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In the strongly connected spanning subgraph () problem, the goal is to find a minimum weight spanning subgraph of a strongly connected directed graph that maintains the strong connectivity. In this paper, we consider the problem for two families of geometric directed graphs; -spanners and symmetric disk graphs. Given a constant , a directed graph is a -spanner of a set of points if, for every two points and in , there exists a directed path from to in of length at most , where is the Euclidean distance between and . Given a set of points in the plane such that each point has a radius , the symmetric disk graph of is a directed graph , such that . Thus, if there exists a directed edge , then exists as well. We present and approximation algorithms for the problem for -spanners and for symmetric disk graphs, respectively. Actually, our approach achieves a -approximation algorithm for all directed graphs satisfying the property that, for every two nodes and , the ratio between the shortest paths, from to and from to in the graph, is at most . [ABSTRACT FROM AUTHOR]

In this paper, we consider constant factor approximation algorithms for a variant of the discrete piercing set problem for unit disks. Here a set of points P is given; the objective is to choose minimum number of points in P to pierce the unit disks centered at all the points in P. We first propose a very simple algorithm that produces 12-approximation result in O(n log n) time. Next, we improve the approximation factor to 4 and then to 3. The worst case running time of these algorithms are O(n8 log n) and O(n15 log n) respectively. Apart from the space required for storing the input, the extra work-space requirement for each of these algorithms is O(1). Finally, we propose a PTAS for the same problem. Given a positive integer k, it can produce a solution with performance ratio (1+1/k)2 in nO(k) time. [ABSTRACT FROM AUTHOR]

Given a set P of n points and a set D of m unit disks on a 2-dimensional plane, the discrete unit disk cover (DUDC) problem is (i) to check whether each point in P is covered by at least one disk in D or not and (ii) if so, then find a minimum cardinality subset D* C D such that the unit disks in D* cover all the points in P. The discrete unit disk cover problem is a geometric version of the general set cover problem which is NP-hard. The general set cover problem is not approximable within c log 1|, for some constant c, but the DUDC problem was shown to admit a constant factor approximation. In this paper, we provide an algorithm with constant approximation factor 18. The running time of the proposed algorithm is O(n logn + mlogm + mn). The previous best known tractable solution for the same problem was a 22-factor approximation algorithm with running time O(m²n4). [ABSTRACT FROM AUTHOR]

The complexity status of the minimum r-star cover problem for orthogonal polygons had been open for many years, until 2004 when Ch. Worman and J. M. Keil proved it to be polynomially tractable (Polygon decomposition and the orthogonal art gallery problem, IJCGA 17(2) (2007), 105-138). However, since their algorithm has Õ(n17)-time complexity, where Õ(·) hides a polylogarithmic factor, and thus it is not practical, in this paper we present a linear-time 3-approximation algorithm. Our approach is based upon the novel partition of an orthogonal polygon into so-called o-star-shaped orthogonal polygons. [ABSTRACT FROM AUTHOR]

Given a set of points in the Euclidean plane, the Euclidean δ -minimum spanning tree (δ -MST) problem is the problem of finding a spanning tree with maximum degree no more than δ for the set of points such the sum of the total length of its edges is minimum. Similarly, the Euclidean δ -minimum bottleneck spanning tree (δ -MBST) problem, is the problem of finding a degree-bounded spanning tree for a set of points in the plane such that the length of the longest edge is minimum. When δ ≤ 4 , these two problems may yield disjoint sets of optimal solutions for the same set of points. In this paper, we perform computational experiments to compare the accuracies of a variety of heuristic and approximation algorithms for both these problems. We develop heuristics for these problems and compare them with existing algorithms. We also describe a new type of edge swap algorithm for these problems that outperforms all the algorithms we tested. [ABSTRACT FROM AUTHOR]

This paper considers a problem of locating the given number of disks into a container so that the area covered by the disks is maximized. In the problem, the radii of the disks can be changed arbitrarily unless they overlap outside of the container, and the disks are allowed to overlap with each other. We present an approximation algorithm for this problem assuming that the container is a convex polygon. Our algorithm achieves approximation ratio (0.78 - ϵ) for any small ϵ > 0. Since the computation time of our algorithm depends on the number of corners of the convex polygon exponentially, we also give a heuristic to reduce the number of corners. [ABSTRACT FROM AUTHOR]

