Weight reduction problems with certain bottleneck objectives

Abstract This paper is concerned with bottleneck weight reduction problems (WRPs) stated as follows. We are given a finite set E , a class F of nonempty subsets of E , a weight w:E→ R + and a cost c:E→ R + . For each e ∈ E , c ( e ) stands for the cost of reducing weight w ( e ) by one unit. For each subset F∈ F , the bottleneck weight of F is w ( F )=min e ∈ F w ( e ). The weight of the family F is the maximum of w ( F ) for all F in F . The problem is to determine new weights x ( e )⩽ w ( e ) such that the weight of 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 NP -hard problems. A WRP can be formulated as a parametric optimization problem over all transversal sets T of the class F . This leads to (strongly) polynomial solution procedures for special systems F . In particular we outline a polynomial algorithm in the case when F is the class of all spanning trees in an undirected graph.

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