Abstract
For two classes of network flow problems a worst-case analysis is given depending on the number of vertices of a pathological graph of Zadeh. Firstly, an exponential number of breakpoints in the optimal value function of the maximal flow problem in generalized networks with parametric capacities is demonstrated. Secondly, it is shown that the bicriterial min-cost flow has, in the worst case. an exponential number of efficient extreme point solutions in the objective space.
Zusammenfassung
Für zwei Klassen von Netzwerkflußproblemen wird eine worst-case Analyse in Abhängigkeit von der Anzahl der Knoten eines pathologischen Graphen von Zadeh vorgenommen. Demnach besitzt das Maximalflußproblem in verallgemeinerten Netzwerken mit parametrischen Kapazitätsbeschränkungen eine exponentielle Anzahl von Knickstellen in der Optimalwertfunktion. und kostenminimale Flüsse haben bereits für zwei Kriterien eine exponentielle Zahl von effizienten Eckpunkten im Bildbereich.
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References
Carstensen PJ (1983) Complexity of some parametric integer and network programming problems. Math Progr 26:64–75
Christofides N (1975) Graph theory — an algorithmic approach. Academic Press, New York London San Francisco
Geoffrion AM (1966) Solving bicriterion mathematical programs. Operations Research 15: 39–54
Hamacher HW, Foulds LR (1986) Algorithms for flows with parametric capacities. Research Report 86-1, University of Florida, Center for Optimization and Combinatorics
Hansen P (1980) Bicriterion path problems. In: Fandel G, Gal T (eds) Multiple criteria decision making — Theory and applications. Lecture Notes in Economics and Mathematical Systems, vol 177:109–127
Ruhe G (1987) Parametric maximal flows in generalized networks — Complexity and algorithms. Appears in: Optimization
Zadeh N (1973) A bad network flow problem for the simplex method and other minimum cost flow algorithms. Math Progr 5:255–266
Zionts S (1985) Multiple criteria mathematical programming: an overview and several approaches. In: Mathematics of multi-objective optimization. CISM Courses and Lectures No. 289, International Center for Mechanical Science. Springer, Wien, pp 85–128
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Ruhe, G. Complexity results for multicriterial and parametric network flows using a pathological graph of Zadeh. Zeitschrift für Operations Research 32, 9–27 (1988). https://doi.org/10.1007/BF01920568
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DOI: https://doi.org/10.1007/BF01920568