complete, Inform. Process. Lett., 12 (1981), 133--137.

....Compute the corresponding complete label placement. The Label Number Maximisation Problem: Find a maximum subset of the features, and for each of these features a label from its set of candidates, such that no two labels overlap. The decision versions of both problems are NP hard in general [5, 4]. In recent years, especially the point labeling problem has achieved some attention in the algorithms community. Forman and Wagner proposed an approximation algorithm that maximises the size of uniform axis parallel square labels. Their algorithm can be applied to technical maps where labels are ....

Robert J. Fowler, Michael S. Paterson, and Steven L. Tanimoto. Optimal packing and covering in the plane are NP-complete. Inform. Process. Lett., 12(3):133-137, 1981.

....decomposed into k histogram polygons, even if the domain is rectilinear. Proof: We give a reduction from the problem Planar 3SAT, which has been shown to be NP complete by Lichtenstein [12] For another application of Planar 3SAT in showing NP hardness of a geometric problem, see Fowler et al. [11]. A 3SAT instance I with n literals and m clauses is considered to be planar if and only if the following bipartite graph G I is a planar graph: The nodes of G I are partitioned into a set corresponding to literals and a set corresponding to clauses; the node for literal x i is adjacent to the ....

R. J. Fowler, M. S. Paterson, and S. L. Tanimoto. Optimal packing and covering in the plane are NP-complete. Inform. Process. Lett., 12(3):133-- 137, 1981.

....2 This is a packing problem with the additional constraint that at most k items be packed. 3 The dual of the RPACK problem is to find a minimum cardinality subset of disjoint rectangles with total weight at least W . Since it is NP hard to find even a feasible solution for this problem (see [FPT81]) it cannot be approximated to any factor and hence is not considered any further. attributes. A histogram is an approximate compact statistic of the database contents and it is used for estimating result sizes of operators. Although other such estimators exist (such as sampling [LNS90] and ....

....the RPACK problem is NP hard. Theorem 4.1. The RPACK problem is NP hard in two dimensions, even in the case that each rectangle in the family intersects at most three other rectangles. Proof. The proof is a reduction from PLANAR 3SAT as in Theorem 2. 1, and is similar to the proof of Theorem 2 in [FPT81]. Our main result in this section is an O(log n) approximation for the RPACK problem. The algorithm uses a decomposition strategy that requires the input rectangles to have a certain geometric structure. We ensure the existence of such a structure by partitioning the input into many distinct ....

[Article contains additional citation context not shown here]

R. Fowler, M. Paterson and S. Tanimoto. Optimal packing and covering in the plane are NP-complete. Information Proc. Letters, 133-137, 1981.

....two 2 This is a packing problem with the additional constraint that at most k items be packed. 3 The dual of the RPACK problem is to nd a minimum cardinality subset of disjoint rectangles with total weight at least W . Since it is NP hard to nd even a feasible solution for this problem (see [FPT81]) it cannot be approximated to any factor and hence is not considered any further. attributes. A histogram is an approximate compact statistic of the database contents and it is used for estimating result sizes of operators. Although other such estimators exist (such as sampling [LNS90] and ....

....the RPACK problem is NP hard. Theorem 4.1. The RPACK problem is NP hard in two dimensions, even in the case that each rectangle in the family intersects at most three other rectangles. Proof. The proof is a reduction from PLANAR 3SAT as in Theorem 2. 1, and is similar to the proof of Theorem 2 in [FPT81]. Our main result in this section is an O(log n) approximation for the RPACK problem. The algorithm uses a decomposition strategy that requires the input rectangles to have a certain geometric structure. We ensure the existence of such a structure by partitioning the input into many distinct ....

R. Fowler, M. Paterson and S. Tanimoto. Optimal packing and covering in the plane are NP-complete. Information Proc. Letters, 133-137, 1981.

....two 2 This is a packing problem with the additional constraint that at most k items be packed. 3 The dual of the RPACK problem is to find a minimum cardinality subset of disjoint rectangles with total weight at least W . Since it is NP hard to find even a feasible solution for this problem (see [FPT81]) it cannot be approximated to any factor and hence is not considered any further. attributes. A histogram is an approximate compact statistic of the database contents and it is used for estimating result sizes of operators. Although other such estimators exist (such as sampling [LNS90] and ....

