J. S. B. Mitchell. Guillotine subdivisions approximate polygonal subdivisions: A simple polynomial-time approximation scheme for geometric TSP, k-MST, and related problems. SIAM J. Comput., 28(4):1298--1309, 1999.  Home/Search   Document Details and Download   Summary   Related Articles   Check

....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 ....

J. S. B. Mitchell. Guillotine subdivisions approximate polygonal subdivisions: A simple polynomial-time approximation scheme for geometric tsp, k-mst, and related problems. SIAM Journal on Computing, 28(4):1298--1309, 1999.

....In this paper, we provide a polynomial time approximation scheme (PTAS) for the SRStA problem. A PTAS for a problem of size n is an algorithm that, for every constant 0, finds an approximate solution with an approximation factor of 1 in time polynomial in n. We apply the method proposed in [3, 7, 8, 9]. For the sake of completeness, we briefly review the results of m guillotine in Section 3. 1.1 Motivations and Applications The SRStA and the RStA problems have a number of applications. An application that is mentioned quite often comes from VLSI design, where a RStA or SRStA is needed to ....

....that any rectangular subdivision with cost L can be converted into a guillotine rectangular subdivision with cost at most 2L by adding a set of new edges whose total length is at most L. Moreover, the cost of the new edges is charged off to the original edge set of the subdivision. Mitchell [7, 8, 9] extended these concepts and ideas by defining m guillotine subdivision and proving that an m guillotine subdivision with cost at most (1 ) Delta L can be obtained from a rectilinear subdivision whose cost is L. With m guillotine subdivision, Mitchell [8, 9] found PTASs for various geometric ....

J. S. B. Mitchell. Guillotine subdivisions approximate polygonal subdivisions: a simple new method for the geometric k-MST problem. Proc. 7th ACM-SIAM Sympos. Discrete Algorithms, 1996, pp. 402-408.

....convergence rate may be derived if a better m dependent analog to the concentration inequality (25) can be found. 4 Convergence Rates for Fixed Partition Approximations Partitioning approximations to minimal graphs have been proposed by many authors, including Karp [5] Ravi etal [25] Mitchell [26], and Arora [27] as ways to reduce computational complexity. The fixed partition approximation is a simple example whose convergence rate has been studied by Karp [5, 28] Karp and Steele [29] and Yukich [2] in the context of a uniform density f . Fixed partition approximations to a minimal graph ....

J. Mitchell, "Guillotine subdivisions approximate polygonal subdivisions: a simple new method for the geometric k-MST problem," in Proc. of ACM-SIAM Symposium on Discrete Algorithms, 1996, pp. 402--408.

....known. This amounts to constructing a shortest traveling salesperson (TSP) tour on the cells. 252 Christian Icking et al. If the polygonal environment contains obstacles, the problem of finding such a minimum length tour is known to be NP hard [25] and there are some approximation schemes [2,3,18,36]. In a simple polygon without obstacles, the complexity of constructing offline a minimum length tour seems to be open. There are, however, some results concerning the related Hamiltonian cycle and path problems [14,44] and approximations [2,38] 3.1 The Competitive Complexity The following ....

J. S. B. Mitchell. Guillotine subdivisions approximate polygonal subdivisions: A simple new method for the geometric k-MST problem. In Proc. 7th ACM-SIAM Sympos. Discrete Algorithms, pages 402--408, 1996.

....the tour. The dynamic programming paradigm for approximating the k TSP is powerful enough to approximate the j WTSP within a constant factor in polynomial time. The geometric component of the k TSP approximation algorithm is unaffected by this reduction, so the approximation bounds proved in Refs. [4,18] hold here as well. Our strategy is to compute, for each tool t, a t subset of approximately minimum average weight, and then return the one of the least average weight. Consider computing this t subset for tool t. We first outline our approach by making the (invalid) assumption that t regions ....

....the t subset of minimumaverage weight, if the j WTSP could be solved exactly. The approximation factor would be O(ln jSj) O(log m logN ) by the well known performance of greedy heuristic. The j WTSP is approximated to within a constant factor in time O( mN ) using the algorithm in Refs. [4,18]. An inspection of the analysis of greedy heuristic for weighted set cover (see, e.g. Refs. 9,15] reveals that this extra constant factor only increases the constant hidden in O(logm log N ) 2 32 Acknowledgments We thank Ajay Joneja of the Industrial Engineering Department at the Hong Kong ....

