S. Khuller, B. Raghavachari, and N. Young. Low degree spanning tree of small weight. SIAM J. Computing, 25:355--368, 1996.  Home/Search   Document Details and Download   Summary   Related Articles   Check

....4 3 for K = 3 and K = 4 respectively. In as early as 1984, Papadimitriou and Vazirani [15] asked whether the geometry of the Euclidean case (besides the triangle inequality) can be exploited to prove better approximation factors for bounded degree spanning trees. Khuller, Raghavachari, and Young [10] took an in depth look into this question and managed to achieve factors 3 2 and 5 4 for K = 3 and K = 4 respectively in the plane. Since then, no improvements have been made, despite frequent references to their work [2, 3, 7, 8, 11, 16, 17] We report the first progress in 8 years: in the ....

....3 and 4 we present the new recursive algorithms and analyses. The study of these worst case ratios in d dimensional Euclidean space is even more vital, because the maximum degree of an MST can be much larger (a constant that depends exponentially on d [18] In their paper, Khuller et al. [10] analyzed a simple algorithm and proved a remarkable 5 3 upper bound for degree 3 spanning trees in any number of dimensions. In Section 5, we mention how the bound can be reduced slightly to # 6 1.633 using essentially the same algorithm. 2. KHULLER, RAGHAVACHARI, AND YOUNG S APPROACH To ....

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S. Khuller, B. Raghavachari, and N. Young. Low degree spanning trees of small weight. SIAM J. Comput., 25:355--368, 1996.

....guarantees. Papadimitriou and Vazirani [23] studied the Euclidean version of this problem for the case when . Monma and Suri [22] showed that for any set of points in the plane, a minimum spanning tree with can be constructed efficiently. Khuller, Raghavachari and Young [17] gave approximation algorithms with performance guarantees of 3 2 and 5 4 for respectively for points in the plane. They also presented an approximation algorithm with a performance guarantee of 5 3 for point sets in higher dimensions when . Iwainsky et al. 16] formulated a ....

....Another interesting question is to investigate the extension of our work to higher connected degree constrained networks without the triangle inequality. In other follow up to our work, the special case of the (U DEGREE, E TOTAL COST, SPANNING TREE) problem in the Euclidean plane was addressed in [17], and improvements to the short cutting scheme of Proposition 6.1 using network flow techniques are presented in [10] Acknowledgments: We thank the referee for several valuable suggestions. We gratefully acknowledge helpful conversations with M. X. Goemans, P. N. Klein, G. Konjevod, S. Krumke, ....

S. Khuller, B. Raghavachari and N. Young, "Low-Degree Spanning Trees of Small Weight," SIAM J. Computing, Vol. 25 No. 2, pp. 355--368, April 1996.

....with bounded degree. Khuller and Vishkin [KV94] Khuller and Raghavachari [KR95] Cheriyan and Thurimella [CT96a] and Khuller [Khu97] approximate the smallest and construct a small k (edge) connected spanning subgraph. Khuller and Raghavachari [KR95] Khuller, Raghavachari, and Young [KRY96] and Goemans and Williamson [GW96] present results for the more general edgeweighted case. The papers of Camerini, Galbiati, and Maffioli [CGM80] and Raghavachari [Rag97] survey related results in this area. As stated we are mainly interested in the sequential complexity of the decision ....

S. Khuller, B. Raghavachari, and N. Young. Low degree spanning trees of small weight. SIAM Journal on Computing, 25(2):355--368, 1996. A preliminary version appeared at STOC`94.

....at most d; see Raghavachari [54] and Bern and Eppstein [14] for a discussion. The salesman path problem is a subcase when d 2. Every minimum spanning tree has degree at most 5, so the problem is trivial for d 5. The case d 4 is NPhard and the status when d 3 is open. Khuller et al. [41] give 1.5and 1.25 approximations for the two problems. The techniques of Section 2 seem very applicable but there is as yet no PTAS. The author has certainly has tried to design one, and maybe others too. Vehicle Routing. This is really a large body of problems in operations research with ....

