S. E. Dreyfus and R. A. Wagner, (1972) The Steiner problem in graphs. Networks, 1, 195--207.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....that for all constants r 2 N the hypergraph H r (N; K) can be constructed in polynomial time: There are less than k subsets of K of cardinality at most r, where k = jKj. For each of these subsets a Steiner minimum tree can be found in log n nm) with the algorithm of Dreyfus and Wagner [8]. A plausible approach for designing an approximation algorithm is therefore to solve the minimum spanning tree problem in H r (N; K) and deduce from that a Steiner tree for K in N as outlined above. This would give an approximation algorithm with ratio ae r . As, however, finding minimum spanning ....

S.E. Dreyfus and R.A. Wagner, The Steiner problem in graphs, Networks 1 (1972), 195--207.

.... fixed parameter tractable [11] That is, can we compute exact minimum cut graphs in time f(g, k) O(1) for some function f The similarity to the Steiner problem o#ers some hope here, since the minimum Steiner tree of k nodes in an n node graph can be computed in O(3 n 2 n ) time [12, 19]. How well can we approximate the minimum cut graph in nearly linear time More generally, is there a simple, practical, O(1) approximation algorithm, like the MST approximation of Steiner trees Unfortunately, the general Steiner tree problem is MAXSNPhard [3] so an e#cient (1 #) approximation ....

S. Dreyfus and R. Wagner. The Steiner problem in graphs. Networks 1:195--207, 1971.

....We quote these cases in the context of the MZWR and MZWP problems, giving the appropriate source. Corollary 1 The MZWR problem can be solved in polynomial time for any of the following restricted classes of instances: 1. The number of terminal regions is bounded above by some fixed integer [5]. 2. The number of regions that are not terminal is bounded above by some fixed integer [10] 3. All of the terminal regions contain a common vertex of G [6] 4. There is a fixed number of vertices of G such that each terminal region contains at least one of these vertices [6] The MZWP ....

....is a fixed number of vertices of G such that each terminal region contains at least one of these vertices [6] The MZWP problem can be solved in polynomial time for any of the following restricted classes of instances: 1. The number of terminal vertices is bounded above by some fixed integer [5]. 2. All of the vertices are contained in a set of terminal regions containing a common vertex of G [6] 14 ....

S. E. Dreyfus and R. A. Wagner. The Steiner problem in graphs. Networks, 2:195-- 207, 1972.

....k. Parameter: k. Question: Is there a set of vertices T V S such that jT j k and G[S[T ] is connected This parameterized version of Steiner Tree is W[2] hard [3] Notice however that the parameterized version in which k is unbounded and jSj is a parameter is xed parameter tractable [12]. Although no membership result for this problem was previously known, we can easily place it in W[2] by devising a multi tape Turing machine that guesses a subset of vertices and checks whether it is a Steiner tree: Theorem 4 The Steiner Tree problem belongs to W[2] Proof. We show a ....

S. Dreyfus, R. Wagner. The Steiner problem in graphs. Networks 1:195-207, 1971.

....are of multicast nature, and thus multicasting has become a very active research area. The multicast problem with one source on undirected graphs becomes the Steiner tree problem. The problem is NP complete and algorithms for the exact solution have computational complexity of at least O(N3 M ) [DW72], which raises the need for good approximations with polynomial complexity. The hierarchical encoding technique enables us to use the resources more eciently. This encoding method has received a lot of attention from the networking community [Gha89] KV89] 1 and is still an active research area. ....

S. E. Dreyfus and R. A. Wagner. The Steiner problem in graphs. Networks, 1:195-207, 1972.

....an integer m. Question: Is there a set of vertices T V S such that jT j m and G[S[T ] is connected Parameter: m W[2] complete (membership: reduction to Short Multi Tape Nondeterministic Turing Machine Computation [38] hardness: reduction from Dominating Set [22, 23] Parameter: k FPT ([62]; solvable in time O(3 k n 2 k n 2 n 3 ) by the DreyfusWagner algorithm [98] Steiner Tree In HyperCubes Instance: Binary sequences X 1 ; X k , where each X i has length q; a positive integer M encoded in binary. Question: Is there a subgraph S of the q dimensional binary ....

