T. Koch and A. Martin, "Solving Steiner tree problems in graphs to optimality", Networks 32 (1998), 207--232.  Home/Search   Document Details and Download   Summary   Related Articles   Check

....Steiner tree in G. Therefore, any algorithm for the STP yields an algorithm for the L GSTP. Even if there exist several contributions on polyhedral aspects (see among others Goemans [24] Goemans and Myung [23] Chopra and Rao [5] 6] and exact methods (see for instance Koch and Martin [33]) for the classical problem, only a few are known, as far as we are aware, for the L GSTP. Polyhedral aspects are studied in Salazar [55] and a lower bounding procedure is described in Gillard and Yang [22] 13 A number of heuristics for the L GSTP have been proposed. Early heuristics for the ....

T. Koch, A. Martin. Solving the Steiner Tree Problems in Graphs to Optimality. Networks 32 (1998) 207--232.

....such as telecommunication network design, VLSI design, and computational biology, among others. The Steiner problem in graphs can be formulated as an integer linear program or a global concave minimization problem. Many exact algorithms for small size problems are based on these formulations [1, 3, 4, 6, 8, 18, 23, 24, 26, 27, 37]. Date: November 1999. Key words and phrases. Combinatorial optimization, global optimization, Steiner problem in graphs, heuristics, local search, GRASP, network design . AT T Labs Research Technical Report: 99.13.1. In memory of Professor P. D. Panagiotopoulos. Several heuristics are ....

T. Koch and A. Martin. Solving Steiner tree problems in graphs to optimality. Networks, 32:207--232, 1998.

....phase (an SMT can be constructed by concatenation of the second and last FST) then v(LP FST ) v(P FST ) 8. Note also that v(LP C ) 7:5 in both cases. The relaxation LP C can be strengthened by additional groups of constraints like the following one, which we call ow balance constraints [4]: X a2 (v i ) y a X a2 (v i ) y a (v i 2 V n R) 5.3) In [7] we prove that these constraints indeed lead to a strictly stronger relaxation, which we call LPC FB . Lemma 5 LP FSC is (strictly) stronger than LP C FB Proof: Consider a vertex v i 2 V n R. Any directed FST T ....

T. Koch and A. Martin. Solving Steiner tree problems in graphs to optimality. Networks, 32:207-232, 1998.

....of the vertices of a weighted network at minimum cost. This is a classical NP hard problem [5] with many important applications in network design in general and VLSI design in particular. Background information on this problem can be found in [4] A key ingredient of the most successful algorithms [1, 6, 10] for the Steiner problem are reduction methods, i.e. methods to reduce the size of a given instance while preserving at least one optimal solution (or the ability to eciently reconstruct one) While classical reduction tests just inspected simple patterns (vertices or edges) recent and more ....

T. Koch and A. Martin. Solving Steiner tree problems in graphs to optimality. Networks, 32:207-232, 1998.

....from scratch. They can be used as stand alone algorithms, providing solutions of reasonable quality within a short amount of time. However, their application is much broader. They are often used as subroutines of more elaborate primal heuristics [2, 10] dual heuristics [8] and exact algorithms [4, 8]. Regardless of the application, constructive heuristics should be as fast as possible, while preserving solution quality. A number of such heuristics are described in the literature (see [3, 12] for surveys) We focus our attention on heuristics that are direct extensions of exact algorithms for ....

....faster than the worst case analysis suggest, since modi cations tend to a ect small portions of the graph. 5 Prim Proposed by Takahashi and Matsuyama [11] the construtive heuristic for the SPG based on Prim s algorithm for the MST problem is probably the most commonly used in the literature [2, 4, 8, 10]. The classic algorithm for the MST problem grows the solution from a root vertex. In each step, the vertex that is closer to the current partial solution is added to it, alongside with the connecting edge. After jV j 1 steps, all vertices are spanned. The corresponding heuristic for the SPG ....

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T. Koch and A. Martin. Solving Steiner tree problems in graphs to optimality. Networks, 32:207-232, 1998.

