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.

....through each polygon. Obviously, the usual TSP is the special case in which each polygon is simply a point. Arkin and Hassin [AH94b] gave an O(1) approximation algorithm for the case of round, approximately equal sized neighborhoods, such as unit disks. In more recent work, Mata and Mitchell [MM95] provide a general framework that gives an O(logk) approximation algorithm for this problem. The same framework yields logarithmic approximation ratios for three other problems that generalize the TSP: 1) the prize collecting TSP, in which we seek a shortest tour visiting k out of n points (see ....

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

....can be solved in polynomial time [42, 58] but the k MST problem is NP complete [97, 116] as are obviously the k TSP and k Steiner tree variants) so one must resort to some form of approximation. In a sequence of many papers, the approximation ratio was reduces to O(k 1 4 ) 97] O(log k) [64, 87], O(log k log log n) 54] O(1) 21] 2 # 2 [89] and, very recently, 1 # (Arora, personal communication) We describe the 2 # 2 approximation algorithm, but the other results use similar methods. For related work on non geometric k MST problems see [13, 22, 30, 97, 116] Mitchell [89] first ....

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

....H is a Hamiltonian path in G 0 ; since every vertex in H has at most in degree 1 and at most out degree 1, H must be a Hamiltonian path in D 0 . 2 4 Point Separation Several authors have considered the question of separating polyhedra, polygons, and point sets by optimal polyhedral surfaces [2, 33, 34, 35]. It is relatively simple to show NP hardness of finding a simple polygon of small perimeter length that separates a blue set of points in the plane from a second red set of points by giving a straightforward reduction of the geometric TSP (see [17] On the other hand, it has been an open ....

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

....n) 6] 5.2 Hardness results for Generalizations We show two simple reductions, that demonstrate that other generalizations of the CDS problem may be as hard to approximate as the set TSP problem for which no approximation algorithms are known. For the Euclidean case, Mata and Mitchell [14] have given approximation algorithms for this problem. c j V j Figure 3: Reduction of set TSP problem to edge weighted CDS Theorem 5.1 A polynomial approximation algorithm for the edge weighted connected dominating set problem with factor f(n) would imply a polynomial approximation algorithm for ....

C. S. Mata and J. S. B. Mitchell "Approximation algorithms for geometric tour and network design problems", Proc. of the 11th Annual Symp. on Computational Geometry, pages 360--369, (1995).

.... subsequently proven to be versatile, with applications in many other problems global illumination [6] shadow generation [10, 11] visibility problems [4, 30] solid modeling [23, 25, 31] geometric data repair [20] ray tracing [22] robotics [5] and approximation algorithms for network design [19] and surface simplification [3] Algorithms have also been developed to construct BSPs for moving objects [1, 12, 24, 32] Informally, a BSP B for a set of objects is a binary tree. Each node v of B is associated with a convex region R v . The regions associated with the children of v are obtained ....

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.

....we are unable to prove an approximation factor better than 4( 2) So, improving the approximation factors remains a challenging problem. In more general settings, i.e. in an environment of polygonal obstacles it is known that the shortest watchman route problem is NP hard. Mata and Mitchell [11] exhibit an O(log n) approximation algorithm using dynamic programming. Perhaps this can be improved. ....

C.S. Mata, J.S.B. Mitchell. Approximation Algorithms for Geometric Tour and Network Design Problems. In Proc. 11th ACM Symposium on Computational Geometry, pages 360369, 1995.

.... BSPs have proven to be versatile, with applications in many other problems global illumination [5] shadow generation [7, 8, 9] visibility problems [3, 25] solid geometry [19, 20, 26] geometric data repair [16] ray tracing [18] robotics [4] and approximation algorithms for network design [15] and surface simpli cation [2] Informally, a BSP B for a set of polygons in R 3 is a binary tree. Each node v of B is associated with a convex region R v . The regions associated with the children of v are obtained by splitting R v with a plane. The regions associated with the leaves of the ....

C. Mata and J. S. Mitchell, Approximation algorithms for geometric tour and network design problems, Proc. 11th Annu. ACM Sympos. Comput. Geom., 1995, pp. 360369.

.... 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 ....

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.

.... subsequently proven to be versatile, with applications in many other problems global illumination [7] shadow generation [11, 12] visibility problems [5, 29] solid modeling [22, 24, 30] geometric data repair [19] ray tracing [21] robotics [6] and approximation algorithms for network design [18] and surface simplification [4] Algorithms have also been developed to construct BSPs for moving objects [2, 13, 23, 31] Informally, a BSP B for a set of (d Gamma 1) dimensional objects in R d is a binary tree. Each node v of B is associated with a convex region R v . The regions associated ....

