S. Arora. Polynomial time approximation schemes for Euclidean traveling salesman and other geometric problems. Journal of the ACM, 45(5): 753--782, 1998. They may not be simple cycles in H, but be glued with a set of other cycles from previous recursive calls.  Home/Search   Document Details and Download   Summary   Related Articles   Check

....reference when comparing heuristics for ESTP. The ratio of the length jSMT (Z)j of an SMT to the length jMST (Z)j of an MST spanning the same set of terminals Z cannot be smaller than ae = 3=2 0:866 [18] An MST is therefore at most 2 p Gamma 1 15:47 longer than an SMT. Recently Arora [3] showed that ESTP belongs to a class of NP hard problems which have a polynomial time approximation scheme, i.e. we can find a solution within a factor 1 ffl from optimum in polynomial time, for any fixed ffl 0. There has been a major breakthrough in the development of exact algorithms for ....

S. Arora. Polynomial Time Approximation Schemes for Euclidean TSP and other Geometric Problems. In Proc. 37th Annual Symp. on Foundations of Computer Science, pages 2--13, 1996.

....can be developed by exploiting the geometry of the problem. There have been several significant developments on geometric graph problems over the last few years, of which the most exciting is an n log n time (1 ) approximation algorithm for the Euclidean traveling salesperson problem [38]; see also [38, 39] We refer the reader to [44] for a survey on approximation algorithms for such geometric optimization problems. ....

....developed by exploiting the geometry of the problem. There have been several significant developments on geometric graph problems over the last few years, of which the most exciting is an n log n time (1 ) approximation algorithm for the Euclidean traveling salesperson problem [38] see also [38, 39]. We refer the reader to [44] for a survey on approximation algorithms for such geometric optimization problems. ....

S. Arora, Polynomial time approximation schemes for Euclidean TSP and other geometric problems, Proc. 37th Annu. IEEE Sympos. Found. Comput. Sci., 1996, pp. 2--11.

....this note, we show 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 ....

....CCR 9504192. While 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 ....

S. Arora, Polynomial time approximation schemes for Euclidean TSP and other geometric problems, Manuscript, March 30, 1996.

....status of the GST problem. 4.1.3 k Minimum Spanning Tree R 0 (e 2 E) the k Minimum Spanning Tree Problem (k MST) asks to find a tree spanning at least k nodes in V of minimum cost. There has been a substantial amount of research on providing upper and lower bounds for the k MST problem (see [26, 13, 2]; currently the best results are those in [2, 3] for both the geometric and nongeometric case) Given an instance I of the k MST problem we can transform it to an instance I ) of BCCMED as follows: Once more, the graph G is identical to G specified in I . The c costs in I are ....

....Tree R 0 (e 2 E) the k Minimum Spanning Tree Problem (k MST) asks to find a tree spanning at least k nodes in V of minimum cost. There has been a substantial amount of research on providing upper and lower bounds for the k MST problem (see [26, 13, 2] currently the best results are those in [2, 3] for both the geometric and nongeometric case) Given an instance I of the k MST problem we can transform it to an instance I ) of BCCMED as follows: Once more, the graph G is identical to G specified in I . The c costs in I are equal to the respective c costs given in I . The ....

S. Arora. Polynomial-time approximation schemes for euclidean TSP and other geometric problems. Journal of the ACM, 45(5):753--782, 1998.

....for the performance of other approximation algorithms, since there is little point in constructing algorithms which produce solutions worse than the MST. It was for a long time an open problem whether there exist approximation algorithms with performance ratios strictly less than . Arora [1] showed that there exists a polynomial time approximation scheme for the Euclidean travelling salesman problem and other geometric problems among these the Euclidean Steiner tree problem. This means that we can find in polynomial time (in the number of terminals but not in 1=ffl) a solution ....

S. Arora. Polynomial Time Approximation Schemes for Euclidean TSP and other Geometric Problems. In Proc. 37th Ann. Symp. on Foundations of Computer Science, pages 2--13, 1996.