S. Dreyfus and R. Wagner. The Steiner problem in graphs. Networks, 1:195-207, 1971.

....Richards and Winter [HRW92] give a good summary of this work. The xed parameter version of this problem, where the number of terminals p is constant, is easily solvable in polynomial time. This result was also published in the rst volume of the journal Networks in 1971, by Dreyfus and Wagner [DW71] and discovered independently by Levin [Lev71] It turns out that a solution to to Steiner Tree can be described by the topology of the solution tree. By topology, we mean the relative order in which the paths between the terminals merge. After the topology of the tree is known, the solution can ....

S. E. Dreyfus and R. A. Wagner. The Steiner problem in graphs. Networks, 1:195-207, 1971. 36

.... exact minimum cut graphs in time f(g; k) n O(1) Optimally Cutting a Surface into a Disk 11 for some function f The similarity to the Steiner problem o ers some hope here, since the minimum Steiner tree of k nodes in an n node graph can be computed in O(3 k n 2 k n 2 n 3 ) time [12, 18]. Our approximation algorithm requires at least quadratic time, which is prohibitive for very large models. How well can we approximate the minimum cut graph in nearly linear time More generally, is there a simple, practical, O(1) approximation algorithm, like the MST approximation of Steiner ....

S. Dreyfus and R. Wagner. The Steiner problem in graphs. Networks 1:195-207, 1971.

....simple Hanan grid graph reduction to the Steiner problem in graphs. This reduction has been by far the most popular approach to computing RSMTs. Various exact algorithms for the Steiner problem in graphs have been tried on grid graphs, including the dynamic programming method of Dreyfus and Wagner [15, 54], Hakimi s method [25] as well as sophisticated branch and cut methods [38, 34] However, even the most sophisticated branch and cut codes fail to solve instances much larger than 40 terminals due to the extreme degeneracy of the Hanan grid graph. In 1996 the author in collaboration with Abilio ....

S. E. Dreyfus and R. A. Wagner. The Steiner problem in graphs. Networks, 1:195--207, 1972.

....harder, because it must decide whether the inclusion of vertices not in M would actually lead to a cheaper solution. The problem is well known in graph theory as the Steiner tree problem which has been shown to be NP complete but is tractable for small networks. See for example Dreyfus and Wagner [10]. Heuristics for the Steiner tree problem have been shown to approach the ideal solution of small networks. See for example Waxman [26] or Rayward Smith [19] Other theoretical work has concentrated on slightly different aspects. BharathKumar and Jaffe consider multicast trees which trade ....

S.E. Dreyfus and R.A. Wagner. The steiner problem in graphs. Networks, 1(3):195--207, 1971.

....problem arises from the existence of an exponential number of spanning trees, and the typical exact solution can be obtained by some enumeration of these trees. Some obvious enumeration schemes can assist in reducing the actual number of trees examined, but the problem still remains intractable [47]. Many approximate solutions to the Steiner tree problem have been devised [48, 49, 39, 50, 51] The Steiner tree has a nice property in that it is possible to find an approximate solution that lies within a constant factor of the optimal cost [52] All the approximation algorithms cited above ....

S. Dreyfus and R. Wagner, "The Steiner problem in graphs," Networks, no. 1, pp. 195--207, 1972.

....Given an undirected weighted graph G(V; E) and a subset of participant nodes whose cardinality we denote by M , we want to find the minimum weight tree that spans the participants. The problem is NP complete and algorithms for the exact solution have computational complexity of at least O(N3 M ) [DW72], which raises the need for good approximations with polynomial complexity. A common measure of the quality of an approximation is the worst case inefficiency (WCI) or competitive ratio) of a multicast algorithm which is defined as the maximum ratio of the cost of the tree of the algorithm over ....

