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Given an assignment of real weights to the ground elements of a matroid, themin-max weight of a ground element $e$ is the minimum, over all circuitscontaining $e$, of the maximum weight of an element in that circuit with theelement $e$ removed. We use this concept to answer the following structuralquestions for the minimum weight basis problem. Which elements are persistentunder a given weighting (belong to all or to none of the optimal bases)? Whatchanges of the weights are allowed while preserving optimality of optimalbases? How does the minimum weight of a basis change when the weight of asingle ground element is changed, or when a ground element is contracted ordeleted? Our answer to this latter question gives the tropical (min,+,-)analogue of Kirchhoff's arithmetic (+,x,/) effective conductance formula forelectrical networks.

Tropical Kirchhoff's Formula and Postoptimality in Matroid Optimization