....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 ....

S. Arora. Polynomial time approximation schemes for Euclidean TSP and other geometric problems. In Proc. 37th Annu. IEEE Sympos. Found. Comput. Sci., pages 2--11, 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 ....

S. Arora. Polynomial time approximation schemes for Euclidean TSP and other pages 2--11, 1996.

....schemes have been invented for this reason. Often the wonderful PCP theorem of Arora et al. ALMSS92] shows that no such approximation exists. But sometimes they do. Let s look at some recent examples, taken from some recent major conferences such as STOC, FOCS and SODA, etc. Arora [Ar96] gave a O(n 3000 ) PTAS for EUCLIDEAN TSP Chekuri and Khanna [CK00] gave a O(n 12(log(1= PTAS for MULTIPLE KNAPSACK Shamir and Tsur [ST98] gave a O(n 1) PTAS for MAXIMUM SUBFOREST Chen and Miranda [CM99] gave a O(n (3mm ) m 1 ) PTAS for GENERAL MULTIPROCESSOR ....

....GENERAL MULTIPROCESSOR JOB SCHEDULING Erlebach et al. EJS01] gave a O(n ( 2 1) 2 2) PTAS for MAXIMUM INDEPENDENT SET for geometric graphs. Table 1 below calculates some running times for these PTAS s with a 20 error. Reference Running Time for a 20 Error Arora [Ar96] O(n 15000 Chekuri and Khanna [CK00] O(n 9;375;000 Shamir and Tsur [ST98] O(n 958;267;391 Chen and Miranda [CM99] O(n 60 (4 Processors) Erlebach et al. EJS01] O(n 523;804 Table 1. The Running Times for Some Recent PTAS s with 20 Error. Now, by anyone s measure, a ....

S. Arora, "Polynomial Time Approximation Schemes for Euclidean TSP and Other Geometric Problems," In: Proceedings of the 37th IEEE Symposium on Foundations of Computer Science, 1996, pp. 2--12.

....where ff denotes the approximation ratio of the best classical TSP heuristic. It was shown in [4] that the (2 Gamma ffl) approximation is NP hard even in the case when only two targets are moving. Thus, combining our approach with the polynomial time approximation scheme for Euclidean TSP [1] yields almost optimal (2 ffl) approximation algorithms for Moving Target TSP when enough of the targets are stationary. Next, in Section 3, we shift our attention to selected variants of Moving Target TSP with Resupply where the pursuer must return to the origin for resupply after ....

S. Arora, Polynomial time approximation schemes for euclidean TSP and other geometric problems, in Proc. IEEE Symp. Foundations of Computer Science, 1996, pp. 2--11.

.... on (exponential time) algorithms for the Euclidean and rectilinear planes with reasonable running times for reasonably small sets (see e.g. 9] and [27] Recently, a polynomial time approximation scheme has been found for the Steiner problem in any normed space of xed nite dimension [2]. The second aspect is geometrical, and nitely solvable in e.g. the cases of Euclidean [24] and rectilinear planes [13] However, the properties of Steiner points in these two planes are very di erent. In particular, in the Euclidean plane Steiner points must have degree three, while in the ....

S. Arora, Polynomial time approximation schemes for Euclidean traveling salesman and other geometric problems, J. ACM 45 (1998), 753-782.

.... locatio with metric spaces [9] The latter problem (and hence our problem(s) admits constant ratio approximation algorithms [9, 21] Moreover, the case in which the service cost is the Euclidean distance (in our model this corresponds to choose c = 1 4) has a polynomial time approximation scheme [1] (in short PTAS) The same paper also gives (with the same technique) a PTAS for the k median problem, while constant factor approximation algorithms for the metric version are presented in [4, 14] Notice that the MAX GAIN problem is a special case of the k median one (while in the facility ....

S. Arora, P. Raghavan, and S. Rao. Polynomial Time Approximation Schemes for Euclidean k-medians and related problems. In Proc. of the lSth ACM STOC, pages 106 113, 1998.

