Tropical Kirchhoff's Formula and Postoptimality in Matroid Optimization

Abstract

Given an assignment of real weights to the ground elements of a matroid, the min-max weight of a ground element $e$ is the minimum, over all circuits containing $e$, of the maximum weight of an element in that circuit with the element $e$ removed. We use this concept to answer the following structural questions for the minimum weight basis problem. Which elements are persistent under a given weighting (belong to all or to none of the optimal bases)? What changes of the weights are allowed while preserving optimality of optimal bases? How does the minimum weight of a basis change when the weight of a single ground element is changed, or when a ground element is contracted or deleted? Our answer to this latter question gives the tropical (min,+,-) analogue of Kirchhoff's arithmetic (+,x,/) effective conductance formula for electrical networks.