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.

.... and was further refined by Fuchs et al. 10] The BSP has been widely used in several areas, including computer graphics (global illumination [4] shadow generation [6, 7] visibility determination [3, 21] and ray tracing [15] solid modeling [16, 22] geometric data repair [12] network design [11], and surface simplification [2] The BSP has A preliminary version of this paper appeared as a communication in the Proceedings of the 13th Annual ACM Symposium on Computational Geometry, 1997, pages 382 384. This author is affiliated with Brown University. Support was provided in part ....

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.

....[7] 5.2 Hardness results for Generalizations We show two simple reductions that demonstrate that other generalizations of the CDS problem may be as hard to approximate as the set TSP problem for which no non trivial approximation algorithms are known. For the Euclidean case, Mata and Mitchell [19] have given approximation algorithms for this problem. c j V j Figure 3: Reduction of set TSP problem to edge weighted CDS. Theorem 5.1 A polynomial approximation algorithm for the edge weighted connected dominating set problem with factor f(n) would imply a polynomial approximation algorithm for ....

C. S. Mata and J. S. B. Mitchell "Approximation algorithms for geometric tour and network design problems", Proc. of the 11th Annual Symp. on Computational Geometry, pages 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.

.... special case of the lawn mowing problem, with a finite (discrete) set of points to be mowed, has been studied by Arkin and Hassin [2] who obtained constant factor approximation methods 2 Figure 2: The milling problem for this and other variants of the TSP with Neighborhoods problem (see also [15]) These problems are clearly NP hard, from the fact that the Euclidean TSP is NP hard. The lawn mowing problem is also closely related to the watchman route problem with limited visibility (or d sweeper problem ) which has been studied by Ntafos [19] How does one sweep the floor of a ....

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

No context found.

<F3.542e+05> C. Mata and J. S. B.<F3.846e+05> Mitchell,<F3.384e+05> Approximation algorithms for geometric tour and network design<F3.846e+05> problems, in Proc. 11th Ann. ACM Sympos. Comput. Geom., Vancouver, Canada, ACM Press, New York, 1995, pp. 360--369.

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.

....using a mower of radius d. This special case of the lawn mowing problem, with a finite (discrete) set of points to be mowed, has been studied by Arkin and Hassin [2] who obtained constant factor approximation methods for this and other variants of the TSP with Neighborhoods problem (see also [15]) These problems are clearly NP hard, from the fact that the Euclidean TSP is NP hard. The lawn mowing problem is also closely related to the watchman route problem with limited visibility (or d sweeper problem ) which has been studied by Ntafos [19] How does one sweep the floor of a ....

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

....k # n, and we are to find a subset of k points of P that has the shortest minimum spanning tree. The problem is NP hard. Ravi et al. 21] give an approximation algorithm with ratio O(k 1 4 ) which was quickly improved to a factor of O(log k) by Garg and Hochbaum [13] and Mata and Mitchell [17]. Eppstein [10] has improved the approximation ratio to O(log k log log n) and has given general techniques to improve the running times (as a function of n) of existing algorithms; further, he shows that the exact k MST problem can be solved in time 2 O(k log k) n O(n log n) which is simply ....

....if S is guillotine with respect to the unit square B. See Figure 3.1 for an example of a guillotine subdivision, where we illustrate the entire tree of perfect cuts. Each perfect cut is indicated with a small arrow. Note that, in contrast with guillotine rectangular subdivisions (see [9, 17]) the guillotine subdivisions we study here are not restricted to have rectangular faces; rather, the faces of a guillotine subdivision are rectilinear polygons. In fact, it is precisely this distinction that permits us to get constant factor approximations, while the previous method of [17] ....

[Article contains additional citation context not shown here]

<F3.761e+05> C. Mata and J. S. B.<F3.826e+05> Mitchell,<F3.789e+05> Approximation algorithms for geometric tour and network design<F3.826e+05> problems, in Proc. 11th Ann. ACM Sympos. Comput. Geom., 1995, pp. 360--369.

....of n simple polygons, having a total of N vertices. For the TSPN, Arkin and Hassin [1] have obtained O(1) approximation algorithms for well behaved (possibly overlapping) regions (e.g. regions are disks or have roughly equal length and parallel diameter segments) while Mata and Mitchell [19] have obtained an O(log n) approximation algorithm for n general (possibly overlapping) regions, based on guillotine rectangular subdivisions . We prove the following theorem, giving the first approximation results for the orienteering version of the problem: Theorem 13 There is an O(log ....

