A. Zelikovsky and D. Lozevanu. Minimal and bounded trees. In Tezele Cong. XVIII Acad. Romano-Americane, pages 25-26, Kishinev, 1993. 14  Home/Search   Document Not in Database   Summary   Related Articles   Check

....by T k the set of all k cardinality trees in G. Then the edge weighted problem (G; w; k) is to nd a k cardinality tree T k 2 T k which minimizes w(T k ) e2E(T k ) w(e) 1) Several authors have proved independently that the edge weighted k cardinality tree problem (1) is NP hard, see [31, 17, 26]. In [26] it has been shown that it is still NP hard if w(e) 2 f1; 2; 3g for all edges e and G = K n , but polynomially solvable if there are only two distinct weights. Several authors have considered special types of graphs. One of the results is that the problem is polynomially solvable if G is ....

A. Zelikovsky and D. Lozevanu. Minimal and bounded trees. In Tezele Cong. XVIII Acad. Romano-Americane, pages 25-26, Kishinev, 1993. 14

....trees in G. Then the node weighted problem is min T k 2T k X v2V (T k ) w(v) 1) and the edge weighted problem is min T k 2T k X e2E(T k ) w(e) 2) 1. 2 INP Hardness Several authors have proved independently that the edge weighted k cardinality tree problem (2) is INP hard, see [32], 14] 30] In [30] it has been shown that the problem (2) is still INP hard if w(e) 2 f1; 2; 3g for all edges e and G = K n , but polynomially solvable if there are only two distinct weights. For the node weighted case (1) INP completeness has been shown in [13] and [9] Several authors ....

A. Zelikovsky and D. Lozevanu. Minimal and bounded trees. In Tezele Cong. XVIII Acad. Romano-Americane, Kishinev, pages 25-26, 1993. 25

....problem of Golden, Levy and Vohra [8] More details of the relation of these problems to the k MST problem can be found in [1] Previous work The k MST problem was shown to be NP hard by R. Ravi, Sundaram, Marathe, Rosenkrantz, and S. S. Ravi [11] and independently by Zelikovsky and Lozevanu [12]. The former paper also presented an O( p k) approximation algorithm for this problem. This was improved by Awerbuch, Azar, Blum and Vempala [1] who gave an O(log 2 k) approximation algorithm. Recently, Rajagopalan and Vazirani [10] obtained an O(log k) approximation algorithm for the k MST ....

A. Zelikovsky and D. Lozevanu. Minimal and bounded trees. In Tezele Cong. XVIII Acad. Romano-Americane, Kishinev, pages 25--26, 1993.

....tree. Up to constant factors in the approximation ratio, it is equivalent to ask for the shortest cycle (the k TSP problem) or shortest Steiner tree. Also inessential is the choice of metric. In fact, there has been a flurry of recent work on non geometric k MST problems [AABV94, CK94, RSM 94, ZL93] In this section, however, we shall focus on the Euclidean k MST formulation. The k MST problem was introduced by Fischetti et al. FHJM94] and Zelikovsky and Lozevanu [ZL93] and proved NP complete by those authors and also by Ravi et al. RSM 94] Ravi et al. devised an approximation ....

....choice of metric. In fact, there has been a flurry of recent work on non geometric k MST problems [AABV94, CK94, RSM 94, ZL93] In this section, however, we shall focus on the Euclidean k MST formulation. The k MST problem was introduced by Fischetti et al. FHJM94] and Zelikovsky and Lozevanu [ZL93] and proved NP complete by those authors and also by Ravi et al. RSM 94] Ravi et al. devised an approximation algorithm with ratio O(k 1 4 ) for the Euclidean k MST. The ratio was improved to O(logk) by Garg and Hochbaum [GH94] O(logk loglogn) by Eppstein [Epp95] O(1) by Blum et al. ....

A.Z. Zelikovsky and D.D. Lozovanu. Minimal and bounded trees. In Tezele Congresului XVIII al Academiei Romano-Americane, pages 25--26. Kishinev, 1993.