J. S. B. Mitchell. Guillotine subdivisions approximate polygonal subdivisions: A simple new method for the geometric k-MST problem. In Proc. 7th ACM-SIAM Sympos. Discrete Algorithms, pages 402--408, 1996.

....2 # times the MST weight for any # 0. However, by using other lower bounds besides the MST weight, it is possible to obtain better approximation guarantees for the traveling salesman problem, as was demonstrated by Christofides [4] factor 3 2) for general metrics, and Arora [1] and Mitchell [13] (factor 1 #) for the Euclidean metric in fixed dimensions. The focus of the present paper is on the following generalization of the traveling salesman path problem (which corresponds to the K = 2 case) given K, find a spanning tree, of minimum weight such that the maximum degree is at most K. ....

J. S. B. Mitchell. Guillotine subdivisions approximate polygonal subdivisions: a simple polynomial-time approximation scheme for geometric TSP, k-MST, and related problems. SIAM J. Comput., 28:1298--1309, 1999.

.... As follows from results of Itai, Papadimitriou, and Swarc ter [17] the Minimum TSP is NP hard for any xed dimension d and any L p or polyhedral norm; see also the earlier results by Garey, Graham, and Johnson [13] and Papadimitriou [23] On the other hand, results of Arora [3] and Mitchell [21] imply that in all these cases a polynomial time approximation scheme (PTAS) exists, i.e. a sequence of polynomial time algorithms A k , 1 k 1, where A k is guaranteed to nd a tour whose length is within a ratio of 1 (1=k) of optimal. The situation for geometric versions of the Maximum TSP ....

Mitchell, J.S.B., \Guillotine subdivisions approximate polygonal subdivisions: Part II { A simple PTAS for geometric k-MST, TSP, and related problems," SIAM J. Comp., 28 (1999), 1298-1309.

....In this paper, we provide a polynomial time approximation scheme (PTAS) for the SRStA problem. A PTAS for a problem of size n is an algorithm that, for every constant e 0, finds an approximate solution with an approximation factor of 1 e in time polynomial in n. We apply the method proposed in [3, 7, 8, 9]. For the sake of completeness, we briefly review the results of m guillotine in Section 3. Computer Science Department, University of Minnesota, Minneapolis, MN 55455, USA. Email: cheng,blu cs.umn.edu. Rutgers University campus at Camden. Email: bhaskar camden.rutgers.edu. rectilinear ....

....that any rectangular subdivision with cost L can be converted into a guillotine rectangular subdivision with cost at most 2L by adding a set of new edges whose total length is at most L. Moreover, the cost of the new edges is charged off to the original edge set of the subdivision. Mitchell [7, 8, 9] extended these concepts and ideas by defining m guillotine subdivision and proving that an m guillotine subdivision with cost at most (1 ) L can be obtained from a rectilinear subdivision whose cost is L. With m guillotine subdivision, Mitchell [8, 9] found PTASs for various geometric ....

[Article contains additional citation context not shown here]

J. S. B. Mitchell. Guillotine subdivisions approximate polygonal subdivisions: Part II - a simple polynomial-time approximation scheme for geometric k-MST, TSP, and related problems. Available at

....In this paper, we provide a polynomial time approximation scheme (PTAS) for the SRStA problem. A PTAS for a problem of size n is an algorithm that, for every constant e 0, finds an approximate solution with an approximation factor of 1 e in time polynomial in n. We apply the method proposed in [3, 7, 8, 9]. For the sake of completeness, we briefly review the results of m guillotine in Section 3. Computer Science Department, University of Minnesota, Minneapolis, MN 55455, USA. Email: cheng,blu cs.umn.edu. Rutgers University campus at Camden. Email: bhaskar camden.rutgers.edu. rectilinear ....

....that any rectangular subdivision with cost L can be converted into a guillotine rectangular subdivision with cost at most 2L by adding a set of new edges whose total length is at most L. Moreover, the cost of the new edges is charged off to the original edge set of the subdivision. Mitchell [7, 8, 9] extended these concepts and ideas by defining m guillotine subdivision and proving that an m guillotine subdivision with cost at most (1 ) L can be obtained from a rectilinear subdivision whose cost is L. With m guillotine subdivision, Mitchell [8, 9] found PTASs for various geometric ....