S. E. Dreyfus and R. A. Wagner. The Steiner problem in graphs. Networks, 1:195--207, 1972.

....can be stated as follows. Given an undirected weighted graph G(V; E) and a subset of participant nodes S V , we want to find the minimum weight tree that spans the participants. The problem is NP complete and algorithms for the accurate solution have at least O(N3 M ) computational complexity [DW72], which raises the need for good polynomial complexity approximations. A common measure of how good such an approximation is the worst case inefficiency (WCI) of a multicast algorithm that is defined as the maximum ratio of the cost of the tree of the algorithm over the optimal one. A number of ....

S.E. Dreyfus and R.A. Wagner. The Steiner problem in graphs. Networks, 1:195--207, 1972.

....of participant nodes whose cardinality we denote by M , while N = jV j. We want to find the minimum weight tree that spans the participants. The problem is NP complete and algorithms for the exact solution have computational complexity exponential in the number of nodes or of the participants [DW72], which raises the need for good approximations with polynomial complexity. Designated Contact Author. E mail: mfalou cs.toronto.edu. Telephone: 416 978 5182. 1 A common measure of the quality of an approximation is the worst case inefficiency (WCI) or competitive ratio of a multicast ....

S.E. Dreyfus and R.A. Wagner. The Steiner problem in graphs. Networks, 1:195--207, 1972.

....have compared their algorithms empirically with other algorithms for computing optimal RSTs. Here we seek to fill this gap in the literature. We present experimental results for many of the best algorithms for computing Preprint submitted to Elsevier Science 17 February 1998 optimal RSTs [1,3,6,8 10,12,21,22,24], including both graph based and geometric algorithms. 2 Graph Algorithms An early result on the RST problem is Hanan s theorem [13] whichprovides a reduction from the RST problem to the graph Steiner tree (GST) problem. Hanan proved that for any instance, an optimal RST exists in whichevery ....

....called the spanning tree enumeration algorithm. Hakimi s algorithm considers every subset S of n;2orfewer nonterminals, computing an MST of T [S for each and taking the minimum to be the optimal Steiner tree. The time complexity of the algorithm is O(n 2 2 m ) Dreyfus and Wagner [6] present a dynamic programming algorithm for the GST problem. The algorithm is based on an elegant decomposition theorem that states that an optimal Steiner tree of a set S of terminals can be decomposed into three subsets A, B,andfvg such that for some nonterminal vertex u, the union of the ....

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S. E. Dreyfus and R. A. Wagner. The Steiner problem in graphs. Networks, 1:195--207, 1972.

....in this chapter were obtained jointly with Philip Klein, and appeared in [91] 4.1 Problem Definition The Steiner tree problem in networks that we introduced in Section 1.1 is a classic hard problem in combinatorial optimization. Much research has been devoted to heuristics for its solution [33, 68, 112, 134, 135, 163]. Despite a slew of new approximation algorithms for this problem and some of its variants, no approximation algorithm has been given for perhaps the most natural variant: the node weighted Steiner tree problem, in which costs can be assigned to nodes as well as edges. Indeed, Winter s survey ....

S. E. Dreyfus, and R. A. Wagner, "The Steiner problem in graphs," Networks, vol. 1 (1971), pp. 195-207.

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S. E. Dreyfus and R. A. Wagner, (1972) The Steiner problem in graphs. Networks, 1, 195--207.

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S. E. Dreyfus and R. A. Wagner. The steiner problem in graphs. Networks, 1:195--207, 1972.

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S. Dreyfus and R. Wagner, "The Steiner Problem in Graphs", NETWORKS, 1, 195207, 1971.

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S. Dreyfus and R. Wagner. The Steiner problem in graphs. Networks, 1:195-207, 1971.

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S. E. Dreyfus and R. A. Wagner. The Steiner problem in graphs. Networks, 1:195--207, 1972. 14

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S. Dreyfus, R. Wagner, The Steiner problem in graphs, Networks 1 (1972), 195--207.

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S. E. Dreyfus and R. A. Wagner. The Steiner problem in graphs. Networks, 1:195-207, 1972.

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S. E. Dreyfus and R. A. Wagner. The Steiner problem in graphs. Networks, 1:195-207, 1972.

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S. E. Dreyfus, and R. A. Wagner, "The Steiner problem in graphs," Networks, vol. 1 (1971), pp. 195-207.

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