We consider a geometric optimization problem that arises in sensor network design. Given a polygon P (possibly with holes) with n vertices, a set Y of m points representing sensors, and an integer k, 1 ≤ k ≤ m. The goal is to assign a sensing range, ri, to each of the sensors yi ∈ Y, such that each point p ∈ P is covered by at least k sensors, and the cost, ${\Sigma _i}\,r_i^\alpha $, of the assignment is minimized, where α is a constant. In this paper, we assume that α = 2, that is, find a set of disks centered at points of Y, such that (i) each point in P is covered by at least k disks, and (ii) the sum of the areas of the disks is minimized. We present, for any constant k ≥ 1, a polynomial-time c1-approximation algorithm for this problem, where c1 = c1(k) is a constant. The discrete version, where one has to cover a given set of n points, X, by disks centered at points of Y, arises as a subproblem. We present a polynomial-time c2-approximation algorithm for this problem, where c2 = c2(k) is a constant. [ABSTRACT FROM AUTHOR]

Given a set T of n points in ℝ2, a Manhattan network on T is a graph G = (V,E) with the property that all the edges in E are vertical or horizontal line segments connecting points in V ⊇ T and for all p, q ∈ T, the graph contains a path having the length exactly L1 distance between p and q. The Minimum Manhattan Network problem is to find a Manhattan network of minimum length, i.e. minimizing the total length of the line segments of the network. In this paper we present a 2-approximation algorithm with time complexity O(n log n), which improves over a recent combinatorial 2-approximation algorithm with running time O(n2). Moreover, compared with other 2-approximation algorithms using linear programming or dynamic programming techniques, we show that a greedy strategy suffices to obtain a 2-approximation algorithm. [ABSTRACT FROM AUTHOR]

Geometric spanner is a fundamental structure in computational geometry and plays an important role in many geometric networks design applications. In this paper, we consider a generalization of the classical geometric spanner problem (called segment spanner): Given a set S of n disjoint 2-D segments, find a spanning network GS with minimum size so that for any pair of points in S, there exists a path in GS with length no more than t times their Euclidean distance. Based on a number of interesting techniques (such as weakly dominating set, strongly dominating set, interval cover, and imaginary Steiner points), we present an efficient algorithm to construct the segment spanner. Our approach first identifies a set Q of Steiner points in S and then constructs a point spanner for the set of Steiner points. Our algorithm runs in O(|Q| + n2log n) time and Q is a constant approximation (in terms of its size) of the optimal solution when S has a constant relative separation ratio. The approximation ratio depends on the stretch factor t and the relative separation ratio of S. [ABSTRACT FROM AUTHOR]

In this paper we develop an efficient implementation for a k-means clustering algorithm. The algorithm is based on a combination of Lloyd's algorithm with random swapping of centers to avoid local minima. This approach was proposed by Mount 30. The novel feature of our algorithms is the use of coresets to speed up the algorithm. A coreset is a small weighted set of points that approximates the original point set with respect to the considered problem. We use a coreset construction described in 12. Our algorithm first computes a solution on a very small coreset. Then in each iteration the previous solution is used as a starting solution on a refined, i.e. larger, coreset. To evaluate the performance of our algorithm we compare it with algorithm KMHybrid 30 on typical 3D data sets for an image compression application and on artificially created instances. Our data sets consist of 300,000 to 4.9 million points. Our algorithm outperforms KMHybrid on most of these input instances. Additionally, the quality of the solutions computed by our algorithm deviates significantly less than that of KMHybrid. We conclude that the use of coresets has two effects. First, it can speed up algorithms significantly. Secondly, in variants of Lloyd's algorithm, it reduces the dependency on the starting solution and thus makes the algorithm more stable. Finally, we propose the use of coresets as a heuristic to approximate the average silhouette coefficient of clusterings. The average silhouette coefficient is a measure for the quality of a clustering that is independent of the number of clusters k. Hence, it can be used to compare the quality of clusterings for different sizes of k. To show the applicability of our approach we computed clusterings and approximate average silhouette coefficient for k = 1,..., 100 for our input instances and discuss the performance of our algorithm in detail. [ABSTRACT FROM AUTHOR]

Given a set S of n points, a weight function w to associate a non-negative weight to each point in S , a positive integer k ≥ 1 , and a real number > 0 , we present algorithms for computing a spanner network G (S , E) for the metric space (S , d w) induced by the weighted points in S. The weighted distance function d w on the set S of points is defined as follows: for any p , q ∈ S , d w (p , q) is equal to w (p) + d π (p , q) + w (q) if p ≠ q , otherwise, d w (p , q) is 0. Here, d π (p , q) is the Euclidean distance between p and q if points in S are in ℝ d , otherwise, it is the geodesic (Euclidean) distance between p and q. The following are our results: (1) When the weighted points in S are located in ℝ d , we compute a k -vertex fault-tolerant (4 +) -spanner network of size O (k n 2 ). (2) When the weighted points in S are located in the relative interior of the free space of a polygonal domain , we detail an algorithm to compute a k -vertex fault-tolerant (4 +) -spanner network with O (k n h + 1 2 lg n) edges. Here, h is the number of simple polygonal holes in . (3) When the weighted points in S are located on a polyhedral terrain , we propose an algorithm to compute a k -vertex fault-tolerant (4 +) -spanner network, and the number of edges in this network is O (k n 2 lg n). [ABSTRACT FROM AUTHOR]