S. Khuller, B. Raghavachari, and N. Young. Low degree spanning tree of small weight. SIAM J. Computing, 25:355--368,

....performance guarantees. Papadimitriou and Vazirani [23] studied the Euclidean version of this problem for the case when d = 3; 4. Monma and Suri [22] showed that for any set of points in the plane, a minimum spanning tree with d = 5 can be constructed efficiently. Khuller, Raghavachari and Young [17] gave approximation algorithms with performance guarantees of 3 2 and 5 4 for d = 3 and d = 4 respectively for points in the plane. They also presented an approximation algorithm with a performance guarantee of 5 3 for point sets in higher dimensions when 4 d = 3. Iwainsky et al. 16] formulated ....

....Another interesting question is to investigate the extension of our work to higher connected degree constrained networks without the triangle inequality. In other follow up to our work, the special case of the (U DEGREE, E TOTAL COST, SPANNING TREE) problem in the Euclidean plane was addressed in [17], and improvements to the short cutting scheme of Proposition 6.1 using network flow techniques are presented in [10] Acknowledgments: We thank the referee for several valuable suggestions. We gratefully acknowledge helpful conversations with M. X. Goemans, P. N. Klein, G. Konjevod, S. Krumke, ....

S. Khuller, B. Raghavachari and N. Young, "Low-Degree Spanning Trees of Small Weight," SIAM J. Computing, Vol. 25 No. 2, pp. 355--368, April 1996.

....a spanning tree T of maximum degree O(B log ( n B ) and total cost at most O(log n) opt B . They generalize their ideas to Steiner trees, generalized Steiner forests and the node weighted case. Another result that is related to our work is given in a paper by Khuller, Raghavachari and Young [11]. The authors show how to compute a spanning tree of n points in the plane that has degree at most 3 (4) and weight at most 1:5 (1:25) that of a minimum weight spanning tree (without any degree constraints) While the approximation factor of O(log n) on the cost of the solution cannot be improved ....

Samir Khuller, Balaji Raghavachari, and Neal Young. Low-degree spanning trees of small weight. SIAM Journal on Computing, 25(2):355-368, April 1996.

....is polynomial time equivalent to the TSP and hence NP hard. The case k = 3 is NP hard; k = 4 is open, and when k 5 the problem can be solved optimally in polynomial 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 ....

S. Khuller, B. Raghavachari, and N. Young. Low degree spanningtree of small weight. In Proc. 26th ACM Symposium on Theory of Computing, 1994.

....a spanning tree T of maximum degree O(B log ( n B ) and total cost at most O(log n) opt B . They generalize their ideas to Steiner trees, generalized Steiner forests and the node weighted case. Another result that is related to our work is given in a paper by Khuller, Raghavachari and Young [11]. The authors show how to compute a spanning tree of n points in the plane that has degree at most 3 (4) and weight at most 1:5 (1:25) that of a minimum weight spanning tree (without any degree constraints) While the approximation factor of O(log n) on the cost of the solution cannot be improved ....

Samir Khuller, Balaji Raghavachari, and Neal Young. Low-degree spanning trees of small weight. SIAM Journal on Computing, 25(2):355-368, April 1996.

....were found to be far less tractable than Euclidean graphs, however, with the CPU time needed to solve standard instances rising from a mean of 0.2 seconds (SD = 0.7s) for n = 30 to a mean of 74.2 seconds for n = 70 (SD = 369.5s) where n is the number of vertices 1 in the graphs. Khuller et al. [6] show that for an arbitrary collection of points in the plane there exists a degree 3 spanning tree of at most 1.5 times the MST and a degree 4 spanning tree of at most 1.25 times the MST. They also give algorithms that compute these trees in O(n) time, given an MST as part of the input. These ....

....can be a very minor factor in comparison to others such as the type, quality, maintainability, speed, and corporate provider of the link in question. 1 In general, therefore, it is valuable to test and compare algorithms for the d MST on nonEuclidean graphs. In such cases, the algorithms of [6] and [7] are not applicable, while those in [5] are cumbersome. 1 We note that these real world considerations may best be tackled using a multicriteria approach but reserve such investigations for future work. 2 We describe a technique designed to achieve fruitful hybridisation of a spanning ....

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S. Khuller, B. Raghavachari, and N. Young, "Low-Degree Spanning Trees of Small Weight," SIAM Journal of Computing, vol. 25, no. 2, pp. 355--368, 1996.

