Abstract
We consider several families of deterministic network optimization problems (NOPs) of particular importance for the design (synthesis) of real-life transportation, communication, and distribution networks. These families of NOPs include determination of optimal spanning and Steiner trees, multiconnected networks, distance bounded networks. and capacitated networks. Problems belonging to these families are formulated in an unified manner, and exact algorithms, heuristics, as well as algorithms for special cases are surveyed. Commonalities within each family as well as across family boundaries are identified. A wide range of open problems is given.
Keywords
- Span Tree
- Minimum Span Tree
- Steiner Tree
- Facility Location Problem
- Network Design Problem
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.
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References
Aneja Y.P. "An integer linear programming approach to the Steiner problem in graphs" Networks 10 (1980) 167–178.
Amborg S. and Proskurowski A. "Linear time algorithms for NP-hard problems on graphs embedded in k-trees" Tech. Report TRITA-NA-8404, Dept. of Numberical Analysis and Computing Science, The Royal Inst. of Technology, Stockholm, Sweden (1984).
Amborg S. and Proskurowski A. "Characterization and recognition of partial ktrees" Tech. Report TRITA-NA-8402, Dept. of Numerical Analysis and Computing Science, The Royal Inst. of Technology, Stockholm, Swden (1984).
Beasley J.E. "An algorithm for the Steiner problem in graphs" Networks 14 (1984) 147–159.
Beasley J.E. "An SST-based algorithm for the Steiner problem in graphs" Technical Report, Dept. of Man. Sciences, Imperial College, London (1985).
Bermond J.C., Bond J., Paoli M. and Peyrat C. "Graphs and interconnection networks: diameter and vulnerability" in Lloyd (ed.) "Surveys in Combinatorics’ London Math. Soc. Lect. Notes 82 (1983) 1–30.
Boesch F.T. "Properties of the distance matrix of a graph", Quart. Appl. Math. 26 (1969) 607–610.
Boesch F.T. "Graph-theoretic models for network reliability studies" Tech. Report 8010, Dept. of Electr. Eng and Comp. Sci., Stevens Inst. of Technology, Hoboken, New Jersey (1980).
Boesch F.T. and Chen S. "A generalization of line connectivity and optimally invulnerable graphs" SIAM J. Appl. Math. 34 (1978) 657–665.
Boesch F.T. and Felzer A. "On the invulnerability of the regular complete k-partite graphs" SIAM J. Appl. Math. 20 (1971) 176–182.
Boyce D.E., Farhi A. and Weischedel R. "Optimal network problem: A branch and bound algorithm" Environment Planning 5 (1973) 519–533.
Camerini P.M. "The min-max spanning tree problem and some extensions" Inf. Proc. Letters 7 (1978) 10–13.
Camerini P.M., Galbiati G. and Maffioli F. "On the complexity of Steiner-like problems" Proc. 7th Ann. Allerton Conf. on Comm., Control and Computing (1979) 969–977.
Chandrasekaran R. "Minimal ratio spanning trees" Networks 7 (1977) 335–342.
Chandrasekaran R., Aneja Y.P. and Nair K.P.K. "Minimal cost-reliability ratio spanning tree" in Hansen P. (ed.) "Studies on Graphs and Discrete Programming" North-Holland (1981) 53–60.
Chandy K.M. and Lo T. "The capacitated minimum spanning tree" Networks (1973) 173–181.
Cheriton D. and Tarjan R.E. "Finding minimum spanning trees" SIAM J. Comput. 5 (1976) 724–742.
Christofides N. and Whitlock C. "Network synthesis with connectivity constraints" in Brans J.P. (ed.) "Operational Research '81" North-Holland, London (1981) 705–723.
Dantzig G.B. "Discrete-variable extremum problems" Operations Research 5 (1957) 266–277.
Dionne R. and Florian M. "Exact and approximate algorithms for optimal network design" Networks 9 (1979) 37–59.
Dreyfus S.E. and Wagner R.A. "The Steiner problem in graphs" Networks 1 (1971) 195–207.
Erickson R.E., Monma C.L. and Veinott A.F.Jr. "Minimum-concave-cost network flows" to appear in Math. of OR.
Frank H. and Frisch I.T. "Analysis and design of survivable networks" IEEE Trans. on Comm. Techn. COM-18 (1970) 501–519.
Frank H. and Frisch I.T. "Communication, Transmission, and Transportation Networks" Addison-Wesley (1971).
Frisch I.T. "Flow variation in multiple min-cut calculations" J. Franklin Inst. 287 (1969) 61–72.