[Article contains additional citation context not shown here]

J. S. B. Mitchell. Guillotine subdivisions approximate polygonal subdivisions: a simple polynomialtime approximation scheme for geometric TSP, k-MST, and related problems. Manuscript, Available at

....In this paper, we provide a polynomial time approximation scheme (PTAS) for the SRStA problem. A PTAS for a problem of size n is an algorithm that, for every constant e 0, finds an approximate solution with an approximation factor of 1 e in time polynomial in n. We apply the method proposed in [3, 7, 8, 9]. For the sake of completeness, we briefly review the results of m guillotine in Section 3. Computer Science Department, University of Minnesota, Minneapolis, MN 55455, USA. Email: cheng,blu cs.umn.edu. Rutgers University campus at Camden. Email: bhaskar camden.rutgers.edu. rectilinear ....

....that any rectangular subdivision with cost L can be converted into a guillotine rectangular subdivision with cost at most 2L by adding a set of new edges whose total length is at most L. Moreover, the cost of the new edges is charged off to the original edge set of the subdivision. Mitchell [7, 8, 9] extended these concepts and ideas by defining m guillotine subdivision and proving that an m guillotine subdivision with cost at most (1 ) L can be obtained from a rectilinear subdivision whose cost is L. With m guillotine subdivision, Mitchell [8, 9] found PTASs for various geometric ....

J. S. B. Mitchell. Guillotine subdivisions approximate polygonal subdivisions: a simple new method for the geometric k-MST problem. Proc. 7th ACM-SIAM Sympos. Discrete Algorithms, 1996, pp. 402-408.

....contains an approach which is able to lead to the similar results. However, one year later, Arora [2] made another big progress that he improved running time from n to n (logn) His new polynomial time approximation scheme also runs randomly in time n(log n) Soon later, Mitchell [20] claimed again that his approach can do a similar thing. We were curious about this piece of history and hence made a study on these two approaches. In this article, we would like to share with readers the result of our investigation and something interesting that we found in their publications. ....

....for crosspoints to O( and this enables us to choose 2m from the O( positions to form a m guillotine cut (m = 1=e) Therefore, the dynamic programming for finding the best such partition runs in time n ) where c is a constant. This is the basic idea of Arora [2] and Mitchell [20]. Arora s work [2, 3] also contains a new technique about the tree structure of partition. Indeed, it is an earlier and better work compared with Mitchell [20] The portal technique cannot apply to the MELRP, the rectilinear Steiner arborescence, and the symmetric rectilinear Steiner ....

[Article contains additional citation context not shown here]

J.S.B. Mitchell, Guillotine subdivisions approximate polygonal subdivisions: Part III - Faster polynomial-time approximation scheme for geometric network optimization, preprint, 1997.

....approximation schemes. More precisely, for any e 0, there exists an approximation algorithm for those problems, running in time n , which produces approximation solution within 1 e from optimal. It made Arora s research be reported in New York Times again. Several weeks later, Mitchell [19] claimed that his earlier work [17] its journal version [18] already contains an approach which is able to lead to the similar results. However, one year later, Arora [2] made another big progress that he improved running time from n to n (logn) His new polynomial time approximation ....

....length(P) and hence Theorem 3.1 holds. Actually, this argument is equivalent to the current proof of Theorem 3.1. In fact, only projections from both sides exist (Case 2) pro j x (P) or pro j y (P) can contribute something against the length of the added segment. 4 m Guillotine Cut Mitchell [19] extended the 1 guillotine cut to the m guillotine cut in the following way: A point p is a horizontal (vertical) m dark point if the horizontal (vertical) line passing through p intersects at least 2m vertical (horizontal) segments of the considered rectangular partition P, among which at least m ....

[Article contains additional citation context not shown here]

J.S.B. Mitchell, Guillotine subdivisions approximate polygonal subdivisions: Part II - A simple polynomial-time approximation scheme for geometric k-MST, TSP, and related problem, SIAM J. Comput. 29 (1999), no. 2, 515--544.

....for any e 0, there exists an approximation algorithm for those problems, running in time n , which produces approximation solution within 1 e from optimal. It made Arora s research be reported in New York Times again. Several weeks later, Mitchell [19] claimed that his earlier work [17] (its journal version [18] already contains an approach which is able to lead to the similar results. However, one year later, Arora [2] made another big progress that he improved running time from n to n (logn) His new polynomial time approximation scheme also runs randomly in time ....

....two approaches. In this article, we would like to share with readers the result of our investigation and something interesting that we found in their publications. 2 Rectangular Partition and Guillotine Cut Let us start from rectangular partition. In fact, before prove his main theorem, Mitchell [17, 18] stated clearly that Our proof is inspired by the proof in [7] where the reference [7] in [17] 9] in [18] is actually a paper of Du, Pan, and Shing [7] on minimum edge length rectangular partition. This paper initiated the idea of using guillotine cut to design approximation algorithms. The ....

