C.S. Mata and J. Mitchell. Approximation Algorithms for Geometric tour and network problems. In Proc. 11th ACM Symp. Comp. Geom., pp 360-369, 1995.  Home/Search   Document Details and Download   Summary   Related Articles   Check

....of Kapoor [2] for finding the shortest path in that domain is based on the Continuous Dijkstra paradigm. If we consider the problem in the polygonal domain with the extension that the path should meet several polygons, it is called TSP with neighborhoods problem which is NP hard and is studied in [3], but if we fix the order of meeting the target polygons, the problem seems to be less complex, yet we do not have any result for it. 3 ....

C. Mata and J. S. Mitchell. Approximation algorithms for geometric tour and network design problems. In Proc. 11th Annu. ACM Sympos. Comput. Geom., pages 360--369, 1995. 4

....tour of shortest length that visits all of the buyers neighborhoods and finally returns to his initial departure point. Both these problems are related to the problem known in the literature as the Traveling Salesperson problem with Neighborhoods (TSPN) and which has been extensively studied [2, 4, 6, 7, 8, 9]. The problem (TSPN) asks for the shortest tour that visits each of the neighborhoods. The problem was recently shown to be APX hard[7] Interesting generalizations of the TSPN problem arise when additional resources (k 1 robots in the sheet cutting problem, or k 1 salespersons in the second ....

C. Mata and J. S. B. Mitchell. Approximation algorithms for geometric tour and network design problems. Proc. 11th Annual ACM Symposium on Computational Geometry, pages 360--369, 1995.

....customers and the service radius represents the distance a customer is willing to travel to meet the salesperson. The goal is to find a minimum length salesperson tour so that all the (customer) nodes are strictly serviced. Restrictions of the problems to geometric instances were considered in [1, 17, 24]. These problems are similar to BCCMED studied here, the primary difference being in the way the location theoretic constraint is formulated. In the current paper, we put a budget on the sum of the costs of the nodes not in the tree while the papers referred to above consider a budget on the ....

C. Mata and J. B. Mitchell. Approximation algorithms for geometric tour and network design problems. In Proceedings of the 11th Annual Symposium on Computational Geometry, pages 360--369. ACM Press, June 1995.

....relative to the salesman s. The salesman wants to find a set p, p) of market places and a tour visiting them that minimizes this cost. The usual Euclidean TSP is the special case where each region is a single point, and so the TSBP is NP hard. The Euclidean TSP with neighborhoods (TSPN) [1, 6, 3] is the special case where 0: the cost of a tour is simply the length of the tour itself, regardless of the maximum travel distances. The TSPN in the plane has been studied recently by Dumitrescu and Mitchell [3] who presented a PTAS for the case of disjoint unit disk neighborhoods, and a ....

Cristian Mata and Joseph S. B. Mitchell. Approximation algorithms for geometric tour and network design problems. In Proc. 11th Annu. ACM Sympos. Cornput. Geom., pages 360-369, 1995.

.... Specifically, geometric divide and conquer appears Karp s dissection heuristic [38] Smith s O( # n) time exact algorithm for TSP [59] and Blum, Chalasani and Vempala s approximation algorithm for k MST [19] and Mata and Mitchell s constantfactor approximations for many geometric problems [47]. Surprisingly, the proofs of the structure theorems for geometric problems are elementary and this survey will describe them essentially completely. We also survey a more recent result of Rao and Smith [55] that improves the running time for some problems. We will be concerned only with ....

....a 2 , an , C , can decide if i # a i C . Arora s paper gave the first PTASs for many of these problems in 1996. A few months later Mitchell independently discovered a similar n time approximation scheme [50] this algorithm used ideas from the earlier paper of Mata and Mitchell [47]. The running time of Arora s and Mitchell s algorithms was n , but Arora later improved the running time of his algorithm to n(log n) Mitchell s algorithm seems to work only in the plane whereas Arora s PTAS works for any constant number of dimensions. If dimension d is not constant but ....

[Article contains additional citation context not shown here]

C.S. Mata and J. Mitchell. Approximation Algorithms for Geometric tour and network problems. In Proc. 11th ACM Symp. Comp. Geom., pp 360-369, 1995.

.... is a hierarchical partitioning of space widely used in several areas, including computer graphics (global illumination [7] shadow generation [10, 11] visibility determination [4, 28] and ray tracing [21] solid modeling [22, 20, 29] geometric data repair [19] robotics [5] network design [18], and surface simplification [3] Key to Support was provided by National Science Foundation research grant CCR 93 01259, by Army Research Office MURI grant DAAH04 96 1 0013, by a Sloan fellowship, by a National Science Foundation NYI award and matching funds from Xerox Corp, and by a grant ....

