Abstract
We review the literature on minimum spanning tree problems with two or more objective functions (MOST) each of which is of the sum or bottleneck type. Theoretical aspects of different types of this problem are summarized and available algorithms are categorized and explained. The paper includes a concise tabular presentation of all the reviewed papers.
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References
Aggarwal, V., Aneja, Y.P., Nair, K.P.K.: Minimal spanning tree subject to a side constraint. Computers and Operations Research 9(4), 287–296 (1982)
Ahuja, R.K., Magnanti, T.L., Orlin, J.B.: Network Flows: Theory, Algorithms, and Applications. Prentice-Hall, Inc., Englewood Cliffs (1993)
Andersen, K.A., Joernsten, K., Lind, M.: On bicriterion minimal spanning trees: an approximation. Computers and Operations Research 23, 1171–1182 (1996)
Camerini, P.M., Galbiati, G., Maffioli, F.: The complexity of multi-constrained spanning tree problems. In: Lovasz, L. (ed.) Theory of Algorithms, pp. 53–101. North-Holland, Amsterdam (1984)
Chen, G., Guo, W., Tu, X., Chen, H.: An improved algorithm to solve the multi-criteria minimum spanning tree problem. J. Softw. 17(3), 364–370 (2006)
Corley, H.W.: Efficient spanning trees. Journal of Optimization Theory and Applications 45(3), 481–485 (1985)
Dell’Amico, M., Maffioli, F.: On some multicriteria arborescence problems: Complexity and algorithms. Discrete Applied Mathematics 65, 191–206 (1996)
Dell’Amico, M., Maffioli, F.: Combining linear and non-linear objectives in spanning tree problems. Journal of Combinatorial Optimiation 4(2), 253–269 (2000)
Duin, C.W., Volgenant, A.: Minimum deviation and balanced optimization: A unified approach. Operations Research Letters 10, 43–48 (1991)
Ehrgott, M.: Multicriteria Optimization, 2nd edn. Springer, Berlin (2005)
Ehrgott, M., Gandibleux, X.: Multiple Criteria Optimization: State of the Art Annotated Bibliographic Surveys. Kluwer Academic Publishers, Boston (2002)
Ehrgott, M., Klamroth, K.: Connectedness of efficient solutions in multiple criteria combinatorial optimization. European Journal of Operational Research 97, 159–166 (1997)
Gabow, H.N.: Two algorithms for generating weighted spanning trees in order. SIAM Journal on Computing 6, 139–150 (1977)
Goemans, M.X., Ravi, R.: A constrained minimum spanning tree problem. In: Proccedings of the Scandinavian Workshop on Algorithmic Theory (SWAT), pp. 66–75. Springer, Heidelberg (1996)
Gorski, J., Klamroth, K., Ruzika, S.: Connectedness of efficient solutions in multiple objective combinatorial optimization. Report in Wirtschaftsmathematik, Nr. 102/2006, University of Kaiserslautern, Department of Mathematics (2006) (submitted for publication)
Gupta, S.K., Punnen, A.P.: Minimum deviation problems. Operations Research Letters 7, 201–204 (1988)
Hamacher, H.W., Ruhe, G.: On spanning tree problems with multiple objectives. Annals of Operations Research 52, 209–230 (1994)
Hassin, R., Levin, A.: An efficient polynomial time approximation scheme for the constrained minimum spanning tree problem using matroid intersection. SIAM Journal on Computing 33(2), 261–268 (2004)
Henn, S.: The weight-constrained minimum spanning tree problem. Master’s thesis, Technische Universität Kaiserslautern (2007)
Hong, S.P., Chung, S.J., Park, B.H.: A fully polynomial bicriteria approximation scheme for the constrained spanning tree problem. Operations Research Letters 32, 233–239 (2004)
Hutson, V.A., ReVelle, C.S.: Maximal direct covering tree problem. Transportation Science 23, 288–299 (1989)
Hutson, V.A., ReVelle, C.S.: Indirect covering tree problems on spanning tree networks. European Journal of Operational Research 65, 20–32 (1993)
Melamed, I.I., Sigal, I.K.: An investigation of linear convolution of criteria in multicriteria discrete programming. Computational Mathematics and Mathematical Physics 35(8), 1009–1017 (1995)
Melamed, I.I., Sigal, I.K.: A computational investigation of linear parametrization of criteria in multicriteria discrete programming. Computational Mathematics and Mathematical Physics 36(10), 1341–1343 (1996)
Melamed, I.I., Sigal, I.K.: Numerical analysis of tricriteria tree and assignment problems. Computational Mathematics and Mathematical Physics 38(10), 1707–1714 (1998)
Melamed, I.I., Sigal, I.K.: Combinatorial optimization problems with two and three criteria. Doklady Mathematics 59(3), 490–493 (1999)
Minoux, M.: Solving combinatorial problems with combined min-max-min-sum objective and applications. Mathematical Programming 45, 361–372 (1989)
Perny, P., Spanjaard, O.: A preference-based approach to spanning trees and shortest paths problems. European Journal of Operational Research 162(3), 584–601 (2005)
Punnen, A.P.: On combined minmax-minsum optimization. Computers and Operations Research 21, 707–716 (1994)
Punnen, A.P., Nair, K.P.K.: A \(\mathcal{O}(m \log n)\) algorithm for the max + sum spanning tree problem. European Journal of Operational Research 89, 423–426 (1996)
Ramos, R.M., Alonso, S., Sicilia, J., Gonzalez, C.: The problem of the optimal biobjective spanning tree. European Journal of Operational Research 111, 617–628 (1998)
Ruzika, S.: On Multiple Objective Combinatorial Optimization. Ph.D thesis, Technische Universität Kaiserslautern (2007)
Schweigert, D.: Linear extensions and efficient trees. Preprint 172, University of Kaiserslautern, Department of Mathematics (1990)
Schweigert, D.: Linear extensions and vector-valued spanning trees. Methods of Operations Research 60, 219–222 (1990)
Sergienko, I.V., Perepelitsa, V.A.: Finding the set of alternatives in discrete multicriterion problems. Kibernetika 5, 85–93 (1987)
Shogan, A.: Constructing a minimal-cost spanning tree subject to resource constraints and flow requirements. Networks 13, 169–190 (1983)
Steiner, S., Radzik, T.: Solving the biobjective minimum spanning tree problem using k-best algorithm. Technical Report TR-03-06, Department of Computers Science, King’s College, London, UK (2003)
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Ruzika, S., Hamacher, H.W. (2009). A Survey on Multiple Objective Minimum Spanning Tree Problems. In: Lerner, J., Wagner, D., Zweig, K.A. (eds) Algorithmics of Large and Complex Networks. Lecture Notes in Computer Science, vol 5515. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-02094-0_6
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DOI: https://doi.org/10.1007/978-3-642-02094-0_6
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