[Article contains additional citation context not shown here]

J.S.B. Mitchell, Guillotine subdivisions approximate polygonal subdivisions: A simple new method for the geometric k-MST problem. Proc. 7th ACM-SIAM Symposium on Discrete Algorithms, (1996) pp. 402-408.

....and suggestions for further research. We should note before proceeding that certain heuristics described elsewhere in this book are for various reasons not covered in this chapter. Perhaps our foremost omission is the approximation schemes for geometric STSP s of Arora, Mitchell, et al. [5, 55, 61], as described in Chapter 5. These heuristics, despite their impressive theoretical guar1 DIMACS is the Center for Discrete Math and Theoretical Computer Science, a collaboration of Rutgers and Princeton Universities with Bell Labs, AT T Labs, NEC Labs, and Telcordia Technologies. This was the 8th ....

J. S. B. Mitchell. Guillotine subdivisions approximate polygonal subdivisions: A simple polynomial-time approximation scheme for geometric TSP, k-MST, and related problems. SIAM Journal on Computing, 28:1298--1309, 1999.

....intersection with the subdivision edges consists of a small number (O(m) of connected components and the subdivisions on either side of the cut are also m guillotine. With a minor change of the proof of [76] Mitchell established a PTAS for minimum length rectangular partition problem. Mitchell [78, 79] further extended this m guillotine subdivision technique to other geometric optimization problems, including Euclidean and rectlinear Steiner tree problems, and obtained PTAS for them. 5 Variations of Steiner Trees Successful researches on classical Steiner tree problems encourage extensive ....

J. S. B. Mitchell, "Guillotine subdivisions approximate polygonal subdivisions: Part III - Faster polynomial-time approximation scheme for geometric network optimization," Proc. ninth Canadian conference on computational geometry, 1997, pp. 229-232.

....is not only on Steiner trees, but also on the design and analysis of aproximation algorithms in combinatorial optimiza13 tion. Let us review these two remarkable techniques in the following. 4. 1 Arora s PTAS It is quite interesting to notice that Arora [3] appeared only one week before Mitchell [77]. Any way, they use very different techniques to reach the same goal. Therefore, both are very interesting. Arora s technique is based on recursive partition. In Jiang et al. [58, 59] although partition can be moved parallelly, the size of each cell is fixed. It cannot be varied according to local ....

....utilized this technique to obtain constant approximations for other geometric optimization problems. With the same technique, Mata 14 [74] obtained a constant factor approximation algorithm for red blue separation problem improving previous result O(logn) Inspired by this success, Mitchell [77] extended guillotine subdivision to m guillotine subdivision, a rectangular polygonal subdivision such that there exists a cut whose intersection with the subdivision edges consists of a small number (O(m) of connected components and the subdivisions on either side of the cut are also ....

J. S. B. Mitchell, "Guillotine subdivisions approximate polygonal subdivisions: Part II - A simple polynomial-time approximation scheme for geometric k-MST, TSP, and related problems," SIAM J. Comp.

....in polynomial time. However, they were only able to show that this guillotine subdivision is an approximation of the minimum length rectangular partition problem with performance ratio two in a special case that the region R is surrounded by a rectangle with some points as holes in it. Mitchell [76] showed that this is actually true in general. He also successfully utilized this technique to obtain constant approximations for other geometric optimization problems. With the same technique, Mata 14 [74] obtained a constant factor approximation algorithm for red blue separation problem ....

....guillotine subdivision, a rectangular polygonal subdivision such that there exists a cut whose intersection with the subdivision edges consists of a small number (O(m) of connected components and the subdivisions on either side of the cut are also m guillotine. With a minor change of the proof of [76], Mitchell established a PTAS for minimum length rectangular partition problem. Mitchell [78, 79] further extended this m guillotine subdivision technique to other geometric optimization problems, including Euclidean and rectlinear Steiner tree problems, and obtained PTAS for them. 5 Variations ....

J. S. B. Mitchell, "Guillotine subdivisions approximate polygonal subdivisions: A simple new method for the geometric k-MST problem," Proc. 7th ACM-SIAM Sympos. Discrete Algorithms, 1996, pp. 402-408.