....in an enlarged graph. In this work we have described a direct ILP model, and have analyzed new classes of facet de ning inequalities. Although we do not provide practical experiences, several computational papers on related problems (e.g. Fischetti, Salazar and Toth [7] or Koch and Martin [13]) using similar inequalities shown that our contributions could be of primary importance for solving the GST problem within a cutting plane approach. ....

T. Koch, A. Martin, \Solving Steiner Tree Problems in Graphs to Optimality", Networks 32 (1998) 207-232.

....inequalities designed for simple fixed charge flow problems. However they still do not take full advantage of the structure of UFC. This explains in part why much previous work has been on the development of specialized algorithms for specific variants of UFC. Chopra et al. 9] and Koch and Martin [24] have developed branch and cut codes for the Steiner tree problem which can be viewed as a special case of UFC in which the flow costs are zero. Several branch and bound algorithms have been described for the uncapacitated fixed charge transportation problem in which the underlying network is ....

....cost of the arc (i, j) is given by c ij = w i w j ) 2 (h i h j ) 2 . Steiner. These undirected instances have been obtained from the Steiner problem Library at ZIB ftp: ftp.zib.de pub Packages mp testdata index.html. A complete description of all these problems can be found in [24]. In our tests we have selected a small subset of the instances: Beasley. Problems 1,2 and 3 of series C described in [21] Instances from series B were not considered as they are too easy. X: Instances brasil and berlin. These have complete graphs and Euclidean weights. Mc7, Mc8 and ....

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Koch T. and Martin A. Solving steiner tree problems in graphs to optimality. Networks, 32(3):207--232, 1998.

....G = V; E) non negative costs c(e) associated to the edges of E and a set T V of terminal vertices, find a Minimum Cost Steiner Tree (MCST) i.e. a tree (V 0 ; E 0 ) with T V 0 minimizing P e2E 0 c(e) The STPG is one of the most widely studied NP hard problems. Koch and Martin [4] presented a very good implementation of an exact algorithm for STPG, consisting of a preprocessing followed by a branch and cut. It could solve most of the instances then available in the literature in reasonable times, including all the classical problems in the OR Library (available at ....

....f = argmin f 0 2ffi(W )nfeg c(f 0 ) be a shortest and second shortest edge in the cut. Suppose e = u; v) with u 2 W and v 2 W . If minfd(u; t 1 ) j t 1 2 T Wg c(e) minfd(v; t 2 ) j t 2 2 T W g c(f) then e is choosable. The tests presented so far are exactly the ones used by [4]. They are quite efficient for many general Steiner instances. For example, e18, the hardest instance in the OR Library, is reduced from 62,500 edges to 5996 edges. But they are almost useless over VLSI instances. For example, diw0559 is only reduced from 7013 edges to 6883 edges. Some improvement ....

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T. Koch and A. Martin, Solving Steiner Tree Problems in Graphs to Optimality, Networks, Vol. 32, 207-232, 1998.

....cut is found or some limit for this family is reached g We want to separate high quality cuts. Besides finding a most violated one, we have a second tie breaking criterion of finding a cut over the minimum number of free variables. A similar idea was successfully used by Koch and Martin [4]. By forcing the objective function , we mean that a cut a T x a 0 must minimize a T w. Even as the third tie breaking criterion, this idea proved to be very useful. It seems important not to forget the original costs while separating. A cut satisfying all such criteria may be found by ....

....procedures are recommended, since such perturbations increase substantially the size of the numbers in the input. The idea of separating many violated cuts by family, as long as their support sets have little or no overlap, is used in some recent branch and cut implementations, including [4]. 5.3.1 p crosses For each face f of G we define a family of cuts and follow the general separation strategy without any limit. In other words, for each face as many non overlapping violated p crosses as possible are added in each cut round. Such policy is efficient because p crosses are (i) ....

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T. Koch and A. Martin, Solving Steiner Tree Problems in Graphs to Optimality, Networks, Vol. 32, 207-232, 1998.

....problem in graphs, see e.g. Duin and Voss [10] Hwang et al. 18] and Voss [33] for recent surveys. Effective preprocessing algorithms based on variable fixation theorems which allow strong graph reductions have been described and implemented by Duin [7] Duin and Volgenant [9] Koch and Martin [20], and Uchoa et al. 30] among others. The shortest path heuristic of Takahashi and Matsuyama [29] is one of the most effective algorithms for computing greedy approximate solutions to SPG. Improvement heuristics based on the insertion of Steiner nodes were proposed by Minoux [24] and Voss [32] ....