Gabow H.N. "Two algorithms for generating weighted spanning trees in order" SIAM J. Comput. 6 (1977) 139–150.
Gabow H.N. "A good algorithm for smallest spanning trees with a degree constraint" Networks 8 (1978) 201–208.
Gabow H.N. and Myers E.W. "Finding all spanning trees of directed and nondirected graphs" SIAM J. Comput. 7 (1978) 280–287.
Gallo G. "A new branch-and-bound algorithm for the network design problem" Report L81-01, Instituto di Elaborazione della Informazione, Pisa, Italy (1981).
Gallo G. "Lower planes for the network design problem" Networks 13 (1983) 411–426.
Garey M.R. and Johnson D.S. "The rectilinear Steiner problem is NP-complete" SIAM J. Appl. Math. 32 (1977) 826–834.
Garey M.R. and Johnson D.S. "Computers and Intractability: A Guide to the Theory of NP-completeness" Freeman, San Francisco (1979).
Garfinkel R.S. and Nemhauser G.L. "Integer Programming" J. Wiley, N.Y. (1972).
Gavish B. "Topological design of centralized computer networks — formulations and algorithms" Networks 12 (1982) 355–377.
Golden B.L. and Magnanti T.L. "Deterministic network optimization: A bibliography" Networks 7 (1977) 149–183.
Gomory R.E. and Hu T.C. "Multi-terminal network flows" J. SIAM 9 (1961) 551–570.
Grotschel M. and Padberg M.W. "On the symmetric travelling salesman problem: I. Inequalities" Mathematical Programming 16 (1979) 265–280.
Groteschel M. and Padberg M.W. "On the symmetric travelling salesman problem: II. Lifting theorems and facets" Mathematical Programming 16 (1979) 281–302.
Hakimi S.L. "Steiner's problem in graphs and its implications" Networks 1 (1971) 113–133.
Hakimi S.L. and Yau S.S. "Distance matrix of a graph and its realizability" Quart. Appl. Math. 22 (1965) 305–317.
Hanan M. "Net wiring for large scale integrated circuits" IBM Res. Report RC1375 (1965).
Harary F. "Graph Theory" Addison-Wesley, Reading (1969).
Hoang H.H. "A computational approach to the selection of an optimal network" Management Science 19 (1973) 488–498.
Hu T.C. "Optimum communication spanning problem" SIAM J. Comput. 3 (1974) 188–195.
Hwang F.K. "On Steiner minimal trees with rectilinear distance" SIAM J. Appl. Math. 30 (1976) 104–114.
Johnson D.S., Lenstra J.K. and Rinnooy Kan A.H.G. "The complexity of the network design problem" Networks 8 (1978) 279–285.
Karp R.M. "Reducibility among combinatorial problems" in Miller and Thatcher (eds.) "Complexity of Computer Computations" Plenum Press, N.Y. (1972) 85–103.
Katoh N., Ibaraki T. and Mine H. "An algorithm for finding k minimum spanning trees" SIAM J. Comput. 10 (1981) 247–255.
Kershenbaum A. "Computing capacitated minimal spanning trees efficiently" Networks 4 (1974) 299–310.
Kou L., Markowsky G. and Berman L. "A fast algorithm for Steiner trees" Acta Informatica 15 (1981) 141–145.
Krarup J. "The generalized Steiner problem" unpublished noted (1978).
Krarup J. and Pruzan P. "Selected families of location problems" Ann. Discrete Math. 5 (1979) 327–387.
Krarup J. and Pruzan P. "The simple plant location problem: survey and synthesis" European J. Operational Research 12 (1983) 36–81.
Kruskal J.B. "On the shortest spanning subtrees of a graph and the travelling salesman problem" Proc. Amer. Math. Soc. 7 (1956) 48–50.
Los M. and Lardinos C. "Combinatorial programming, statistical optimization and the optimal transportation network problem" Transportation Research 16B (1980) 89–124.
Magnanti T.L., Mireault P. and Wong R.T. "Tailoring Benders decomposition for network design" Technical Report OR 125-83, Operations Research Center, MIT (1983).
Magnanti T.L. and Wong R.T. "Accelerating Benders decomposition: Algorithmic enhancement and model selection criteria" Operations Research 29 (1981) 464–484.
Magnanti T.L. and Wong R.T. "Network design and transportation planning: Models and algorithms" Transportation Science 18 (1984) 1–55.
Martello S. and Toth P. "Finding minimum equivalent graph of a digraph" Networks 12 (1982) 89–100.
Matula D.W. "k-components, clusters, and slicings in graphs" SIAM J. Appl. Math. 22 (1972) 459–480.
