G. N. Frederickson and J. JaJa. On the relationship between the biconnectivity augmentation and Traveling Salesman Problem. Theoret. Comput. Sci., 19(2):189--201, 1982.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....bound is tight, by using the same family of graphs as our conjectured worst case instance for the Held Karp heuristic. Furthermore, they include a result of Cunningham that shows that the optimal solution to the Subtour LP has cost no greater than that of the minimum cost 2Frederickson and Ja Ja [10] showed earlier that this was true for planar graphs. CHAPTER 2. THE SYMMETRIC CASE WITH TRIANGLE INEQUALITY k nodes (b) c) 1 2 Figure 2.3: Conjectured Worst Case for the Held Karp Heuristic 37 biconnected graph. We draw an additional connection to the minimum cost biconnected graph ....

G.N. Frederickson, J. Ja'Ja' (1982). On the relationship between the biconnectivity augmentation and traveling salesman problems. Theoret. Cornput. Sci. 19, 189-201.

....approximate solutions to NP hard problems. These algorithms typically involve a short cutting step where the triangle inequality is used to bound the cost of the obtained solution. Examples include Christofides heuristic for the traveling salesperson problem [3] biconnectivity augmentation [8], approximate weighted matching [11] prize collecting traveling salesperson [2] and bounded degree subgraphs which have low weight and small bottleneck cost [16] A question of general interest is how to obtain improved approximation algorithms for such problems when the points come from a ....

G. N. Frederickson and J. J' aJ' a, On the relationship between the biconnectivity augmentation and traveling salesman problems, Theoret. Comp. Sci., 19 (1982), pp. 189--201.

....running time (a constant factor provided that d and are constant) one can also obtain a deterministic (1 ) approximation. Since for any set of points in a metric space the minimum cost biconnected graph G spanning this set is a minimum cost two edge connected graph spanning it (see, e.g. FJ82] our PTAS yields also the corresponding PTAS for the Euclidean minimum cost two edge connectivity. We present a fast randomized polynomial time approximation scheme for nding a minimum cost k edge connected multigraph spanning a set of points in a d dimensional Euclidean space. The running ....

....d r) 4. 3 Exchanging two edge connectivity for biconnectivity In this section we observe that any two edge connected multigraph on a set of points in an arbitrary metric space can be easily transformed into a biconnected graph on the same point set without any increase in the cost (cf. also [FJ82, Section 3] The proof of the following theorem is in the Appendix. Theorem 5 A two edge connected multigraph on a nite set of points in a metric space can be transformed in linear time into a biconnected graph on the same set of points without increasing the total cost. 5 Connectivity in ....

G. N. Frederickson and J. JaJa. On the relationship between the biconnectivity augmentation and Traveling Salesman Problem. Theoretical Computer Science, 19(2):189-201, 1982.

....special form of the Euclidean Traveling Salesman Problem to it. The ETSP involves nding the minimum Euclidean length circuit through a given set of points on the Euclidean plane. The Traveling Salesman Problem has always been closely associated with 2 connected network problems (see, for example [4]) as a TSP circuit through a set of points is also a feasible though not necessarily optimal solution to 2SNPP for that set of points. We rst show that under special conditions the TSP solution is in fact the optimal solution to 2SNPP. Lemma 2 Let Z be a set of k integer lattice points. ....

G.N. Frederickson and J. Ja'Ja', 1982. On the relationship between the biconnectivity augmentation and traveling salesman problems, Theor. Comp. Sci. 19, pp. 189-201.

.... (that is, when the weight function satisfies the triangle inequality) and uniform weights (that is, when the weight of every edge is the same) and there has been extensive recent research on approximation algorithms for this and related problems with uniform weights and with metric weights, see [1, 3, 7, 9, 10]. For metric weights, Khuller and Raghavachari [10] developed a (2 2(k Gamma 1) n) approximation algorithm for the minimum weight k node connected subgraph problem. For uniform weights, Cheriyan and Thurimella [3] gave a (1 1 k ) approximation algorithm. We design approximation algorithms ....

G.L.Frederickson and J.Ja'Ja', "On the relationship between the biconnectivity augmentation and traveling salesman problems," Theor. Comp. Sci. 19 (1982), 189--201.

....connectivity degree can be seen as a measure of the tolerance to failures occurring in the nodes of the network. For more motivations see e.g. 8] The k MWVC problem is NP hard, for k 2, even in presence of unweighted edges. Various approximation algorithms for its solution have been proposed in [10, 14, 4]. An approximation algorithm with factor ff, is a polynomial time algorithm that returns a solution whose cost is within a factor ff from the cost of an optimal solution. The best approximation algorithm for the k MWVC problem is due to Ravi and Williamson [14] and it obtains a factor 2H(k) where ....