....may be significantly reduced by preprocessing. We applied the classical tests from Duin and Volgenant [9] and a new test proposed by Uchoa et al. 30] An outline of the preprocessing procedure follows. We first apply the Nearest Special Vertex (NSV) test [9] also known as Terminal Distance test [20], for the fixation of short edges at one) until no further reduction is possible. Next, we apply the Special Distance (SD) test [9] once for each edge (for the fixation of long edges at zero) and the simple Degree (D) test [9] which is actually a group of tests consisting of the elimination of ....

[Article contains additional citation context not shown here]

T. Koch and A. Martin, "Solving Steiner tree problems in graphs to optimality", Networks 32 (1998), 207--232.

....a set T V of terminal vertices, find a Minimum Steiner Tree (MST) i.e. a subtree (V 0 ; E 0 ) of G with T V 0 minimizing P e2E 0 c(e) The SPG is one of the most widely studied NP hard problems, see Maculan [10] Winter [14] and Hwang et al. 7] for good surveys. Koch and Martin [8] presented an exact algorithm for SPG, consisting of preprocessing followed by a branch and cut procedure. They obtained good practical results and could solve most of the instances then available in the literature in reasonable times, including all classical problems in the OR Library [4] They ....

....) c(g) c(f ) Removing g from R creates two subtrees, one containing t 1 and the other containing t 2 . Connecting this subtrees by a shortest path from t 1 to t 2 passing through e, we obtain a new Steiner tree R 0 using e such that c(R 0 ) c(R) The above tests are exactly those used in [8]. They are quite effective for many general Steiner instances. For example, they allow the reduction of e18, the hardest instance in the OR Library, from 62,500 to 5,996 edges. However, they are almost useless for VLSI instances. For example, instance diw0559 is only reduced from 7,013 to 6,883 ....

[Article contains additional citation context not shown here]

T. Koch and A. Martin, "Solving Steiner tree problems in graphs to optimality", Networks 32 (1998), 207--232.

.... D, and E from the OR Library [1] Each series contains randomly generated instances, with edge weights taken from a uniform distribution in the interval [1; 10] The second set is made up by instances defined over grid graphs with holes, extracted from real VLSI layout problems by Koch and Martin [7]. Finally, the third set is formed by incidence instances created by Duin and Voss [3] so as to make the reduction tests ineffective. Different preprocessing strategies were applied to the OR Library and to the VLSI instances, as described in [13, 15] All instances are available from the SteinLib ....

T. Koch and A. Martin, "Solving Steiner tree problems in graphs to optimality", Networks 32 (1998), 207--232.

....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 Lucena solved several of the 100 terminal instances from the OR library. ....

....algorithm whose significant details were presented. Empirical results show that on both rectilinear and Euclidean Steiner minimal tree problems the new FST concatenation algorithm vastly out performs all other algorithms in existence. Its nearest rectlinear competitors seem to be Martin and Koch [34] (up to 40 terminals) and Fomeier and Kaufmann [16] 70 terminals, but at least one instance of 100 terminals) For the Euclidean problem, Winter and Zachariasen [62] is the closest competitor at 150 terminals. 100 101 Provided a suitable FST generator is available, this method is applicable to ....

T. Koch and A. Martin. Solving Steiner tree problems in graphs to optimality. Technical Report SC 96--42, Konrad-Zuse-Zentrum fur Informationstechnik, Berlin, Germany, 1996.

....be used for the Steiner tree subproblems and solved using branch and bound with LP relaxations in the nodes. However, it turns out that this approach is inecient for anything but the smallest of the test problems. An ecient algorithm for solving Steiner tree problems is given by Koch and Martin [KM98], where the authors report solving very large Steiner tree problems to optimality. Their algorithm is a branch and cut method based on the directed formulation (4.4) of the tree constraints (4.2) We follow the strategies that they propose when solving our Steiner tree problems. The general ....