....e. when the environment is already known. This amounts to constructing a shortest traveling salesperson (TSP) tour on the cells. If the polygonal environment contains obstacles, the problem of finding such a minimum length tour is known to be NP hard [8] and there are some approximation schemes [1, 2, 6, 9]. In a simple polygon without obstacles, the complexity of constructing o# line a minimum length tour seems to be open. There are, however, some results concerning the related Hamiltonian cycle and path problems [4, 12] and approximations [1, 11] For our online exploration problem we are not ....

J. S. B. Mitchell. Guillotine subdivisions approximate polygonal subdivisions: A simple new method for the geometric k-MST problem. In Proc. 7th ACM-SIAM Sympos. Discrete Algorithms, pages 402--408, 1996.

....1.0.0.1 History. The current paper evolved out of preliminary results obtained in January 1996, culminating in a submission to IEEE FOCS 1996 in April 1996 [3] The running time of the algorithm then was n O(c) in # 2 and n O(c d 1 log d 2 n) in # d . A few weeks later, Mitchell [49] independently discovered an n O(c) time approximation scheme for points in # 2 . His algorithm useds ideas from his earlier constant factor approximation algorithm for k MST [48] It relies on the geometry of the plane and does not seem to generalize to higher dimensions. In January 1997 the ....

J. Mitchell. Guillotine subdivisions approximate polygonal subdivisions: Part II- A simple PTAS for geometric k-MST, TSP, and related problems. Preliminary manuscript, April 30, 1996. To appear in SIAM J. Computing.

....the algorithm then was n O(c) in # 2 and n O(c d 1 log d 2 n) in # d . A few weeks later, Mitchell [49] independently discovered an n O(c) time approximation scheme for points in # 2 . His algorithm useds ideas from his earlier constant factor approximation algorithm for k MST [48]. It relies on the geometry of the plane and does not seem to generalize to higher dimensions. In January 1997 the author discovered the nearly linear time algorithm described in this paper. The key ingredient of this algorithm is Theorem 2, which the author had originally conjectured to be false. ....

....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 [23] Blum, Chalasani, and Vempala [14] gave the first O(1) factor approximation algorithm for points in # 2 and Mitchell [48] improved this factor to 2 # 2. Euclidean Min Cost Perfect Matching: EMCPM) Given 2n points in # 2 (or # d in general) find the minimum cost set of nonadjacent edges that cover all vertices. This problem can be solved in polynomial time (even for nongeometric instances) Vaidya shows how ....

J. Mitchell. Guillotine subdivisions approximate polygonal subdivisions: A simple new method for the geometric k-MST problem. In Proc. ACM-SIAM Symposium on Discrete Algorithms, pp. 402-208, 1996.

....that runs in polynomial time and show that its performance ratio is at most 5. Since the proof is complex, we rst prove a simpler bound of 6.5; this proof contains some of the basic ideas. For geometrical instances, such as points in the plane, the algorithms of Arora [4, 5] and Mitchell [20]) can be used to obtain an (1 ) approximation of the TSP, and this leads to an approximation factor of 4(1 ) for these instances. We also describe a simple algorithm that nds a preemptive tour whose length is at most 5 times the length of an optimal preemptive tour. We will shortly ....

....For the algorithm, we will assume that an approximation of TSP tour is used, and therefore the weight of the tour is at most TOPT C 0 k . If Christo des algorithm is used, then 1:5. For geometric instances, such as points in the plane, the algorithms of Arora [4, 5] and Mitchell [20] can be used to obtain a (1 ) approximation of the TSP tour. In this case, 1 , for any constant 0. Using the same ideas as in the above lemma we can show that there is always a valid starting point on this tour, such that we never run out of pegs. In Step 5c we nd a matching M on ....

[Article contains additional citation context not shown here]

J. Mitchell, \Guillotine subdivisions approximate polygonal subdivisions: A simple polynomial-time approximation scheme for geometric TSP, k-MST and related problems, " SIAM J. Comput., 28(4):1298-1309, (1999). 18

....distance matrix requires (n 2 ) entries; even with today s computing power, it is hopeless to store and use the distance matrix for instances with, say, n = 10 6 . The study of geometric instances has resulted in a number of powerful theoretical results. Most notably, Arora [2] and Mitchell [16] have developed a general framework that results in polynomial time approximation schemes (PTASs) for many geometric versions of graph optimization problems: Given any constant , there is a polynomial algorithm that yields a solution within a factor of (1 ) of the optimum. However, these ....

J. S. B. Mitchell. Guillotine subdivisions approximate polygonal subdivisions: A simple polynomial-time approximation scheme for geometric TSP, k-MST, and related problems. SIAM J. Comput., 28 (1999), 1298--1309.

