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Weight reduction problems with certain bottleneck objectives. (English) Zbl 1137.90689
Summary: This paper is concerned with bottleneck weight reduction problems (WRPs) stated as follows. We are given a finite set \(E\), a class \(\mathcal F\) of nonempty subsets of \(E\), a weight \(w : E \to \mathbb R^+\) and a cost \(c: E \to \mathbb R^+\) . For each \(e\in E\), \(c(e)\) stands for the cost of reducing weight \(w(e)\) by one unit. For each subset \(F \in \mathcal F\), the bottleneck weight of \(F\) is \(w(F)=\min_{e\in F}\;w(e)\). The weight of the family \(\mathcal F\) is the maximum of \(w(F)\) for all \(F\) in \(\mathcal F\). The problem is to determine new weights \(x(e) \leqslant w(e)\) such that the weight of \(\mathcal F\) is minimized under the constraint that the overall reduction cost does not exceed a given budget \(B\). Similarly to capacity expansion problems, WRPs include \(\mathcal{NP}\)-hard problems. A WRP can be formulated as a parametric optimization problem over all transversal sets \(T\) of the class \(\mathcal F\). This leads to (strongly) polynomial solution procedures for special systems . In particular we outline a polynomial algorithm in the case when \(\mathcal F\) is the class of all spanning trees in an undirected graph.

MSC:
90C35 Programming involving graphs or networks
90C31 Sensitivity, stability, parametric optimization
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