Abstract
In multiple criteria optimization an important research topic is the topological structure of the set X e of efficient solutions. Of major interest is the connectedness of X e , since it would allow the determination of X e without considering non-efficient solutions in the process. We review general results on the subject, including the connectedness result for efficient solutions in multiple criteria linear programming. This result can be used to derive a definition of connectedness for discrete optimization problems. We present a counterexample to the connectivity conjecture, namely that the sets of efficient solutions of the shortest path problem and the minimal spanning tree problem are connected. We will also present a general method to construct non-connected efficiency graphs for these problems.
partially supported by the Deutsche Forschungsgemeinschaft (DFG) and grant ERBCHRXCT930087 of the European HC&M Programme
supported by the Deutsche Forschungsgemeinschaft (DFG)
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© 1997 Springer-Verlag Berlin Heidelberg
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Ehrgott, M., Klamroth, K. (1997). Non-connected Efficiency Graphs in Multiple Criteria Combinatorial Optimization. In: Caballero, R., Ruiz, F., Steuer, R. (eds) Advances in Multiple Objective and Goal Programming. Lecture Notes in Economics and Mathematical Systems, vol 455. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-46854-4_15
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DOI: https://doi.org/10.1007/978-3-642-46854-4_15
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