One-dimensional minimal fillings with negative edge weights

TL;DR

This paper extends the concept of minimal fillings to include negative edge weights, proving that the minimal filling with possibly negative weights has the least total weight among all such generalized fillings.

Contribution

It introduces generalized minimal fillings allowing negative weights and proves their minimality property for any finite metric space.

Findings

Minimal fillings with negative weights exist and are minimal among all generalized fillings.

The minimal filling with negative weights has the least total weight.

The approach generalizes previous non-negative weight models.

Abstract

Ivanov and Tuzhilin started an investigation of a particular case of Gromov Minimal Fillings problem (generalized to the case of stratified manifolds). Weighted graphs with non-negative weight function were used as minimal fillings of finite metric spaces. In the present paper we introduce generalized minimal fillings, i.e. minimal fillings where the weight function is not necessarily non-negative. We prove that for any finite metric space its minimal filling has the minimum weight in the class of all generalized fillings of the space.