....the LP solution as capacities in the graph and consider the minimal (r k ; j) cut: if it is less than one, then a violated cut inequality (4.4a) has been found. If all minimal (r k ; j) cuts are greater than or equal to one, then there are no violated inequalities. As suggested by Koch and Martin [KM98], a creep ow strategy is used to improve the quality of the cut inequalities. Typically there will be a large number of capacity minimal cuts since most of the LP solution variables will be zero. The sharpest of these are likely to be the cuts that are also arc minimal. To nd an arc minimal ....

T. Koch and A. Martin. Solving Steiner tree problems in graphs to optimality. Networks, 32, 207-232, 1998.

....issues. We cannot give a complete list of all relevant publications here. Let us just mention the papers [CR94a, CR94b] where the polyhedral structure of the classical Steiner tree polyhedron is studied, Goe94, GM93] where several alternative mixed integer formulations are discussed, and [KM98], which focuses on computational aspects of the problem. In the rst part of this paper we present three mixed integer programming formulations for this problem. The rst (natural) formulation uses only one integer capacity variable for each edge and and one binary tree variable for each ....

....working with the directed cut formulation 3.4, all violated directed Steiner cut inequalities (5) need to be separated. This can be done by solving a sequence of jT k j 1 maximum ow problems for each commodity k 2 N . We use a special variant of this separation method which was described in [KM98] for the (single) Steiner tree problem: Let ( y; x) be a fractional point. For some tiny value (we used = 10 6) we compute the minimal (r k ; t) dicut for all t 2 T k with respect to the capacities x k . The advantage of adding this so called creep ow to the fractional Steiner ....

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T. Koch and A. Martin, Solving Steiner tree problems in graphs to optimality, Networks 32 (1998), 207 - 232.

....horizontal lines through all terminals in Z. Since an RSMT always exists in this grid [10] it can be found by solving the Steiner tree problem in the corresponding Hanan grid graph. However, it is well known that these graph problems are difficult to solve by using, e.g. branch and cut methods [16]. 2.2 Rectilinear Full Steiner Trees An RSMT is a union of full Steiner trees (FSTs) in which every leaf is a terminal (having degree one) and all other nodes (having degree three or more) are Steiner points. Hwang [13] proved that there always exists an RSMT for which every FST has one of the ....

.... uses an undirected integer programming (IP) formulation of the Steiner tree in hypergraph problem; the IP is solved using LP relaxation and branch and cut [29] An overview of different formulations of the Steiner tree problem is given in [8] Jack3 Steiner tree in graph solver by Koch and Martin [16]. This code uses a directed IP formulation of the Steiner tree problem that is also solved by LP relaxation and branch and cut. In the following we first present detailed results for the POWER3 instances; in total 64 problem instances with 11 to 99 groups. For each problem size we picked one ....

T. Koch and A. Martin. Solving Steiner Tree Problems in Graphs to Optimality. Networks, 33:207--232, 1998.

....problem in graphs, see e.g. Duin and Voss [11] Hwang et al. 21] and Voss [39] for recent surveys. E#ective preprocessing algorithms based on variable fixation theorems which allow strong graph reductions have been described and implemented by Duin [9] Duin and Volgenant [10] Koch and Martin [24], and Uchoa et al. 36] among others. Constructive methods were proposed, e.g. by Choukmane [5] Kou et al. 25] Plesnk [32] # October 2, 2000 Department of Computer Science, Catholic University of Rio de Janeiro, Rio de Janeiro, RJ 22453 900, Brazil. Work of this author was sponsored by ....

....be significantly reduced by preprocessing. We applied the classical tests from Duin and Volgenant [10] and a new test proposed by Uchoa et al. 36] An outline of the preprocessing procedure follows. We first apply the Nearest Special Vertex (NSV) test [10] also known as Terminal Distance test [24], for the fixation of short edges at one) until no further reduction is possible. Next, we apply the Special Distance (SD) test [10] once for each edge (for the fixation of long edges at zero) and the simple Degree (D) test [10] which is actually a group of tests consisting of the elimination of ....