....di Salerno, 84081 Baronissi (Italy) e mail: fauletta,parenteg dia.unisa.it. 2 an approximation algorithm that obtains a factor not greater than 2 2(k Gamma 1) n when the edge weights satisfy the triangle inequality. Better algorithms have been proposed for particular values of k. In [14] and [4] two algorithms are described that obtain a factor 3 for the case k = 2. In [10] Khuller and Raghavachari improved on this result obtaining a factor of 2 1=n, where n is the number of vertices in the graph. Recently, Penn and Shasha Krupnik have given in [13] announced to appear) a 2 ....

G. N. Frederickson, J. J'aJ'a, On the relationship between the biconnectivity augmentation and travelling salesman problem Theoretical Computer Science, 19(2), 189--201, (1982).

....approximate solutions to NP hard problems. These algorithms typically involve a short cutting step where the triangle inequality is used to bound the cost of the obtained solution. Examples include Christofides heuristic for the traveling salesperson problem [3] biconnectivity augmentation [8], approximate weighted matching [11] prize collecting traveling salesperson [2] and bounded degree subgraphs which have low weight and small bottleneck cost [16] A question of general interest is how to obtain improved approximation algorithms for such problems when the points come from a ....

G. N. Frederickson and J. J'aJ'a. On the relationship between the biconnectivity augmentation and traveling salesman problems. Theoret. Comp. Sci., 19 (2): 189--201, 1982.

....E) such that the cost c(E 0 ) P e2E 0 c e is minimum. Khuller Vishkin [6] pointed out that a 2 approximation guarantee can be obtained via the weighted matroid intersection algorithm. When the edge weights satisfy the triangle inequality (i.e. when c is a metric) Frederickson and Ja ja [3] gave a 1.5 approximation algorithm, and this is still the best approximation guarantee known. In fact, they proved that the TSP tour found by the Christofides heuristic achieves an approximation guarantee of 1.5; there are simpler proofs of this result due to Cunningham (see [8] and Goemans ....

G. L. Frederickson and J. Ja'Ja', "On the relationship between the biconnectivity augmentation and traveling salesman problems," Theor. Comp. Sci. 19 (1982), 189--201.

.... min size k outconnected subgraph problem) and for metric costs (i.e. edge costs satisfying the triangle inequality) There has been extensive recent research on approximation algorithms for NP hard network design problems with uniform costs and with metric costs, see the survey in [8] and see [1, 3, 5, 9], etc. For the uniform cost multi root problem, we improve the approximation guarantee from 2q to minf2; k 2q Gamma1 k g (this implies a 1 1 k approximation algorithm for the uniform cost single root problem) and for the metric cost multi root problem, we improve the approximation guarantee ....

G.L.Frederickson and J.Ja'Ja', "On the relationship between the biconnectivity augmentation and traveling salesman problems," Theor. Comp. Sci. 19 (1982), 189--201.

....of 2 1=n. Not much more is known about the k vertex connectivity problem for the special case when the weights satisfy the triangle inequality. For k = 2, it is easy to show that the TSP algorithm of doubling the minimum spanning tree has a performance guarantee of 2. Frederickson and J aJ a [10] proved that Christofides algorithm [4] for the TSP problem) has a performance guarantee of 1.5 for the minimum weight biconnectivity problem as well. The analysis for the biconnectivity algorithm is more complicated since the relationship between the weight of a minimum weight matching and the ....

....incident vertices. We describe an algorithm that finds a k vertex connected subgraph whose weight is within a factor 2 2(k Gamma 1) n of a minimum weight k vertex connected subgraph in G. Previously, an algorithm achieving a factor of 1:5 for the case k = 2 was given by Frederickson and J aJ a [10]. Many algorithms for graphs satisfying triangle inequality are based on simple ideas for shortcutting. For example, taking two copies of a minimum spanning tree and then shortcutting it suitably yields a 2 approximation for 2 connectivity. One may wonder whether such a simple algorithm exists for ....

G. N. Frederickson and J. J'aJ'a, On the relationship between the biconnectivity augmentation and traveling salesman problems, Theoret. Comput. Sci., 19 (2), pp. 189--201, (1982).

....degree can be seen as the tolerance to faults occurring in the vertices. For more motivations see e.g. 8] The k MWVC problem is NP hard [7] even in presence of unweighted edges or even when the weights can be just 1 or 2. Various approximation algorithms for its solution have been proposed, [9, 12, 5], and their approximation factors are summarized in Figure 1. An approximation algorithm with factor ff, is a polynomial time algorithm that returns a solution whose cost is no greater than ff times the cost of an optimial solution. The best approximation algorithm for the general case is due to ....

....whose cost is no greater than ff times the cost of an optimial solution. The best approximation algorithm for the general case is due to Ravi and Williamson [12] and it obtains a factor 2H(k) where H(k) P k i=1 1=i. Better algorithms have been proposed for particular values of k. In [12] and [5] two algorithms are described that obtain a factor 3 for the case k = 2. In [9] Khuller and Raghavachari improved on this result obtaining a factor of 2 1=n, where n is the number of vertices in the graph. Recently, Penn and Nutow in [11] have obtained an approximation factor 2. They have also ....

