Solution of large linear systems with embedded network structure for a non-homogeneous network flow programming problem

TL;DR

This paper presents a direct solution method for large, sparse linear systems with embedded network structures, common in non-homogeneous network flow programming, utilizing graph-based decomposition techniques.

Contribution

The paper introduces a novel direct algorithm tailored for solving underdetermined systems with embedded network structures, leveraging graph theory for decomposition.

Findings

Effective solution for large sparse systems with network structure

Algorithm demonstrates efficiency on example problems

Provides theoretical basis for system decomposition

Abstract

In the paper we consider the linear underdetermined system of a special type. Systems of this type appear in non-homogeneous network flow programming problems in the form of systems of constraints and can be characterized as systems with a large sparse submatrix representing the embedded network structure. We develop a direct method for finding solutions of the system. The algorithm is based on the theoretic-graph specificities for the structure of the support and properties of the basis of a solution space of a homogeneous system. One of the key steps is decomposition of the system. A simple example is regarded at the end of the paper.