....[57] and Arora [2] Goemans and Williamson [57] show how to use semide nite programming to give better approximation algorithms for MAX 2SAT and MAX CUT. Arora [2] has discovered a polynomial time approximation scheme (PTAS) for Euclidean TSP and Euclidean Steiner tree problem. Mitchell [80] independently discovered similar results a few months later. These were two notable problems not addressed by our hardness result in this paper since they were not known to be MAX SNP hard. Arora s result nally resolves the status of these two important problems. 8.1 Future Directions Thus far ....

J. Mitchell. Guillotine subdivisions approximate polygonal subdivisions: Part II- A simple PTAS for geometric k-MST, TSP, and related problems. Preliminary manuscript, April 30, 1996. To appear in SIAM J. Computing.

....O(k 1 4 ) approximation for the problem (2)in the plane. Improved algorithms, with ratio of O(log k) are due to [18] and [29] Further decrease to O( log k log log n ) 12] and a rst constant factor approximation ( 4] with the constant not speci ed) followed. Using guillotine subdivisions [25] develops a 2 p 2 approximation for the l 2 metric, and a 2 approximation for the l 1 metric. Finally, a polynomial time approximation scheme has been given in [2] 1.4 Local Search and Metaheuristics More recently, authors have successfully applied local search methods to the k cardinality ....

J.S.B. Mitchell. Guillotine subdivisions approximate polygonal subdivisions: A simple new method for the geometric k-mst problem. In Proceedings of the 7th Annual ACMSIAM Symposium on Discrete Algorithms, pages 402-408. SIAM, 1996.

....to get a worst case lower bound proportional to d in d dimensions, for d 3. It remains an interesting open problem if an approximation ratio independent of d is possible for general d. We note that our algorithm only generates hierarchical tilings, also known as Guillotine Subdivisions [7]. Such tilings can be computed optimally by dynamic programming in Theta(N 2 ) time. The dynamic programming algorithm is an order of magnitude slower than the greedy algorithm, and it is not known if the former achieves a better ratio in the worst case than the greedy algorithm. 14 ....

J. S. B. Mitchell. Guillotine Subdivisions Approximate Polygonal Subdivisions: A Simple New Method for the Geometric k-MST Problem. Proc. 7th Symposium on Discrete Algorithms, pp. 402--408, 1996.

....tree and the perfect matching is constructed. The solution found this way is 3 2 approximative. For the special case that the vertices in the input graph corresponds to points in the Euclidean plane and the edge lengths are given by the Euclidean distances, Arora [2] and independently Mitchell [37] have devised polynomial time approximation schemes. 4 However, no practical implementation of these fairly complicated algorithms has been reported yet. Real time applications require that answers are computed online and within tight time windows. The length of this time window is closely ....

J. S. B. Mitchell, Guillotine subdivisions approximate polygonal subdivisions: A simple polynomial-time approximation scheme for geometric tsp, k-mst, and related problems, SIAM Journal on Computing 28 (1999), no. 4, 1298--1309.

....TSP cannot be approximated better than by a factor of 2 ( p n) by a polynomial time algorithm unless P=NP, even if the maximum speed is bounded. The n denotes the size of the input instance. Especially the last result is surprising in the light of existing polynomial time approximation schemes [1, 2, 8, 9] for the static version of the problem. In the next section, we state de nitions and give preliminary results concerning Kinetic TSP. Speci cally, we give an overview of the original reduction of Garey et al. 5] proving the NP hardness of the Euclidean TSP, since it plays an important role in ....

....instance into an instance of the Euclidean MHP using the bijective mapping f v . Given this new instance, we compute a Hamiltonian path APXE such that C(APXE ) 1 )C(OPTE ) using a PTAS for the Euclidean MHP problem. Such schemes can be constructed by modifying any PTAS for the Euclidean TSP [1, 2, 8, 9]. Here OPTE denotes a minimal Hamiltonian path in the Euclidean instance. Let APX T and OPT T denote the corresponding paths in the translational MHP instance. Observe that OPT T is an optimal salesman path by Lemma 3. We have C(APX T ) v(y n y 1 ) 1 v 2 C(APXE ) 1) v(y n y 1 ) 1 v ....

J. S. B. Mitchell. Guillotine subdivisions approximate polygonal subdivisions: Part II  a simple polynomial-time approximation scheme for geometric TSP, k-MST, and related problems. To appear in SIAM J. Computing.

