Summary
We present in this paper a Lagrangean relaxation of a particular formulation for the shortest Hamiltonean path problem in a directed graph. The dual problem defined gives a lower bound. This relaxation has not the integrality property. To solve the dual problem we have to find a shortest path (not constrained to be elementary) with a fixed number of arcs, and solve a classical shortest path problem with nonnegative lengths.
Zusammenfassung
Wir geben eine Lagrange-Relaxation für das Problem der Bestimmung kürzester Hamilton'scher Pfade in gerichteten Graphen. Diese Relaxation erfüllt nicht die Ganzzahligkeitsbedingung, das entsprechende duale Problem definiert jedoch eine untere Schranke. Das duale Problem besteht in der Bestimmung kürzester Pfade — die nicht unbedingt einfach sein müssen — mit einer vorgegebenen Anzahl von Kanten. Dazu ist ein klassisches Kürzeste-Wege-Problem mit nicht negativen Pfadlängen zu lösen.
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Maculan, N., Salles, J.J.C. A lower bound for the shortest Hamiltonean path in directed graphs. OR Spektrum 13, 99–102 (1991). https://doi.org/10.1007/BF01719934
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DOI: https://doi.org/10.1007/BF01719934