....finding a traveling salesman path can be viewed as a degree constrained minimum spanning tree problem, in which each vertex has degree at most two. With this in mind, it becomes interesting to consider weaker bounds on the vertex degree of a minimum spanning tree. Khuller, Raghavachari, and Young [KRY94] consider this problem for degree bounds of three and four; they show that one can find constrained spanning trees with length 1.5 and 1.25 times the MST length, respectively. It is no constraint to require the degree to be at most five, as there always exists an MST satisfying this requirement. ....

S. Khuller, B. Raghavachari, and N. Young. Low-degree spanning trees of small weight. In Proc. 26th ACM Symp. Theory of Computing, pages 412--421, 1994.

....to higher dimensional traveling salesman problems as well, but the time bounds grow to quasipolynomial (exponential in a polylogarithmic function of n, with the exponent in the polylogarithm depending on dimension) One can also consider degree bounds between two and five. Khuller et al. [75] consider this problem for degree bounds of three and four; they show that one can find constrained spanning trees 11 Figure 3. a) 6 point minimum spanning tree. b) Guillotine partition . with length 3 2 and 5 4 times the MST length respectively. They also show that in any dimension one can ....

S. Khuller, B. Raghavachari, and N. Young. Low degree spanning trees of small weight. Proc. 26th ACM Symp. Theory of Computing, 1994, pp. 412--421.

....degree 4 in the case of planar L 1 distances. In 1984, Papdimitriou and Vazirani [4] showed that finding a minimum T 3 in those cases is NP hard and conjectured that this is also the case for finding a minimum T 4 under Euclidean distances. In their recent paper, Khuller, Raghavachari and Young [2] presented the first approximation factors smaller than 2 by showing how to obtain a T 3 and a T 4 from a T min such that w(T 3 ) w(Tmin ) 3 2 and w(T 4 ) w(Tmin ) 5 4 hold for the case of Euclidean distances in R 2 . They also showed that for any set of points in R D with d 3, ....

....implications of our results for the case of point sets in R D under L 1 distances. In particular, it follows for a planar point set that we can find a T 3 that satisfies w(T 3 ) w(Tmin ) 3 2 . So far, this is the best proven factor for this case; a direct implementation of the approach of [2] for L 1 distances yields a factor of 5 3 . 2 The Lower Bound In this section, we show that for any k and m, the ratio w(T k ) w(T k m ) between the weights of a minimum T k and a minimum T k m in a graph G with a distance function satisfying triangle inequality can be arbitrarily close to ae ....

[Article contains additional citation context not shown here]

S.Khuller, B.Raghavachari and N.Young, Low Degree Spanning Trees of Small Weight, Proc. 26th ACM Symposium on the Theory of Computing, 412-421, 1994.

....the minimum degree, a trade off has to be made here regarding the percentage increase in weight, say for example, over that of the MST or some other cut off value determined by the owner of the problem. An Euclidean version of the MW MDST problem was examined in a recent paper by Khuller et al. [4]. They show that given a set S of n points in the plane, there exists a tree spanning S of degree 3 with weight at most 1.5 times the weight of the Euclidean Minimum Spanning Tree. The graphs we have considered have random edge weights. Thus, the weight of the solution is dependent on the topology ....

S. Khuller, B. Raghavachari and N. Young, Low Degree Spanning Trees of Small Weight, 26 th ACM Symp. on the Theory of Computing, May 1994, 412--421.

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S. Khuller, B. Raghavachari, and N. Young. Low-degree spanning trees of small weight. SIAM J. Computing, 25:355--368, 1996.

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S. Khuller, B. Raghavachari, and N. Young, Lowdegree spanning trees of small weight, SIAM J. Comput. , pp. 355-368, 1996.

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S. Khuller, B. Raghavachari, and N. Young, Low-degree spanning trees of small weight, SIAM J. Comput., pp. 355-368, 1996.

....[2] Computational results for some heuristics for the general problem are presented in [14, 19, 21] Papadimitriou and Vazirani [15] raised the problem of finding the complexity of computing a minimum weight degree 4 spanning tree of points in the plane. Some geometric aspects are considered in [10, 13, 17]. In this paper, we consider modifying a given spanning tree T , to meet the degree constraints. We introduce a novel network flow based algorithm that does this optimally in the following sense: if for some algorithm a worst case performance guarantee can be proved that is solely a function of ....

