Modern Heuristic Search Methods for the Steiner Tree Problem in Graphs

Given an edge-weighted graph, the Steiner tree problem in graphs is to determine a minimum cost subgraph spanning a set of specified vertices. More specifically, consider an undirected connected graph G = (V, E) with vertex set V, edge set E, and nonnegative weights associated with the edges. Given a set Q ⊆ V of basic vertices Steiner’s problem in graphs (SP) is to find a minimum cost subgraph of G such that there exists a path in the subgraph between every pair of basic (or required) vertices. In order to achieve this minimum cost subgraph additional vertices from the set S:= V— Q, socalled Steiner vertices, may be included. Since all edge weights are assumed to be nonnegative, there is an optimal solution which is a tree, a so-called Steiner tree.

[1]  C. W. Duin,et al.  HEURISTIC METHODS FOR THE RECTILINEAR STEINER ARBORESCENCE PROBLEM , 1993 .

[2]  José D. P. Rolim,et al.  Randomization and Approximation Techniques in Computer Science , 2002, Lecture Notes in Computer Science.

[3]  Anoop Ghanwani.  Neural and delay based heuristics for the Steiner problem in networks , 1998, Eur. J. Oper. Res..

[4]  Victor J. Rayward-Smith,et al.  On finding steiner vertices , 1986, Networks.

[5]  Billy E. Gillett,et al.  A Comparison of Two Simulated Annealing Algorithms Applied to the Directed Steiner Problem on Networks , 1991, INFORMS J. Comput..

[6]  Y.-B. Ji,et al.  A new scheme for the Steiner problem in graphs , 1988, 1988., IEEE International Symposium on Circuits and Systems.

[7]  M. Gendreau,et al.  A tabu search heuristic for the Steiner Tree Problem , 1999, Networks.

[8]  Anantaram Balakrishnan,et al.  Problem reduction methods and a tree generation algorithm for the steiner network problem , 1987, Networks.

[9]  Panos M. Pardalos,et al.  A test problem generator for the Steiner problem in graphs , 1993, TOMS.

[10]  Alfredo Candia-Véjar,et al.  An approach for the Steiner problem in directed graphs , 1991, Ann. Oper. Res..

[11]  Fred W. Glover,et al.  Using tabu search to solve the Steiner tree-star problem in telecommunications network design , 1996, Telecommun. Syst..

[12]  M. R. Rao,et al.  Solving the Steiner Tree Problem on a Graph Using Branch and Cut , 1992, INFORMS J. Comput..

[13]  Panos M. Pardalos,et al.  Network Design: Connectivity and Facilities Location , 1998 .

[14]  C. C. De Souza,et al.  A tight worst case bound for the performance ratio of heuristics for the minimum rectilinear steiner tree problem , 1990 .

[15]  Pawel Winter,et al.  Steiner problem in networks: A survey , 1987, Networks.

[16]  A. Volgenant,et al.  Reduction tests for the steiner problem in grapsh , 1989, Networks.

[17]  Kurt Mehlhorn,et al.  A Faster Approximation Algorithm for the Steiner Problem in Graphs , 1988, Inf. Process. Lett..

[18]  Stefan Voß,et al.  Steiner's Problem in Graphs: Heuristic Methods , 1992, Discret. Appl. Math..

[19]  R. Prim.  Shortest connection networks and some generalizations , 1957 .

[20]  Reinhard Männer,et al.  Optimization of Steiner Trees Using Genetic Algorithms , 1989, International Conference on Genetic Algorithms.

[21]  J. Plesnik.  WORST-CASE RELATIVE PERFORMANCES OF HEURISTICS FOR THE STEINER PROBLEM IN GRAPHS , 1991 .

[22]  Henrik Esbensen,et al.  Computing near-optimal solutions to the steiner problem in a graph using a genetic algorithm , 1995, Networks.

[23]  G. D. Smith,et al.  Solving the Graphical Steiner Tree Problem Using Genetic Algorithms , 1993 .

[24]  Celso C. Ribeiro,et al.  Greedy randomized adaptive search procedures for the Steiner problem in graphs. , 1997 .

[25]  Abilio Lucena,et al.  A Branch and Cut Algorithm for the Steiner Problem in Graphs , 1998 .

[26]  K. Dowsland.  HILL-CLIMBING, SIMULATED ANNEALING AND THE STEINER PROBLEM IN GRAPHS , 1991 .

[27]  Choukhmane El-Arbi.  Une heuristique pour le problème de l'arbre de Steiner , 1978 .

[28]  Piotr Berman,et al.  Improved approximations for the Steiner tree problem , 1992, SODA '92.

[29]  Panos M. Pardalos,et al.  A heuristic for the Steiner problem in graphs , 1996, Comput. Optim. Appl..

[30]  Marek Karpinski,et al.  New Approximation Algorithms for the Steiner Tree Problems , 1997, J. Comb. Optim..

[31]  Kenneth Steiglitz,et al.  Combinatorial Optimization: Algorithms and Complexity , 1981 .

[32]  Rolf Floren.  A Note on "A Faster Approximation Algorithm for the Steiner Problem in Graphs" , 1991, Inf. Process. Lett..

[33]  Reinhard Männer,et al.  On Steiner Trees and Genetic Algorithms , 1989, Parallelism, Learning, Evolution.

[34]  Marco Dorigo,et al.  Ant system: optimization by a colony of cooperating agents , 1996, IEEE Trans. Syst. Man Cybern. Part B.

