# Dual Linear Programming Problem and One-Dimensional Gromov Minimal Fillings of Finite Metric Spaces

## TL;DR

This paper investigates minimal parametric fillings of finite metric spaces using linear programming, improving estimates, providing alternative proofs, and deriving explicit formulas for spaces with 5 and 6 points.

## Contribution

It offers new bounds on multi-tours, an alternative proof of the minimal filling formula, and explicit formulas for small finite metric spaces.

## Key findings

- Improved estimate on multi-tour multiplicity
- Alternative proof of minimal filling weight formula
- Explicit formulas for 5- and 6-point finite spaces

## Abstract

The present paper is devoted to studying of minimal parametric fillings of finite metric spaces (a version of optimal connection problem) by methods of Linear Programming. The estimate on the multiplicity of multi-tours appearing in the formula of weight of minimal fillings is improved, an alternative proof of this formula is obtained, and also explicit formulas for finite spaces consisting of $5$ and $6$ points are derived.