Monma C.L., Munson B.S. and Pulleyblank W.R. "Minimum-weight two-connected spanning networks" unpublished paper.
Petersen B.E. "A cut-flow procedure for transportation network optimization" Networks 10 (1980) 33–43.
Plesnik J. "The complexity of designing a network with minimum diameter" Networks 11 (1981) 77–85.
Plesnik J. "A bound for the Steiner tree problem in graphs" Math. Slovaca 31 (1981) 155–163.
Plesnik J. "A note on the complexity of finding regular subgraphs" Discrete Math. 49 (1984) 161–167.
Preparata F.P. and Shamos M.I. "Computational Geometry: An Introduction" Springer-Verlag, Berlin (1985).
Prim R.C. "Shortest connection networks and some generalizations" Bell System Tech. J. 36 (1957) 1389–1401.
Rayward-Smith V.J. "The computation of nearly minimal Steiner trees in graphs" Int. J. Math. Educ. Sci. Technol. 14 (1983) 15–23.
Rayward-Smith V.J. and Clare A. "On finding Steiner vertices" Networks 16 (1986) 283–294.
Schumacher U. "An algorithm for construction of a k-connected graph with minimum number of edges and quasiminimal diameter" Networks 14 (1984) 63–74.
Scott A.J. "The optimal network problem: some computational procedures" Transportation Research 3 (1969) 201–210.
Shore M.L., Foulds L.R. and Gibbons P.B. "An algorithm for the Steiner problem in graphs" Networks 12 (1982) 323–333.
Smith J.M. "Generalized Steiner network problems in engineering design" in Gero J.S. (ed.) "Design Optimization" Academic Press (1985) 119–161.
Steiglitz K., Weiner P. and Kleitman D.J. "The design of minimum cost survivable networks" IEEE Trans. on Circuit Theory CT-16 (1969) 455–460.
Takahashi H. and Matsuyama A. "An approximate solution for the Steiner problem in graphs" Math. Japonica 24 (1980) 573–577.
Tansel B.C., Francis R.L. and Lowe T.L. "Location on networks: A survey. Part I: The p-center and p-median problems" Management Science 29 (1983) 482–497.
Tansel B.C., Francis R.L. and Lowe T.L. "Location on networks: A survey. Part II: Exploiting tree network structure" Management Science 29 (1983) 498–511.
Toueg S. and Steiglitz K. "The design of small diameter networks by local search" IEEE Trans. Comput. C-28 (1979) 537–542.
Wald J.A. and Colbourn C.J. "Steiner trees, partial 2-trees, and minimum IFI networks" Networks 13 (1983) 159–167.
White K., Farber M. and Pulleyblank W. "Steiner trees, connected domination and strongly chordal graphs" Networks 15 (1985) 109–124.
Wilkov R.S. "Analysis and design of reliable computer networks" IEEE Trans. Commun. COM-20 (1972) 660–678.
Winter P. "Network Synthesis: Spanning Trees" Tech. Report, Inst. of Datalogy, Univ. of Copenhagen, Denmark (1983).
Winter P. "Steiner problem in Halin networks" to appear in Discrete Applied Mathematics.
Winter P. "An algorithm for the Steiner problem in the Euclidean plane" Networks 15 (1985) 323–345.
Winter P. "Generalized Steiner problem in outerplanar networks" BIT 25 (1985) 485–496.
Winter P. "Steiner problem in networks — a survey" to appear in Networks.
Winter P. "Generalized Steiner problem in series parallel networks" to appear in J. of Algorithms.
Winter P. "An algorithm for the enumeration of spanning trees" BIT 26 (1986) 44–62.
Winter P. "Generalized Steiner problem in Halin networks" presented at XII Math. Progr. Symp., Boston (1985).
Wong R.T. "Worst-case analysis of network design problem heuristics" SIAM J. Alg. Disc. Meth. 1 (1980) 51–63.
Wong R.T. "A dual ascent approach for Steiner tree problems on a directed graph" Math. Programming 28 (1984) 271–287.
Wu Y.F., Widmeyer P. and Wong C.K. "A faster approximation algorithm for the Steiner problem in graphs" Acta Informatica 23 (1986) 223–229.
Yao A.C. "An O(|E|loglog|V|) algorithm for finding minimum spanning trees" Inf. Proc. Letters 4 (1975) 21–23.
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Winter, P. (1989). Topological network synthesis. In: Simeone, B. (eds) Combinatorial Optimization. Lecture Notes in Mathematics, vol 1403. Springer, Berlin, Heidelberg. https://doi.org/10.1007/BFb0083472
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DOI: https://doi.org/10.1007/BFb0083472
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