.... ) times the minimum spanning tree weight always exists. Our algorithms modify the given tree by performing a sequence of adoptions. Our polynomial time algorithm performs an optimal sequence of adoptions. Adoptions have been previously used to obtain bounded degree trees in weighted graphs [10, 16, 18]. The main contributions of this paper are a careful analysis of the power of adoptions and a network flow technique for selecting an optimal sequence of adoptions. The method yields a stronger performance guarantee and may yield better results in practice. The analysis of adoptions shows that ....

[Article contains additional citation context not shown here]

S. Khuller, B. Raghavachari and N. Young. Low degree spanning trees of small weight. Proc. of 26th Annual ACM Symp. on the Theory of Computing, pp. 412--421, May 1994. To appear in SIAM J. Comput.

....[2] Computational results for some heuristics for the general problem are presented in [14, 19, 21] Papadimitriou and Vazirani [15] raised the problem of finding the complexity of computing a minimum weight degree 4 spanning tree of points in the plane. Some geometric aspects are considered in [10, 13, 17]. In this paper, we consider modifying a given spanning tree T , to meet the degree constraints. We introduce a novel network flow based algorithm that does this optimally in the following sense: if for some algorithm a worst case performance guarantee can be proved that is solely a function of ....

.... ) times the minimum spanning tree weight always exists. Our algorithms modify the given tree by performing a sequence of adoptions. Our polynomialtime algorithm performs an optimal sequence of adoptions. Adoptions have been previously used to obtain bounded degree trees in weighted graphs [10, 16, 18]. The main contributions of this paper are a careful analysis of the power of adoptions and a network flow technique for selecting an optimal sequence of adoptions. The method yields a stronger performance guarantee and may yield better results in practice. The analysis of adoptions shows that ....

[Article contains additional citation context not shown here]

S. Khuller, B. Raghavachari and N. Young. Low degree spanning trees of small weight. Proc. of 26th Annual Symp. on Theory of Computing, pp. 412--421, May 1994. To appear in SIAM Journal on Computing.

....[2] Computational results for some heuristics for the general problem are presented in [14, 19, 21] Papadimitriou and Vazirani [15] raised the problem of finding the complexity of computing a minimum weight degree 4 spanning tree of points in the plane. Some geometric aspects are considered in [10, 13, 17]. In this paper, we consider modifying a given spanning tree T , to meet the degree constraints without increasing its weight considerably. We introduce a novel network flow based algorithm that does this optimally in the following sense: if for some algorithm a worst case performance guarantee ....

.... ) times the minimum spanning tree weight always exists. Our algorithms modify the given tree by performing a sequence of adoptions. Our polynomial time algorithm performs an optimal sequence of adoptions. Adoptions have been previously used to obtain bounded degree trees in weighted graphs [10, 16, 18]. The main contributions of this paper are a careful analysis of the power of adoptions and a networkflow technique for selecting an optimal sequence of adoptions. The method yields a stronger performance guarantee and may yield better results in practice. The analysis of adoptions shows that ....

[Article contains additional citation context not shown here]

S. Khuller, B. Raghavachari and N. Young. Low degree spanning trees of small weight. SIAM J. Comput. 25 (1996), pp. 355--368.

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S. Khuller, B. Raghavachari, and N. Young. Low degree spanning tree of small weight. SIAM J. Computing, 25:355--368, 1996.

No context found.

S. Khuller, B. Raghavachari, and N. Young. Low degree spanning trees of small weight. SIAM J. Comput., 25:355-368, 1996.

No context found.

Samir Khuller, Balaji Raghavachari, and Neal Young. Low-degree spanning trees of small weight. SIAM Journal of Computing, 25(2):355-368, 1996.

No context found.

S. Khuller, B. Raghavachari, and N. Young. Low degree spanning tree of small weight. SIAM J. Computing, 25:355--368,

No context found.

S. Khuller, B. Raghavachari, and N. Young. Low degree spanning tree of small weight. In Proc. 26th ACM Symposium on Theory of Computing, 1994.

No context found.

S. Khuller, B. Raghavachari, and N. Young. Low degree spanning trees of small weight. SIAM Journal on Computing, 25(2):355--368, 1996.

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