....where the goal is to select p facilities so as to optimize a certain cost measure defined on the set of selected facilities. Problems that can be cast in this framework include the p dispersion problem [RRT94, Ta91, EN89] and the k minimum spanning tree problem [RR 94, GH94, AA 94, BCV95, ZL93] In contrast, not much work has been done in finding optimal location of facilities when there is more than one objective. A notable work in this direction is by Bar Ilan, Kortsarz and Peleg [BKP93] who considered the problem of assigning network centers, with a bound imposed on the number of ....

A. A. Zelikowsky and D. D. Lozevanu. Minimal and bounded trees. Acad. Romano-Americane, Kishinev, 1993, pp. 25--26 26

....to the problem of finding a path connecting k points (the k TSP problem) or a Steiner tree connecting k points. The choice of Euclidean metric is also not critical. However we will use the k MST formulation for simplicity. The k MST problem was introduced independently by Zelikovsky and Lozevanu [116], and by Ravi et al. 97] Many similar k point selection problems with other optimization criteria can be solved in polynomial time [42, 58] but the k MST problem is NP complete [97, 116] as are obviously the k TSP and k Steiner tree variants) so one must resort to some form of approximation. ....

....the k MST formulation for simplicity. The k MST problem was introduced independently by Zelikovsky and Lozevanu [116] and by Ravi et al. 97] Many similar k point selection problems with other optimization criteria can be solved in polynomial time [42, 58] but the k MST problem is NP complete [97, 116] (as are obviously the k TSP and k Steiner tree variants) so one must resort to some form of approximation. In a sequence of many papers, the approximation ratio was reduces to O(k 1 4 ) 97] O(log k) 64, 87] O(log k log log n) 54] O(1) 21] 2 # 2 [89] and, very recently, 1 # (Arora, ....

[Article contains additional citation context not shown here]

A. A. Zelikovsky and D. D. Lozevanu. Minimal and bounded trees. Proc. Tezele Cong. XVIII Acad. Romano-Americane, Kishinev, 1993, pp. 25--26. 35

....to asking for a path connecting k points (the k TSP problem) or a Steiner tree connecting k points. The choice of Euclidean metric is also not critical. However we will continue to use the k MST formulation for simplicity. The k MST problem was introduced independently by Zelikovsky and Lozevanu [9], and by Ravi et al. 8] Ravi et al. described an approximation algorithm with approximation ratio O(k 1 4 ) this was quickly improved to O(log k) by Garg and Hochbaum [7] and to O(1) by Blum et al. 3] Many similar k point selection problems with other optimization criteria can be solved in ....

.... algorithm with approximation ratio O(k 1 4 ) this was quickly improved to O(log k) by Garg and Hochbaum [7] and to O(1) by Blum et al. 3] Many similar k point selection problems with other optimization criteria can be solved in polynomial time [5, 6] but the k MST problem is NP complete [8, 9] (as are obviously the k TSP and k Steiner tree variants) Zelikovsky and Lozevanu [9] also consider a dual problem in which, given a length bound, one wishes to find the largest set of points with MST length within the bound; this formulation has di#erent approximation behavior than the k MST ....

[Article contains additional citation context not shown here]

A. A. Zelikovsky and D. D. Lozevanu. Minimal and bounded trees. In Tezele Cong. XVIII Acad. Romano-Americane, Kishinev, pages 25--26, 1993. 12

....k out of n cities, and other combinatorial optimization problems. For example, the Euclidean minimal k point spanning tree (kMST) is the minimum weight tree spanning any k of the points. As pointed out in Eppstein [3] the planar k MST problem was shown to be NP complete by Zelikovsky and Lozevanu [4] and Ravi, Sundaram, Marathe, Rosenkrantz and Ravi [1] Ravi etal proposed a polynomial time approximation algorithm for the planar k MST with approximation ratio O(k 1=4 ) which has been successively improved to O(log(k) by Garg and Hochbaum [5] O(log(k) log log(n) by Eppstein [3] O(1) by ....

A. A. Zelikovsky and D. D. Lozevanu, "Minimal and bounded trees," in Proceedings of Tezele Congres XVIII Acad. Romano-Americaine, pp. 25--26, Kishinev, 1993.

....k MST problem, in which we are given a graph on n vertices with nonnegative distances on the edges, and an integer k n, and our goal is to find a tree of least total weight that spans some subset of k vertices. The problem is known to be NP hard, both in general graphs and in the Euclidean plane [12, 28, 31]. The current best approximation algorithm for general edge weighted graphs is a 3 approximation by Garg [14] which applies also to the rooted case (the tree is required to include a given node) this has been improved to a 2.5 approximation, by Arya and Ramesh [6] if the tree is not rooted. ....