....of unless P = NP. Since these initial works on probabilistically checkable proofs, we have seen a large number of outstanding papers improving the proof systems and getting stronger hardness of approximation results. H astad [H as97] gets tight results for some approximation problems. Arora [Aro98] after failing to achieve lower bounds for traveling salesman in the plane, has developed a polynomial time approximation algorithm for this and related problems. A series of results due to Cai, Condon, Lipton, Lapidot, Shamir, Feige and Lov asz [CCL92, CCL90, CCL91, Fei91, LS91, FL92] have ....

S. Arora. Polynomial time approximation schemes for Euclidean traveling salesman and other geometric problems. Journal of the ACM, 45(5):753-782, September 1998.

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S. Arora, P. Raghavan, and S. Rao. Polynomial time approximation schemes for the euclidean k-medians problem. In Proceedings of the 30th Annual ACM Symposium on Theory of Computing, pages 106-113, 1998.

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S. Arora. Polynomial-time approximation schemes for Euclidean TSP and other geometric problems. JACM 45(5) 753-782, 1998.

....return an almost perfect matching as a solution, which [ST93] could not. Previous work on PTAS s. PTAS s are known for relatively few problems; knapsack [IK75] and bin packing [FL81, KK82] are the only two well known examples. Recently, a PTAS has also been discovered for Euclidean TSP (Arora [A96]) for Multiple Knapsack Problem (Chekuri and Khanna [CK00] and some Scheduling Problems (Afrati et al. [A99] In fact, a result by Arora, Lund, Motwani, Sudan and Szegedy 5 [ALM 92] shows that if P 6= NP, then PTAS s do not exist for a large body of problems the so called MAX SNP hard ....

S. Arora. Polynomial Time Approximation Schemes for Euclidean Traveling Salesman and other Geometric Problems. JACM 45(5): 753-782, 1998.

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S. Arora. Polynomial time approximation schemes for Euclidean traveling salesman and other geometric problems. Journal of the ACM, 45(5): 753--782, 1998. They may not be simple cycles in H, but be glued with a set of other cycles from previous recursive calls.

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S. Arora. Polynomial time approximation schemes for Euclidean traveling salesman and other geometric problems. J. Assoc. Comput. Mach., 45(5):753--782, 1998.

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Arora, S., Polynomial time approximation schemes for Euclidean traveling salesman and other geometric problems, JACM, 45(5), 753, 1998.

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S. Arora. Polynomial time approximation schemes for Euclidean TSP and other geometric problems. In Proc. 37th Annu. IEEE Sympos. Found. Comput. Sci., pages 2--11, 1996.

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S. Arora. Polynomial time approximation schemes for Euclidean TSP and other geometric problems. In Proc. 37th Annu. IEEE Sympos. Found. Comput. Sci., pages 2--11, 1996. 47

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Sanjeev Arora. Polynomial time approximation schemes for Euclidean traveling salesman and other geometric problems. Journal of the ACM, 45(5):753-782, 1998.

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S. Arora. Polynomial time approximation schemes for Euclidean traveling salesman and other geometric problems. Journal of the ACM, 45(5):753--782, 1998. JAN REMY AND ANGELIKA STEGER

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S. Arora. Polynomial time approximation schemes for Euclidean TSP and other geometric problems. In 37th Annual Symposium on Foundations of Computer Science, pages 2--11, 1996.

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S. Arora. Polynomial time approximation schemes for Euclidean traveling salesman and other geometric problems. Journal of the ACM, 45(5):753--782, 1998.

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S. Arora. Polynomial time approximation schemes for Euclidean TSP and other geometric problems. In 37th Annual Symposium on Foundations of Computer Science, pages 2--11, 1996.

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S. Arora. Polynomial-time approximation schemes for euclidean TSP and other geometric problems. JACM, 45(5):753--782, 1998.

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S. Arora. Polynomial time approximation schemes for euclidean traveling salesman and other geometric problems. Journal of the ACM, 45(5):753--782, 1998.