C. Mata and J. S. Mitchell, Approximation algorithms for geometric tour and network design problems, Proc. 11th Annu. ACM Sympos. Comput. Geom., 1995, pp. 360--369.

....vertex of Q is a bridge edge and a bridge line is formed by extending a bridge edge as far as possible without intersecting P or Q. It turns out that the extreme vertices of the shortest non wrapping watchman route must lie on legs or on bridge lines. This is stated in the following lemma. Lemma 11 The extreme vertices of any shortest non wrapping watchman route must lie on legs or on bridge lines. In the shortest non wrapping watchman routes shown in Figures 11b, 11c, and 11e, all extreme vertices lie on legs; but in the one shown in Figure 11d, one of the extreme vertices (e 1 ) lies on a ....

....shown in Figures 11b, 11c, and 11e, all extreme vertices lie on legs; but in the one shown in Figure 11d, one of the extreme vertices (e 1 ) lies on a bridge line. Let U(q i ; p j ) denote the shortest non wrapping watchman route that spans the wedges of q i and p j . Observation 6 and Lemma 11 suggest the following direct approach for computing the shortest non wrapping watchman route: compute U(q i ; p j ) for all pairs of matching wedges and the shortest among them is the shortest non wrapping watchman route. Since O(n) time may be required to compute each U(q i ; p j ) and since ....

[Article contains additional citation context not shown here]

Cristian S. Mata and Joseph S. B. Mitchell, "Approximation Algorithms for Geometric Tours and Network Design Problems", 11th Annual Symposium on Computational Geometry, 1995.

....to the salesman s. The salesman wants to nd a set fp 1 ; pk g of market places and a tour visiting them that minimizes this cost. The usual Euclidean TSP is the special case where each region is a single point, and so the TSBP is NP hard. The Euclidean TSP with neighborhoods (TSPN) [1, 6, 3] is the special case where = 0: the cost of a tour is simply the length of the tour itself, regardless of the maximum travel distances. The TSPN in the plane has been studied recently by Dumitrescu and Mitchell [3] who presented a PTAS for the case of disjoint unit disk neighborhoods, and a ....

Cristian Mata and Joseph S. B. Mitchell. Approximation algorithms for geometric tour and network design problems. In Proc. 11th Annu. ACM Sympos. Comput. Geom., pages 360-369, 1995.

....of Zelikovsky[64] has provided better algorithms, with an approximation ratio around 1.143 [65] The metric case is MAX SNP hard [13] k TSP: Given n nodes in # d and an integer k 1, find the shortest tour that visits at least k nodes. An approximation algorithm due to Mata and Mitchell [46] K # 8) is PLS complete [41] This strongly suggests that no polynomial time algorithm can find such a local optimum; see [35] Many variants of Lin Kernighan are also PLS complete [51] 2 It appears that this problem was first posed by Gauss in a letter to Schumacher [29] Approximation ....

C.S. Mata and J. Mitchell. Approximation Algorithms for Geometric tour and network problems. In Proc. 11th ACM Symp. Comp. Geom., pp 360-369, 1995.

....the subdivision based on it. Extending the algorithm to the case in which we want to see multiple points from the path is not easy since it introduces multiple regions (neighborhoods) which must be touched by the shortest path. The problem is similar to TSP with neighborhoods which is studied in [4]. Since the problem is NP hard, finding an approximation algorithm seems to be the most feasible approach for solving it. ....

C. S. Mata and J. S. B. Mitchell, Approximation Algorithms for Geometric Tour and Network Design Problems, In Proc. 11th Annu. Sympos. Comput. Geom., pp. 360-369, 1995.

....[9, 14] Arkin and Hassin [1] gave an O(1) approximation algorithm for the special case in which the neighborhoods all have diameter segments that are parallel to a common direction, and the ratio between the longest and the shortest diameter is bounded by a constant. Recently, Mata and Mitchell [11] provided a general framework that gives an O(log k) approximation algorithm for the general case, when no start point is given, with polynomial time complexity (n 5 ) in the worst case. In this paper we give several results: First we show a simple and practical algorithm that produces a TSPN ....

....at least one of the following two tasks (depending on the instance) 1) It outputs in time O(n log n) a TSPN tour of length O(log k) times the optimum. 2) It outputs a TSPN tour of length less than (1 ) times the optimum in cubic time. The rst part of our method builds upon the idea in [11], in that our logarithmic approximation algorithm produces a guillotine subdivision. However, we produce a quite di erent guillotine partition (partly inspired from [7, 10] and show that it has some nice sparseness properties, which guarantee the O(log k) approximation bound. The method ....