A. Zelikovsky and D. Lozevanu. Minimal and bounded trees. In Tezele Cong. XVIII Acad. Romano-Americane, Kishinev, pages 25--26, 1993.

....problem, we are given a graph on n vertices with nonnegative distances on the edges, and an integer k n, and our goal is to find a tree of least total weight that spans some subset of k vertices. The k MST problem was introduced independently by Fischetti et al. 11] Zelikovsky and Lozevanu [22], and Ravi et al. 21] In those papers, the problem is shown to be NP complete, and Ravi et al. give an approximation algorithm with factor O( p k) Algorithms with improved approximation factors have since been discovered: Awerbuch et al. 2] obtain factor O(log 2 k) Rajagopalan and ....

A. Zelikovsky and D. Lozevanu, Minimal and bounded trees, Tezele Cong. XVIII Acad. Romano-Americane, Kishinev (1993), pp. 25--26.

....k MST problem, in which we are given a graph on n vertices with nonnegative distances on the edges, and an integer k n, and our goal is to find a tree of least total weight that spans some subset of k vertices. The problem is known to be NP hard, both in general graphs and in the Euclidean plane [12, 29, 26]. The current best approximation algorithm for general edge weighted graphs is a 3 approximation by Garg [14] which applies also to the rooted case (the tree is required to include a given node) this has been improved to a 2.5 approximation, by Arya and Ramesh [6] if the tree is not rooted. ....

A. Zelikovsky and D. Lozevanu. Minimal and bounded trees. In Tezele Cong. XVIII Acad. Romano-Americane, Kishinev, pages 25--26, 1993.

....[11] More details of the relation of these problems to the k MST problem can be found in [2] Previous work The k MST problem was shown to be NP hard by R. Ravi, Sundaram, Marathe, Rosenkrantz, and S. S. Ravi [15] and independently by Fischetti et al. 6] and also by Zelikovsky and Lozevanu [16]. Ravi et al. 15] also presented an O( p k) approximation algorithm for this problem. This was improved by Awerbuch, Azar, Blum and Vempala [2] who gave an O(log 2 k) approximation algorithm. Recently, Rajagopalan and Vazirani [14] obtained an O(log k) approximation algorithm for the k MST ....

A. Zelikovsky and D. Lozevanu. Minimal and bounded trees. In Tezele Cong. XVIII Acad. Romano-Americane, Kishinev, pages 25--26, 1993.

....the goal is to select p facilities so as to optimize a certain cost measure defined on the set of selected facilities. Problems that can be cast in this framework include the p dispersion problem [RRT94, Ta91, EN89] and the k minimum spanning tree problem [RR 94, GH94, AA 94, BCV95, ZL93] In contrast, not much work has been done in finding optimal location of facilities when there is more than one objective. A notable work in this direction is by Bar Ilan, Kortsarz and Peleg [BKP93] who considered the problem of assigning network centers, with a bound imposed on the number of ....

A. A. Zelikowsky and D. D. Lozevanu. Minimal and bounded trees. Acad. Romano-Americane, Kishinev, 1993, pp. 25--26

....problem, we are given a graph on n vertices with nonnegative distances on the edges, and an integer k n, and our goal is to find a tree of least total weight that spans some subset of k vertices. The k MST problem was introduced independently by Fischetti et al. 11] Zelikovsky and Lozevanu [22], and Ravi et al. 21] In those papers, the problem is shown to be NP complete, and Ravi et al. give an approximation algorithm with factor O( p k) Algorithms with improved approximation factors have since been discovered: Awerbuch et al. 2] obtain factor O(log 2 k) and Rajagopalan and ....

A. Zelikovsky and D. Lozevanu, Minimal and bounded trees, Tezele Cong. XVIII Acad. Romano-Americane, Kishinev (1993), pp. 25--26.

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A. A. Zelikovsky and D. D. Lozevanu, \Minimal and bounded trees," in Proceedings of Tezele Congres XVIII Acad. Romano-Americaine, pp. 25-26, (Kishinev), 1993.

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