Tropical Kirchhoff's Formula and Postoptimality in Matroid Optimization

TL;DR

This paper introduces a tropical analogue of Kirchhoff's formula for matroid optimization, providing structural insights into optimal bases and their stability under weight changes.

Contribution

It develops a tropical (min,+,-) framework for analyzing matroid bases, extending classical electrical network formulas to combinatorial optimization.

Findings

Characterizes persistent elements in optimal bases.

Describes weight change conditions preserving optimality.

Provides a formula for basis weight variation under element modifications.

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.