In this paper, we consider the k-center problem for streaming points in ℝ d. More precisely, we consider the single-pass streaming model, where each point in the stream is allowed to be examined only once and a small amount of information can be stored in a device. Since the size of memory is much smaller than the size of the data in the streaming model, it is important to develop an algorithm whose space complexity does not depend on the number of input data. We present an approximation algorithm for k = 2 that guarantees a (2 + ε)-factor using O(d/ε) space and update time in arbitrary dimensions for any metric. We show that our algorithm can be extended to approximate an optimal k-center within factor (2 + ε) for k > 2. [ABSTRACT FROM AUTHOR]

Given a simple (cuttable) polygon Q drawn on a piece of planar material R, we cut Q out of R by a (small) circular saw with a total number of cuts no more than twice the optimal. This improves the previous approximation ratio of 2.5 obtained by Demaine et al in 2001. [ABSTRACT FROM AUTHOR]

Data clustering is a fundamental problem arising in many practical applications. In this paper, we present new geometric approximation and exact algorithms for the density-based data clustering problem in d-dimensional space ℝd (for any constant integer d ≥ 2). Previously known algorithms for this problem are efficient only when the specified range around each input point, called the δ-neighborhood, contains on average a constant number of input points. Different distributions of the input data points have significant impact on the efficiency of these algorithms. In the worst case when the data points are highly clustered, these algorithms run in quadratic time, although such situations might not occur very frequently on real data. By using computational geometry techniques, we develop faster approximation and exact algorithms for the density-based data clustering problem in ℝd. In particular, our approximation algorithm based on the ∊-fuzzy distance function takes O(n log n) time for any given fixed value ∊>0, and our exact algorithms take sub-quadratic time. The running times and output quality of our algorithms do not depend on any particular data distribution. We believe that our fast approximation algorithm is of considerable practical importance, while our sub-quadratic exact algorithms are more of theoretical interest. We implemented our approximation algorithm and the experimental results show that our approximation algorithm is efficient on arbitrary input point sets. [ABSTRACT FROM AUTHOR]

In this paper, we study a 3D geometric problem originated from computing neural maps in the computational biology community: Given a set S of n points in 3D, compute k cylindrical segments (with different radii, orientations, lengths and no segment penetrates another) enclosing S such that the sum of their radii is minimized. There is no known result in this direction except when k=1. The general problem is strongly NP-hard and we obtain a polynomial time approximation scheme (PTAS) for any fixed k>1 in O(n3k-2/δ4k-3) time by returning k cylindrical segments with sum of radii at most (1+δ) of the corresponding optimal value, if exist. Our PTAS is built upon a simple (though slower) approximation algorithm for the case when k=1. [ABSTRACT FROM AUTHOR]

Data clustering is an important theoretical topic and a sharp tool for various applications. It is a task frequently arising in geometric computing. The main objective of data clustering is to partition a given data set into clusters such that the data items within the same cluster are "more" similar to each other with respect to certain measures. In this paper, we study the pairwise data clustering problem with pairwise similarity/dissimilarity measures that need not satisfy the triangle inequality. By using a criterion, called the minimum normalized cut, we model the general pairwise data clustering problem as a graph partition problem. The graph partition problem based on minimizing the normalized cut is known to be NP-hard. For an undirected weighted graph of n vertices, we present a ((4+o(1))Inn)-approximation polynomial time algorithm for the minimum normalized cut problem; this is the first provably good approximation polynomial time algorithm for the problem. We also give a more efficient algorithm for this problem by sacrificing the approximation ratio slightly. Further, our scheme achieves a ((2+o(1))Inn)-approximation polynomial time algorithm for computing the sparsest cuts in edge-weighted and vertex-weighted undirected graphs, improving the previously best known approximation ratio by a constant factor. Some applications and implementation work of our approximation normalized cut algorithms are also discussed. [ABSTRACT FROM AUTHOR]