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Arora, S. , 1998, "Polynomial Time Approximation Schemes for Euclidean Traveling Salesman and Other Geometric Problems." Journal of the ACM 45(5), p 753--782.

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S. Arora. Polynomial-time approximation schemes for euclidean TSP and other geometric problems. Proc. of 37th FOCS, pages 2-12, 1996.

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Sanjeev Arora "Polynomial Time Approximation Schemes for Euclidean Traveling Salesman and other Geometric Problems," J. ACM 45(5), pp. 753-782, 1998.

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S. Arora. Polynomial-time approximation schemes for Euclidean traveling salesman and other geometric problems. J. of the ACM, 45(5):753--782, 1998.

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S. Arora, Polynomial time approximation schemes for Euclidean TSP and other geometric problems. In: Proceedings of the 37th IEEE Symposium on Foundations of Computer Science, 1996, pp. 2-12.

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S. Arora. Polynomial-time approximation schemes for Euclidean TSP and other geometric problems. Journal of the ACM, 45:753-782, 1998. 14

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Sanjeev Arora "Polynomial Time Approximation Schemes for Euclidean Traveling Salesman and other Geometric Problems," J. ACM 45(5), pp. 753-782, 1998.

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S. Arora. Polynomial time approximation schemes for euclidean tsp and other geometric problems. J. Assoc. Comput. Mach., 45(5):753--782, Sep 1998.

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S. Arora. Polynomial time approximation schemes for euclidean tsp and other geometric problems. J. Assoc. Comput. Mach., 45(5):753-782, Sep 1998.

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Sanjeev Arora. Polynomial time approximation schemes for euclidean tsp and other geometric problems. In Proc. of the 37th Annual Symposium on Foundations of Computer Science, 1996.

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S. Arora. Polynomial-time approximation schemes for Euclidean TSP and other geometric problems. JACM, 45:753--782, 1998.

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S. Arora. Polynomial time approximation schemes for Euclidean traveling salesman and other geometric problems. J. ACM, 45:753-782, 1998.

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S. Arora, Polynomial-time approximation schemes for Euclidean TSP and other geometric problems, JACM 45, pp. 753-782, 1998.

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Arora, S.: Polynomial Time Approximation Scheme for Euclidean TSP and Other Geometric Problems, in Proc. 37th Ann. IEEE Symp. on Foundations of Comput. Sci., IEEE Computer Society, 2-11, 1996.

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S. Arora. Polynomial-time approximation schemes for Euclidean TSP and other geometric problems. Journal of the ACM, 45:753-782, 1998.

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S. Arora, "Polynomial-time approximation schemes for Euclidean traveling salesman and other geometric problems," Journal of the ACM 45, no. 5, p. 753, 1998.

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S. Arora. Polynomial time approximation schemes for Euclidean TSP and other geometric problems. In Proceedings of the 37th IEEE Symposium on Foundations of Computer Science, 1996.

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S. Arora, Polynomial-time approximation schemes for Euclidean TSP and other geometric problems, Journal of the ACM 45, pp. 753-782, 1998.

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S. Arora, Polynomial Time Approximation Schemes for Euclidean TSP and other Geometric Problems, Proceedings of the 37th Symposium on the Foundations of Computer Science, 2-11 (1996).

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S. Arora, Polynomial time approximation schemes for Euclidean TSP and other geometric problems, Manuscript, March 30, 1996.

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S. Arora. Polynomial-time approximation scheme for Euclidean TSP and other geometric problems. In Proceedings of the Symposium on Foundations of Computer Science, pages 2--11, 1996.

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Sanjeev Arora. Polynomial time approximation schemes for Euclidean traveling salesman and other geometric problems. Journal of the ACM, 45(5):753--782, 1998.

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S. Arora, "Polynomial Time Approximation Schemes for Euclidean TSP and Other Geometric Problems," In: Proceedings of the 37th IEEE Symposium on Foundations of Computer Science, 1996.

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