[Article contains additional citation context not shown here]

C. S. Mata and J. S. B. Mitchell. Approximation algorithms for geometric tour and network design problems. In SoCG'95, pages 360-369, 1995.

....that prescribes a budget on the service distance for each node not in the tree. The goal is to find a minimum length salesperson tour (or a tree as may be the case) so that all the (customer) nodes are strictly serviced. Restrictions of the problems to geometric instances were considered in [1,12,19]. Finally, the problem BCCMED can be seen as a generalization of the classical k Median Problem, where we require the set of medians to be connected. 4. HARDNESS RESULTS Theorem 3. The problem BCCMED is weakly NP hard even on trees. This result continues to hold even if we require the two cost ....

C. Mata and J. B. Mitchell, Approximation algorithms for geometric tour and network design problems, Proceedings of the 11th Annual Symposium on Computational Geometry, ACM Press, June 1995, pp. 360--369.

....time. For the cases k = 3; 4 in 2 , a constant factor approximation algorithm is given by Khuller, Raghavachari, and Young [27] k TSP: Given n nodes in d and an integer k 1, find the smallest tour that visits at least k nodes. An approximation algorithm due to Mata and Mitchell [32] achieves a constant factor approximation in 2 . k MST: Given n nodes in d and an integer k 2, find k nodes with the shortest MST. Blum, Chalasani, and Vempala [7] gave the first O(1) factor approximation algorithm for points in 2 ; there has been much other work before and since. ....

C.S. Mata and J. Mitchell. Approximation Algorithms for Geometric tour and network problems. In Proc. 11th ACM Symp. Comp. Geom., pp 360-369, 1995.

.... is a hierarchical partitioning of space widely used in several areas, including computer graphics (global illumination [9] shadow generation [12, 13] visibility determination [4, 33] and ray tracing [24] solid modeling [27, 25, 34] geometric data repair [23] robotics [5] network design [21], and surface simpli cation [3] Key to the BSP s success is that it serves both as a model for an object (or a set of objects) and as a data structure for querying the object. Informally, a BSP B for a set of objects is a binary tree, where each node v is associated with a convex region v . ....

C. Mata and J. S. B. Mitchell, Approximation algorithms for geometric tour and network design problems, Proc. 11th Annu. ACM Sympos. Comput. Geom., 1995, pp. 360{ 369.

....[9, 14] Arkin and Hassin [1] gave an O(1) approximation algorithm for the special case in which the neighborhoods all have diameter segments that are parallel to a common direction, and the ratio between the longest and the shortest diameter is bounded by a constant. Recently, Mata and Mitchell [11] provided a general framework that gives an O(log k) approximation algorithm for the general case, when no start point is given, with polynomial time complexity# (n 5 ) in the worst case. In this paper we give several results: First we show a simple and practical algorithm that produces a TSPN ....

....at least one of the following two tasks (depending on the instance) 1) It outputs in time O(n log n) a TSPN tour of length O(log k) times the optimum. 2) It outputs a TSPN tour of length less than (1 #) times the optimum in cubic time. The first part of our method builds upon the idea in [11], in that our logarithmic approximation algorithm produces a guillotine subdivision. However, we produce a quite di#erent guillotine partition (partly inspired from [7, 10] and show that it has some nice sparseness properties, which guarantee the O(log k) approximation bound. The method ....

[Article contains additional citation context not shown here]

C. S. Mata and J. S. B. Mitchell. Approximation algorithms for geometric tour and network design problems. In SoCG'95, pages 360--369, 1995.

.... BSPs have subsequently proven to be versatile, with applications in many problems apart from hiddensurface removal [8, 101] global illumination [23] shadow generation [29, 30] solid modelling [75, 78, 102] ray tracing [74] robotics [12] and approximation algorithms for network design [66] and surface simplification [7] The efficiency of our model repair and hidden surface removal algorithms algorithms as well as the applications mentioned above inherently depends on the size and height of the BSP. In Chapters 4 6, we describe our algorithms for constructing BSPs of small size. ....

.... the same is true of algorithms that use the BSP in a variety of different applications: hidden surface removal itself [8, 101] global illumination [23] shadow generation [29, 30] solid modelling [75, 78, 102] ray tracing [74] robotics [12] and approximation algorithms for network design [66] and surface simplification [7] As a result, there has been a lot of effort to construct BSPs of small size. While several simple heuristics have been developed for constructing BSPs of reasonable sizes [8, 24, 47, 73, 101, 102] provable bounds were first obtained by Paterson and Yao. They show ....

