Abstract
The generalized Steiner problem (GSP) is concerned with the determination of a minimum cost subnetwork of a given network where some (not necessarily all) vertices satisfy certain pairwise (vertex or edge) connectivity requirements.
The GSP has applications to the design of water and electricity supply networks, communication networks and other large-scale systems where connectivity requirements ensure the communication between the selected vertices when some vertices and/or edges can become inoperational due to scheduled maintenance, error, or overload.
The GSP is known to beNP-complete. In this paper we show that if the subnetwork is required to be biconnected or respectively edge-biconnected, and the underlying network is outerplanar, the GSP can be solved in linear time.
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References
A. V. Aho, J. E. Hopcroft, J. D. Ullman,The Design and Analysis of Computer Algorithms, Addison-Wesley, Reading (1974).
T. Beyer, W. Jones, S. L. Mitchell,Linear algorithms for isomorphism of maximal outerplanar graphs, J. ACM 26 (1979) 603–610.
N. Christofides, C. A. Whitlock:Network synthesis with connectivity constraints — a survey, in J. P. Brans (ed.),Operational Research '81, North-Holland Publ. Comp. (1981) 705–723.
M. R. Garey, D. S. Johnson:Computers and Intractability: A Guide to the Theory of NP-completeness, Freeman, San Francisco (1979).
S. L. Hakimi:Steiner's problem in graphs and its implications, Networks 1 (1971) 113–133.
F. Harary:Graph Theory, Addison-Wesley, Reading (1969).
J. Krarup:The generalized Steiner problem, unpublished note (1978).
J. B. Kruskal:On the shortest spanning subtree of a graph and the traveling salesman problem, Proc. Amer. Math. Soc. 7 (1956) 48–50.
E. L. Lawler:Combinatorial Optimization: Networks and Matroids, Holt, Rinehart and Winston, N.Y. (1976).
R. C. Prim:Shortest connection networks and some generalizations, Bell System Tech. J. 36 (1957) 1389–1401.
M. M. Syslo:Outerplanar graphs: characterization, testing, coding, and counting, Bull. Acad. Polon. Sci. 26 (1978) 675–684.
D. T. Tang:Bi-path networks and multicommodity flows, IEEE Trans. Circuit Theory CT-11 (1964) 468–474.
R. E. Tarjan:Depth-first search and linear graph algorithms, SIAM J. Comput. 1 (1972) 146–159.
J. A. Wald, C. J. Colbourn:Steiner trees in outerplanar graphs, Congr. Numerantium, 36 (1982) 15–22.
J. A. Wald, C. J. Colbourn:Steiner trees, partial 2-trees, and minimum IFI networks, Networks 13 (1983) 159–167.
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Winter, P. Generalized steiner problem in outerplanar networks. BIT 25, 485–496 (1985). https://doi.org/10.1007/BF01935369
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DOI: https://doi.org/10.1007/BF01935369