[Article contains additional citation context not shown here]

T. Koch and A. Martin, "Solving Steiner tree problems in graphs to optimality", Networks 32 (1998), 207--232.

....ffl Incidence instances: Those are the instances of type incidence problems described by Duin and Voss [7] with 80 (instances DV 80) 160 (instances DV160) and 320 (instances DV 320) vertices. They were generated so as to make the reduction tests ineffective, as confirmed by Koch and Martin [17]. In each case, 20 classes of problems were generated, combining different densities of terminal nodes and edges. There are five problems in each class and a total of 300 instances, organized in series of 100 test problems characterized by their number of nodes jV j = 80; 160, and 320. The graphs ....

T. Koch and A. Martin, "Solving Steiner tree problems in graphs to optimality", Networks 32 (1998), 207-232.

....cut is found or some limit for this family is reached g We want to separate high quality cuts. Besides finding a most violated one, we have a second tie breaking criterion of finding a cut over the minimum number of free variables. A similar idea was successfully used by Koch and Martin [5]. By forcing the objective function , we mean that a cut a T x a 0 must minimize a T w. Even as the third tie breaking criterion, this idea proved to be very useful. It seems important not to forget the original costs while separating. A cut satisfying all such criteria may be found by ....

....procedures are recommended, since such perturbations increase substantially the size of the numbers in the input. The idea of separating many violated cuts by family, as long as their support sets have little or no overlap, is used in some recent branch and cut implementations, including [5]. 5.3.1. p crosses For each face f of G we define a family of cuts and follow the general separation strategy without any limit. In other words, for each face as many non overlapping violated p crosses as possible are added in each cut round. Such policy is efficient because p crosses are (i) ....

[Article contains additional citation context not shown here]

T. Koch and A. Martin, Solving Steiner Tree Problems in Graphs to Optimality, Networks, Vol. 32, 207-232, 1998.

....such as telecommunication network design, VLSI design, and computational biology, among others. The Steiner problem in graphs can be formulated as an integer linear program or a global concave minimization problem. Many exact algorithms for small size problems are based on these formulations [1, 3, 4, 6, 8, 18, 23, 24, 26, 27, 37]. Date: November 1999. Key words and phrases. Combinatorial optimization, global optimization, Steiner problem in graphs, heuristics, local search, GRASP, network design . AT T Labs Research Technical Report: 99.13.1. In memory of Professor P. D. Panagiotopoulos. 1 2 S.L. MARTINS, M.G.C. ....

T. Koch and A. Martin. Solving Steiner tree problems in graphs to optimality. Networks, 32:207--232, 1998.

....C w i = CW ) w i =W . The total contribution is bounded by P z i 2Znfrg w i =W = 1; consequently, the tree constructed must have minimum length as edge weights were assumed to be integer valued. The branch and cut algorithm used to solve the problem is basically the one by Koch and Martin [9], but without the pre processing algorithm for reducing the size of the problem. The traditional branching strategy which branches on variables is used; a fractional edge variable with LP value closest to 0.5 is selected. Note that it is enough to ensure that all edge variables x uv have integer ....

....edge variable with LP value closest to 0.5 is selected. Note that it is enough to ensure that all edge variables x uv have integer value. When this is the case the ow variables will be set accordingly. Computational results for this algorithm will be presented in Section 6. It is well known [9] that solving Steiner tree problems in the Hanan grid graph is computationally dicult due to high degree of symmetry. Graph reduction methods for the ordinary Steiner tree problem on the Hanan grid graph were proposed by Winter [13] however, not all the proposed reduction tests and 10 in ....

T. Koch and A. Martin. Solving Steiner Tree Problems in Graphs to Optimality. Networks, 33:207-232, 1998.

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T. Koch and A. Martin, "Solving Steiner tree problems in graphs to optimality", Networks 32 (1998), 207--232.

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T. Koch and A. Martin. Solving Steiner tree problems in graphs to optimality. Networks, 32:207--232, 1998.

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T. Koch, A. Martin. Solving Steiner tree problems in graphs to optimality. Networks, 32, 3:207--232, 1998.

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Koch, T., A. Martin. 1998. Solving Steiner tree problems in graphs to optimality, Networks

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