....on decompositions of k connected Euclidean graphs combined with the general framework proposed recently by Arora [Aro98] for designing PTAS for Euclidean versions of TSP, Minimum Steiner Tree, Min Cost Perfect Matching, k TSP, and k MST. For another related framework for geometric PTAS see [Mit97, Mit99] In contrast to all previous applications of Arora s framework using Steiner points in the so called patching procedures [Aro98, CL98] one avoids introducing Steiner points in [CL99] Steiner points of degree at least three are dicult to remove in case of k connectivity, k 2, contrary to the ....

J. S. B. Mitchell. Guillotine subdivisions approximate polygonal subdivisions: A simple polynomial-time approximation scheme for geometric TSP, k-MST, and related problems. SIAM Journal on Computing, 28(4):270-283, 1999.

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J. S. B. Mitchell, Guillotine subdivisions approximate polygonal subdivisions: Part III -- Faster polynomial-time approximation schemes for geometric network optimization, Manuscript, April 1997, available at http://www.ams.sunysb.edu/~jsbm/jsbm.html

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J. S. B. Mitchell, Guillotine subdivisions approximate polygonal subdivisions: A simple new method for the geometric k-MST problem, Proc. 7th ACM-SIAM Sympos. Discrete Algorithms (1996), pp. 402--408.

....how a modification to our earlier results on guillotine subdivisions leads to an n time (deterministic) PTAS for Euclidean versions of various geometric network optimization problems on a set of n points in the plane. This improves the previous n time algorithms of Arora [1] and Mitchell [10]. Arora [2] has recently obtained even better bounds than our own: He obtains a randomized algorithm with expected running time O(n log n) However, our new theorem on approximating guillotine subdivisions may be interesting in its own right, and may lead to yet further improvements or ....

....running time O(n log n) However, our new theorem on approximating guillotine subdivisions may be interesting in its own right, and may lead to yet further improvements or generalizations. Our methods are based on the concept of an m guillotine subdivision , which were introduced by Mitchell [9, 10]. Roughly speaking, an m guillotine subdivision is a polygonal subdivision with the property that there exists a line ( cut ) whose intersection with the subdivision edges consists of a small number (O(m) of connected components, and the subdivisions on either side of the line are also ....

[Article contains additional citation context not shown here]

J. S. B. Mitchell, Guillotine subdivisions approximate polygonal subdivisions: Part II --- A simple polynomial-time approximation scheme for geometric k-MST, TSP, and related problems, SIAM J. Comp., to appear.

....running time O(n log n) However, our new theorem on approximating guillotine subdivisions may be interesting in its own right, and may lead to yet further improvements or generalizations. Our methods are based on the concept of an m guillotine subdivision , which were introduced by Mitchell [9, 10]. Roughly speaking, an m guillotine subdivision is a polygonal subdivision with the property that there exists a line ( cut ) whose intersection with the subdivision edges consists of a small number (O(m) of connected components, and the subdivisions on either side of the line are also ....

....the geometric optimization problems considered here are known to be NP hard, polynomialtime approximation algorithms have been known previously that get within a constant factor of optimal. Further, polynomial time approximation schemes were discovered last year, by Arora [1] and by Mitchell [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 ....

[Article contains additional citation context not shown here]

J. S. B. Mitchell, Guillotine subdivisions approximate polygonal subdivisions: A simple new method for the geometric k-MST problem, Proc. 7th ACM-SIAM Sympos. Discrete Algorithms (1996), pp. 402--408.

.... 32 and heur = 16s 38, for a performance ratio arbitrarily close to 4 3 for large s. 5.6 PTAS. Here we outline a PTAS for the problem of minimizing a weighted average of the two cost criteria: length and number of turns. Our technique is based on using the theory of m guillotine subdivisions [28], properly extended to handle turn costs. We prove the following result: Theorem 5.10. For any xed 0, there is a (1 ) approximation algorithm with running time O(2 h N O(C) that computes a milling tour for an integral orthogonal polygon P with h holes, where the cost of the tour ....

....h N O(C) that computes a milling tour for an integral orthogonal polygon P with h holes, where the cost of the tour is its length plus C times the number of (90 degree) turns, and N is the number of pixels in P . Proof. sketch) Let T be a minimum cost tour. Following the notation of [28], we rst use the main structure theorem to show that there exists an m guillotine subdivision, TG , obtained from T , with length at most (1 1 m ) times the length of T . Note that part of TG may lie outside the pocket P , since we added m spans to make it m guillotine. We then ....