....sites, while having weight at most B. Our approximation algorithm is to tabulate, for each value of k, k = 1; n, a shortest possible connected guillotine rectangular subdivision spanning k or more regions. This can be done in time polynomial in n and N , using dynamic programming, as in [19]. We then select, from among these, the subdivision that visits the maximum number k of regions, subject to its weight being less than the bound B. We now have to prove that, among the subdivisions tabulated, there must be one, T , whose weight is less than B, while the cardinality, k, of the ....

[Article contains additional citation context not shown here]

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.

....the plane. Open Problem 21 Is there an O(1) approximation algorithm for the group Steiner problem on a set of points in the Euclidean plane If the groups of points are in fact connected sets (e.g. polygons) in the plane, then an O(logk) approximation algorithm is given by Mata and Mitchell [272]. The special case in which the groups are intervals that lie on two parallel lines has a polynomial time algorithm by Ihler [220] The group Steiner problem is closely related to the one of a set traveling salesperson problem (TSP) and the TSP with neighborhoods, which are discussed below in ....

....even for points in the Euclidean plane; see [158, 333] A series of approximation results have been obtained for this problem. Ravi et al. 333] give an approximation algorithm with ratio O(k 1=4 ) which was improved to a factor of O(log k) by Garg and Hochbaum [166] and Mata and Mitchell [272]. Eppstein [149] has improved the approximation ratio to O(logk= log log n) and has given general techniques to improve the running times (as a function of n) of existing algorithms; further, he shows that the exact k MST problem can be solved in time 2 O(k log k) n O(n log n) which is ....

[Article contains additional citation context not shown here]

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.

....k n, and we are to find a subset of k points of P that has the shortest minimum spanning tree. The problem is NP hard. Ravi et al. 21] give an approximation algorithm with ratio O(k 1=4 ) which was quickly improved to a factor of O(logk) by Garg and Hochbaum [13] and Mata and Mitchell [17]. Eppstein [10] has improved the approximation ratio to O(log k= log log n) and has given general techniques to improve the running times (as a function of n) of existing algorithms; further, he shows that the exact k MST problem can be solved in time 2 O(k log k) n O(n log n) which is ....

....if S is guillotine with respect to the unit square, B. See Figure 3.1 for an example of a guillotine subdivision, where we illustrate the entire tree of perfect cuts. Each perfect cut is indicated with a small arrow. Note that, in contrast with guillotine rectangular subdivisions (see [9, 17]) the guillotine subdivisions we study here are not restricted to have rectangular faces; rather, the faces of a guillotine subdivision are rectilinear polygons. In fact, it is precisely this distinction that permits us to get constant factor approximations, while the previous method of [17] ....

[Article contains additional citation context not shown here]

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.

....of n simple polygons, having a total of N vertices. For the TSPN, Arkin and Hassin [1] have obtained O(1) approximation algorithms for well behaved (possibly overlapping) regions (e.g. regions are disks or have roughly equal length and parallel diameter segments) while Mata and Mitchell [19] have obtained an O(logn) approximation algorithm for n general (possibly overlapping) regions, based on guillotine rectangular subdivisions . We prove the following theorem, giving the first approximation results for the orienteering version of the problem: Theorem 15 There is an ....

....while having weight at most B. Our approximation algorithm is to tabulate, for each value of k, k = 1; n, a shortest possible connected guillotine rectangular subdivision spanning k or more regions. This can be done in time polynomial in n and N , using dynamic programming, as in [19]. We then select, from among these, the subdivision that visits the maximum number k of regions, subject to its weight being less than the bound B. We now have to prove that, among the subdivisions tabulated, there must be one, T , whose weight is less than B, while the cardinality, k, of the ....

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.

....k n, and we are to find a subset of k points of P that has the shortest minimum spanning tree. The problem is NP hard. Ravi et al. 21] give an approximation algorithm with ratio O(k 1=4 ) which was quickly improved to a factor of O(logk) by Garg and Hochbaum [13] and Mata and Mitchell [17]. Eppstein [10] has improved the approximation ratio to O(log k= log log n) and has given general techniques to improve the running times (as a function of n) of existing algorithms; further, he shows that the exact k MST problem can be solved in time 2 O(k logk) n O(n log n) which is simply ....

....if S is guillotine with respect to the unit square, B. See Figure 3.1 for an example of a guillotine subdivision, where we illustrate the entire tree of perfect cuts. Each perfect cut is indicated with a small arrow. Note that, in contrast with guillotine rectangular subdivisions (see [9, 17]) the guillotine subdivisions we study here are not restricted to have rectangular faces; rather, the faces of a guillotine subdivision are rectilinear polygons. In fact, it is precisely this distinction that permits us to get constant factor approximations, while the previous method of [17] ....

[Article contains additional citation context not shown here]

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.

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.

No context found.

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.

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.

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