C. Mata and J. S. B. Mitchell, Approximation algorithms for geometric tour and network design problems, Proc. 11th Annu. ACM Sympos. Comput. Geom., 1995, pp. 360--369.

....an easy lower bound for the area, which is met i no new grid points are encountered. As a consequence, we get a close connection to problems arising from the separation of point sets by means of polygonal curves have been considered by Mitchell and Suri [14] Mitchell [12] Mata and Mitchell [13], and Aggarwal and Suri [1] When dealing with structures of small area, one encounters speci c diculties. Most notably, edges in a polygon with small area need not be short. This makes it dicult to restrict potential neighbors of a point in a good polygonalization, inhibiting local search methods ....

C. Mata and J. S. B. Mitchell. Approximation algorithms for geometric tour and network design problems. Proc. Eleventh Annual ACM Sympos. on Computational Geometry, 1995, 360-369.

....of Zelikovsky[63] has provided better algorithms, with an approximation ratio around 1.143 [64] The metric case is MAX SNP hard [13] k TSP: Given n nodes in # d and an integer k 1, find the shortest tour that visits at least k nodes. An approximation algorithm due to Mata and Mitchell [45] achieves a constant factor approximation in # 2 . k MST: Given n nodes in # d and an integer k # 2, find k nodes with the shortest Minimum Spanning Tree. The problem is NP hard [22] Blum, Chalasani, and Vempala [14] gave the first O(1) factor approximation algorithm for points in # 2 ....

C.S. Mata and J. Mitchell. Approximation Algorithms for Geometric tour and network problems. In Proc. 11th ACM Symp. Comp. Geom., pp 360-369, 1995.

No context found.

C. Mata and J. S. B. Mitchell, Approximation algorithms for geometric tour and network design problems, Proc. 11th Annu. ACM Sympos. Comput. Geom. (1995), pp. 360--369.

.... [9, 10] This paper represents a continuation of our previous work on guillotine subdivisions ( 9, 10, 11] which in turn is based on the concept of division trees introduced by Blum, Chalasani, and Vempala [3, 11] and the guillotine rectangular subdivision methods of Mata and Mitchell [8]. Here, we obtain substantially better bounds than before, and we generalize our previous results to apply to a much broader class of problem instances that include those weighted graphs whose edge lengths correspond to shortest path lengths among obstacles in the plane. 2 Grid Rounded ....

C. Mata and J. S. B. Mitchell, Approximation algorithms for geometric tour and network design problems, Proc. 11th Annu. ACM Sympos. Comput. Geom. (1995), pp. 360--369.

.... the last ten years; see [3, 4, 5, 11, 12, 13] In the case of guarding a polygonal domain (with holes ) the Euclidean version of the watchman route problem is NP hard (from Euclidean TSP) 4] and there is an O(log n) approximation algorithm for a rectilinear version with restricted visibility [9]. In the case of guarding a simple polygon (no holes) Chin and Ntafos [5] gave an algorithm claimed to have time complexity O(n ) for the fixed watchman route through a specified starting point; their algorithm, as well as some of its successors, were later found to be flawed ( 8] and new ....

C. Mata and J. S. B. Mitchell. Approximation algorithms for geometric tour and network design problems. In Proc. 11th Annu. ACM Sympos. Comput. Geom., pages 360--369, 1995.

....translates of a connected region, and more generally, for regions which have diameter segments that are parallel to a common direction, and the ratio between the longest and the shortest diameter is bounded by a constant. For the general case of connected polygonal regions, Mata and Mitchell [11] obtained an O(log n) approximation algorithm, based on guillotine rectangular subdivisions , with time bound O(m 5 ) where m is the total complexity of the n regions. Gudmundson and Levcopoulos [7] have recently obtained a faster method, which, for any xed 0, is guaranteed to perform at ....

C. Mata and J. S. B. Mitchell. Approximation algorithms for geometric tour and network design problems. In Proc. 11th Annu. ACM Sympos. Comput. Geom., pages 360-369, 1995.

No context found.

C.S. Mata and J. Mitchell. Approximation Algorithms for Geometric tour and network problems. In Proc. 11th ACM Symp. Comp. Geom., pp 360-369, 1995.

No context found.

C. Mata and J. B. Mitchell. Approximation algorithms for geometric tour and network design problems. In Proceedings of the 11th Annual Symposium on Computational Geometry, pages 360--369, New York, NY, USA, June 1995. ACM Press.

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C. Mata and J. S. Mitchell. Approximation algorithms for geometric tour and network design problems. In Proc. 11th Annu. ACM Sympos. Comput. Geom. (1995) 360--369.

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