J. S. B. Mitchell. Guillotine subdivisions approximate polygonal subdivisions: A simple polynomial-time approximation scheme for geometric TSP, k-MST, and related problems. SIAM J. Comput., 28(4), pp. 1298{ 1309, 1999.

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J. S. B. Mitchell. Guillotine subdivisions approximate polygonal subdivisions: A simple polynomial-time approximation scheme for geometric TSP, k-MST, and related problems. SIAM J. Comput., 28(4):1298--1309, 1999.

No context found.

J. S. B. Mitchell. Guillotine subdivisions approximate polygonal subdivisions: A simple polynomial-time approximation scheme for geometric TSP, k-MST, and related problems. SIAM J. Comput., 28:1298--1309, 1999.

No context found.

J. S. B. Mitchell. Guillotine subdivisions approximate polygonal subdivisions: A simple new method for the geometric k-MST problem. In Proc. 7th ACM-SIAM Sympos. Discrete Algorithms, pages 402--408, 1996.

No context found.

J. S. B. Mitchell. Guillotine subdivisions approximate polygonal subdivisions: A simple new method for the geometric k-MST problem. In Proceedings of the 7th Annual ACM-SIAM Symposium on Discrete Algorithms, pages 402--408, 1996.

No context found.

J. S. B. Mitchell. Guillotine subdivisions approximate polygonal subdivisions: A simple polynomial-time approximation scheme for geometric TSP, k-MST and related problems. SIAM Journal on Computing, 28(4):1298--1309, 1999.

No context found.

J. S. B. Mitchell. Guillotine subdivisions approximate polygonal subdivisions: A simple new method for the geometric k-MST problem. In Proceedings of the 7th Annual ACM-SIAM Symposium on Discrete Algorithms, pages 402--408, 1996.

No context found.

J. S. B. Mitchell. Guillotine subdivisions approximate polygonal subdivisions: A simple polynomial-time approximation scheme for geometric TSP, k-MST and related problems. SIAM Journal on Computing, 28(4):1298--1309, 1999.

No context found.

J. S. B. Mitchell. Guillotine subdivisions approximate polygonal subdivisions: Part II--a simple polynomial-time approximation scheme for geometric TSP, k-MST, and related problems. SIAM J. Comp., 28:1298--1309, 1999.

No context found.

J. S. B. Mitchell. Guillotine subdivisions approximate polygonal subdivisions: A simple polynomial-time approximation scheme for geometric TSP, k-MST, and related problems. SIAM Journal on Computing, 28:1298-1309, 1999.

No context found.

J. Mitchell. Guillotine subdivisions approximate polygonal subdivisions: A simple PTAS for geometric k-MST, TSP, and related problems. SIAM J. Comp., 28, 1999.

No context found.

J.S.B. Mitchell. Guillotine subdivisions approximate polygonal subdivisions: Part II---a simple polynomial-time approximation scheme for geometric TSP, k-MST, and related problems. SIAM J. Computing, 28:1298--1309, 1999.

No context found.

J. S. B. Mitchell. Guillotine subdivisions approximate polygonal subdivisions: a simple polynomial-time approximation scheme for geometric TSP, k-MST, and related problems. SIAM J. Comput., 28:1298-1309, 1999.

No context found.

J.S.B. Mitchell, Guillotine Subdivisions Approximate Polygonal Subdivisions: Part II { A simple polynomialtime approximation scheme for geometric TSP, kMST, and related problems, SIAM J. Computing, 28, pp. 1298-1309, 1999. 4

No context found.

J.S.B. Mitchell, Guillotine Subdivisions Approximate Polygonal Subdivisions: Part II { A simple polynomial-time approximation scheme for geometric TSP, k-MST, and related problems, SIAM J. Computing, 28, pp. 1298-1309, 1999.

No context found.

J. Mitchell, Guillotine Subdivisions Approximate Polygonal Subdivisions: A Simple Polynomial-Time Approximation Scheme for Geometric TSP, k-MST, and Related Problems, SIAM Journal on Computing 28, 1298-1309 (1999).

No context found.

J. Mitchell, "Guillotine subdivisions approximate polygonal subdivisions: a simple new method for the geometric k-MST problem," in Proc. of ACM-SIAM Symposium on Discrete Algorithms, 1996, pp. 402--408.

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J. S. B. Mitchell. Guillotine subdivisions approximate polygonal subdivisions: A simple new method for the geometric k-MST problem. In Proceedings of the Seventh Annual ACMSIAM Symposium on Discrete Algorithms, pages 402--408, Atlanta, Georgia, 28--30 January 1996.

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