....the RPACK problem is NP hard. Theorem 4.1. The RPACK problem is NP hard in two dimensions, even in the case that each rectangle in the family intersects at most three other rectangles. Proof. The proof is a reduction from PLANAR 3SAT as in Theorem 2. 1, and is similar to the proof of Theorem 2 in [FPT81]. Our main result in this section is an O(log n) approximation for the RPACK problem. The algorithm uses a decomposition strategy that requires the input rectangles to have a certain geometric structure. We ensure the existence of such a structure by partitioning the input into many distinct ....

[Article contains additional citation context not shown here]

R. Fowler, M. Paterson and S. Tanimoto. Optimal packing and covering in the plane are NP-complete. Information Proc. Letters, 133-137, 1981.

....two 2 This is a packing problem with the additional constraint that at most k items be packed. 3 The dual of the RPACK problem is to find a minimum cardinality subset of disjoint rectangles with total weight at least W . Since it is NP hard to find even a feasible solution for this problem (see [FPT81]) it cannot be approximated to any factor and hence is not considered any further. attributes. A histogram is an approximate compact statistic of the database contents and it is used for estimating result sizes of operators. Although other such estimators exist (such as sampling [LNS90] and ....

....the RPACK problem is NP hard. Theorem 4.1. The RPACK problem is NP hard in two dimensions, even in the case that each rectangle in the family intersects at most three other rectangles. Proof. The proof is a reduction from PLANAR 3SAT as in Theorem 2. 1, and is similar to the proof of Theorem 2 in [FPT81]. Our main result in this section is an O(log n) approximation for the RPACK problem. The algorithm uses a decomposition strategy that requires the input rectangles to have a certain geometric structure. We ensure the existence of such a structure by partitioning the input into many distinct ....

[Article contains additional citation context not shown here]

R. Fowler, M. Paterson and S. Tanimoto. Optimal packing and covering in the plane are NP-complete. Information Proc. Letters, 133-137, 1981.

No context found.

R. Fowler, M. Paterson and S. Tanimoto, "Optimal packing and covering in the plane are NP complete", Inf. Proc. Letters, Vol. 12, No. 3, pp. 133-137, 1981.

No context found.

Robert J. Fowler, Michael S. Paterson, and Steven L. Tanimoto. Optimal packing and covering in the plane are NP-complete. Information Processing Letters, 12(3):133--137, 1981.

No context found.

Robert J. Fowler, Michael S. Paterson, and Steven L. Tanimoto. Optimal packing and covering in the plane are NP-complete. Information Processing Letters, 12(3):133--137, 1981.

No context found.

Fowler, R.F., Paterson, M.S., Tanimoto, S.L.: Optimal packing and covering in the plane are NP-complete. Information Processing Letters 12 (1981) 133--137

No context found.

R.J Fowler, M.S. Paterson and S.L. Tanimoto, \Optimal packing and covering in the plane are NP-complete", Information Processing Letters 12, 133-137, 1987.

No context found.

Fowler, R. J., Paterson R. M., Tanimoto S. T. (1981). Optimal packing and covering in the plane are NP-complete. Information Processing Letters 12, 133--137.

No context found.

R. J. Fowler, M. S. Paterson, and S. L. Tanimoto. Optimal packing and covering in the plane are NP-complete. Inform. Process. Lett., 12:133-137, 1981.

No context found.

R. J. Fowler, M. S. Paterson, and S. L. Tanimoto, Optimal packing and covering in the plane are NPcomplete, Inform. Process. Lett., 12 (1981), 133--137.

No context found.

R. Fowler, M. Paterson, and S. Tanimoto. Optimal packing and covering in the plane are np-complete. Information Proc. Letters, 12, 133--137, 1981.

No context found.

R. J. Fowler, M. S. Paterson, and L. Tanimoto. "Optimal packing and covering in the plane are NP-complete," Inform. Process. Lett., 12 (1981), 133--137.

No context found.

Robert J. Fowler, Michael S. Paterson, and Steven L. Tanimoto. Optimal packing and covering in the plane are NP-complete. Information Processing Letters, 12(3):133-137, 1981.

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