[35]  Stefan Voß,et al.  Generic metaheuristics application to industrial engineering problems , 1999 .

[36]  Stefan Voss,et al.  Steiner-Probleme in Graphen , 1990 .

[37]  A. Volgenant,et al.  An edge elimination test for the steiner problem in graphs , 1989 .

[38]  David L. Woodruff,et al.  Proposals for chunking and tabu search , 1998, Eur. J. Oper. Res..

[39]  V. J. Rayward-Smith,et al.  The computation of nearly minimal Steiner trees in graphs , 1983 .

[40]  Stefan Voß,et al.  A chunking based genetic algorithm for the Steiner tree problem in graphs , 1997, Network Design: Connectivity and Facilities Location.

[41]  Ding-Zhu Du,et al.  On better heuristics for Steiner minimum trees , 1992, Math. Program..

[42]  Sanjeev Arora,et al.  Polynomial time approximation schemes for Euclidean TSP and other geometric problems , 1996, Proceedings of 37th Conference on Foundations of Computer Science.

[43]  Josep M. Basart,et al.  A High Performance Approximate Algorithm for the Steiner Problem in Graphs , 1998, RANDOM.

[44]  Gilbert Syswerda,et al.  Uniform Crossover in Genetic Algorithms , 1989, ICGA.

[45]  Ján Plesník.  On Some Heuristics for the Steiner Problem in Graphs , 1992 .

[46]  Gen-Huey Chen,et al.  The steiner problem in distributed computing systems , 1993, Inf. Sci..

[47]  Celso C. Ribeiro,et al.  An object-oriented framework for local search heuristics , 1998, Proceedings. Technology of Object-Oriented Languages. TOOLS 26 (Cat. No.98EX176).

[48]  Ján Plesník.  Heuristics for the Steiner Problem in Graphs , 1992, Discret. Appl. Math..

[49]  Siavash Vahdati Daneshmand,et al.  Improved algorithms for the Steiner problem in networks , 2001, Discret. Appl. Math..

[50]  J. David Schaffer,et al.  Proceedings of the third international conference on Genetic algorithms , 1989 .

[51]  Celso C. Ribeiro,et al.  A Parallel GRASP for the Steiner Problem in Graphs , 1998, IRREGULAR.

[53]  V. J. Rayward-Smith,et al.  Effective Local Search Techniques for the Steiner Tree Problem , 2000 .

[54]  D. Preßmar,et al.  Operations research proceedings , 1990 .

[55]  Marshall W. Bern,et al.  The Steiner Problem with Edge Lengths 1 and 2 , 1989, Inf. Process. Lett..

[56]  Victor J. Rayward-Smith,et al.  Modern Heuristic Search Methods , 1996 .

[57]  Emile H. L. Aarts,et al.  Parallel local search , 1995, J. Heuristics.

[58]  Michel Gendreau,et al.  A tabu search heuristic for the Steiner Tree Problem , 1998, Networks.

[59]  S. Voss,et al.  The pilot method; a strategie of heuristic repetition with application to the Steiner problem in graphs , 1996 .

[60]  Peter Widmayer,et al.  An Approximation Algorithms for Steiner's Problem in Graphs , 1986, WG.

[61]  Alexander Zelikovsky.  An 11/6-Approximation Algorithm for the Steiner Problem on Graphs , 1992 .

[62]  John E. Beasley,et al.  OR-Library: Distributing Test Problems by Electronic Mail , 1990 .

[63]  J. Plesník.  A bound for the Steiner tree problem in graphs , 1981 .

[64]  J. D. Becker,et al.  Parallelism, Learning, Evolution , 1991, Lecture Notes in Computer Science.

[65]  Stefan Voß,et al.  Worst-Case Performance of Some Heuristics for Steiner's Problem in Directed Graphs , 1993, Inf. Process. Lett..

[66]  C. W. Duin.  Preprocessing the Steiner problem in graph , 2000 .

[67]  Richard T. Wong,et al.  A dual ascent approach for steiner tree problems on a directed graph , 1984, Math. Program..

[68]  Pawel Winter.  Reductions for the rectilinear steiner tree problem , 1995, Networks.

[69]  Alex Zelikovsky.  A Faster Approximation Algorithm for the Steiner Tree Problem in Graphs , 1993, Inf. Process. Lett..

[71]  S. Voß,et al.  Efficient path and vertex exchange in steiner tree algorithms , 1997 .

[72]  A. Volgenant,et al.  Reducing the hierarchical network design problem , 1989 .

[73]  M. Minoux.  Efficient Greedy Heuristics For Steiner Tree Problems Using Reolptimization And Super Modularity , 1990 .

[74]  J. Kruskal.  On the shortest spanning subtree of a graph and the traveling salesman problem , 1956 .

[75]  Panos M. Pardalos,et al.  Randomization Methods in Algorithm Design , 1998 .

[76]  Victor O. K. Li,et al.  Minimum-weight vertex cover problem for two-class resource connection graphs , 1993, Inf. Sci..

[77]  Makoto Imase,et al.  Worst-Case Performance of Rayward-Smith's Steiner Tree Heuristic , 1988, Inf. Process. Lett..

[78]  Robert E. Tarjan,et al.  Efficient algorithms for finding minimum spanning trees in undirected and directed graphs , 1986, Comb..

[79]  Henry A. Kautz,et al.  Solving Problems with Hard and Soft Constraints Using a Stochastic Algorithm for MAX-SAT , 1995 .