G. N. Frederickson, J. J'aJ'a, On the relationship between the biconnectivity augmentation and traveling salesman problem Theoretical Computer Science, 19(2), 189--201, (1982).

....This section shows that the well known 4 3 conjecture for the metric TSP (due to Cunningham (1986) and others) implies that there is a 4 3 approximation algorithm for a minimum size 2 ECSS, see Theorem 18. Almost all of the results in this section are well known, except possibly Fact 13, see [1, 3, 5, 7, 11, 13]. The details are included to make the paper self contained. In the metric TSP (traveling salesman problem) we are given a complete graph G 0 = Kn and edge costs c 0 that satisfy the triangle inequality (c 0 vw c 0 vu c 0 uw ; 8v; w; u 2 V ) The goal is to compute c 0 TSP , the ....

....E) such that the cost c(E 0 ) P e2E 0 c e is minimum. Khuller Vishkin [8] pointed out that a 2 approximation guarantee can be obtained via the weighted matroid intersection algorithm. When the edge costs satisfy the triangle inequality (i.e. when c is a metric) Frederickson and Ja Ja [5] gave a 1.5 approximation algorithm, and this is still the best approximation guarantee known. In fact, they proved that the TSP tour found by the Christofides heuristic achieves an approximation guarantee of 1.5. Simpler proofs of this result based on Theorem 16 were found later by Cunningham ....

G. L. Frederickson and J. Ja'Ja', "On the relationship between the biconnectivity augmentation and traveling salesman problems," Theor. Comp. Sci. 19 (1982), 189--201.

....of 2 1=n. Not much more is known about the k vertex connectivity problem for the special case when the weights satisfy the triangle inequality. For k = 2, it is easy to show that the TSP algorithm of doubling the minimum spanning tree has a performance guarantee of 2. Frederickson and J aJ a [9] proved that Christofides algorithm [4] for the TSP problem) has a performance guarantee of 1.5 for the minimum weight biconnectivity problem as well. The analysis for the biconnectivity algorithm is more complicated since the relationship between the weight of a minimum weight matching and the ....

....triangle inequality. We describe an algorithm that finds a k vertex connected subgraph whose weight is within a factor 2 2(k Gamma1) n of a minimum weight k vertex connected subgraph in G. Previously, an algorithm achieving a factor of 1:5 for the case k = 2 was given by Frederickson and J aJ a [9]. Many algorithms for graphs satisfying triangle inequality are based on simple ideas for shortcutting. For example, taking two copies of a minimum spanning tree and then shortcutting it suitably yields a 2 approximation for 2 connectivity. One may wonder whether such a simple algorithm exists for ....

G. N. Frederickson and J. J'aJ'a, On the relationship between the biconnectivity augmentation and traveling salesman problems, Theoret. Comput. Sci., 19 (2), pp. 189-- 201, (1982).

....approximation factor in O(m n log n) time by [KT93] Actually, the problem solved is of increasing the connectivity of an existing network from 1 to 2, but it can be used for an approximation factor of 3 as well. There is extensive literature on the k edge connected spanning subgraph problem [FJ82, GB93, GMS92, MK89, SWK69], for the case where the edge weights satisfy the triangle inequality (and the underlying feasibility graph is a clique) For the case of edge or vertex connectivity when the underlying feasibility graph is a clique (any edge can be added at unit cost) one can solve the problem of the smallest ....

G. N. Frederickson and J. J'aJ'a, "On the relationship between the biconnectivity augmentation and traveling salesman problems," Theoretical Computer Science, Vol. 19, No. 2, pp. 189--201, (1982).

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G. N. Frederickson and J. JaJa. On the relationship between the biconnectivity augmentation and Traveling Salesman Problem. Theoret. Comput. Sci., 19(2):189--201, 1982.

No context found.

G.N. Frederickson and J. JaJa, On the relationship between the biconnectivity augmentation and traveling salesman problems, Theoretical Computer Science 19(2), 189-201, 1982.

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G.N. Frederickson and J. JaJa. On the relationship between the biconnectivity augmentation and traveling salesman problems. Theoretical Computer Science, 19(2):189--201, 1982.

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G.N. Frederickson and J. J'aJ'a (1982), "On the relationship between the biconnectivity augmentation and traveling salesman problem," Theoretical Computer Science 19, 189201.

No context found.

G. N. Frederickson and J. JaJa. On the relationship between the biconnectivity augmentation and Traveling Salesman Problem. Theoret. Comput. Sci., 19(2):189--201, 1982.

No context found.

G. L. Frederickson and J. Ja'Ja', "On the relationship between the biconnectivity augmentation and traveling salesman problems," Theor. Comp. Sci. 19 (1982), 189--201.

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G. N. Frederickson and J. J'aJ'a, On the relationship between biconnectivity augmentation and the travelling salesman problem, Theoretical Computer Science, 19 (1982), pp. 189-201.

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