We investigate the problem of computing a minimal-volume container for the non-overlapping packing of a given set of three-dimensional convex objects. Already the simplest versions of the problem are 𝒩𝒫-hard so that we cannot expect to find polynomial time algorithms to determine the exact solution. We give constant ratio approximation algorithms for packing axis-parallel (rectangular) cuboids under translation into an axis-parallel (rectangular) cuboid as container, for packing cuboids under rigid motions into an axis-parallel cuboid or into an arbitrary convex container, and for packing convex polyhedra under rigid motions into an axis-parallel cuboid or arbitrary convex container. This work gives the first approximability results for the computation of minimum volume containers for the objects described. [ABSTRACT FROM AUTHOR]

Numerous approximation algorithms for problems on unit disk graphs have been proposed in the literature, exhibiting a sharp trade-off between running times and approximation ratios. We introduce a variation of the known shifting strategy that allows us to obtain linear-time constant-factor approximation algorithms for such problems. To illustrate the applicability of the proposed variation, we obtain results for three well-known optimization problems. Among such results, the proposed method yields linear-time -approximations for the maximum-weight independent set and the minimum dominating set of unit disk graphs, thus bringing significant performance improvements when compared to previous algorithms that achieve the same approximation ratios. Finally, we use axis-aligned rectangles to illustrate that the same method may be used to derive linear-time approximations for problems on other geometric intersection graph classes. [ABSTRACT FROM AUTHOR]

Motivated by a crane assignment problem, we consider a Euclidean bipartite matching problem with edge-crossing constraints. Specifically, given red points and blue points in the plane, we want to construct a perfect matching between red and blue points that minimizes the length of the longest edge, while imposing a constraint that no two edges may cross each other. We show that the problem cannot be approximately solved within a factor less than 1:277 in polynomial time unless . We give simple dynamic programming algorithms that solve our problem in two special cases, namely (1) the case where the red and blue points form the vertices of a convex polygon and (2) the case where the red points are collinear and the blue points lie to one side of the line through the red points. [ABSTRACT FROM AUTHOR]

Let P be a d-dimensional n-point set. A partition of P is called a Tverberg partition if the convex hulls of all sets in intersect in at least one point. We say that is t- tolerant if it remains a Tverberg partition after deleting any t points from P. Soberón and Strausz proved that there is always a t-tolerant Tverberg partition with ⌈n/(d + 1)(t + 1)⌉ sets. However, no nontrivial algorithms for computing or approximating such partitions have been presented so far. For d ≤ 2, we show that the Soberón-Strausz bound can be improved, and we show how the corresponding partitions can be found in polynomial time. For d ≥ 3, we give the first polynomial-time approximation algorithm by presenting a reduction to the regular Tverberg problem (with no tolerance). Finally, we show that it is coNP-complete to determine whether a given Tverberg partition is t-tolerant. [ABSTRACT FROM AUTHOR]

Let A and B be two sets of n resp. m disjoint unit disks in the plane, with m ≥ n. We consider the problem of finding a translation or rigid motion of A that maximizes the total area of overlap with B. The function describing the area of overlap is quite complex, even for combinatorially equivalent translations and, hence, we turn our attention to approximation algorithms. We give deterministic (1 - ∊)-approximation algorithms for translations and for rigid motions, which run in O((nm/∊2) log(m/∊)) and O((n2m2/∊3) log m)) time, respectively. For rigid motions, we can also compute a (1 - ∊)-approximation in O((m2n4/3Δ1/3/∊3) log n log m) time, where Δ is the diameter of set A. Under the condition that the maximum area of overlap is at least a constant fraction of the area of A, we give a probabilistic (1 - ∊)-approximation algorithm for rigid motions that runs in O((m2/∊4) log2(m/∊) log m) time and succeeds with high probability. Our results generalize to the case where A and B consist of possibly intersecting disks of different radii, provided that (i) the ratio of the radii of any two disks in A ∪ B is bounded, and (ii) within each set, the maximum number of disks with a non-empty intersection is bounded. [ABSTRACT FROM AUTHOR]

In the Euclidean traveling salesman problem with discrete neighborhoods, we are given a set of points P in the plane and a set of n connected regions (neighborhoods), each containing at least one point of P. We seek to find a tour of minimum length which visits at least one point in each region. We give (i) an O(α)-approximation algorithm for the case when the regions are disjoint and α-fat, with possibly varying size; (ii) an O(α3)-approximation algorithm for intersecting α-fat regions with comparable diameters. These results also apply to the case with continuous neighborhoods, where the sought TSP tour can hit each region at any point. We also give (iii) a simple O(log n)-approximation algorithm for continuous non-fat neighborhoods. The most distinguishing features of these algorithms are their simplicity and low running-time complexities. [ABSTRACT FROM AUTHOR]