# Solving Splitted Multi-Commodity Flow Problem by Efficient Linear Programming Algorithm

## TL;DR

This paper introduces two novel algorithms, locSolver and incSolver, that significantly improve the efficiency of solving linear equations in column generation for multi-commodity flow problems by exploiting sparsity and solution reuse.

## Contribution

The paper presents new algorithms for solving sparse linear systems more efficiently within column generation, reducing computational time in multi-commodity flow problem solving.

## Key findings

- incSolver is at least 37 times faster than LAPACK.
- Algorithms effectively exploit sparsity and solution similarity.
- Preliminary experiments demonstrate substantial speedups.

## Abstract

Column generation is often used to solve multi-commodity flow problems. A program for column generation always includes a module that solves a linear equation. In this paper, we address three major issues in solving linear problem during column generation procedure which are (1) how to employ the sparse property of the coefficient matrix; (2) how to reduce the size of the coefficient matrix; and (3) how to reuse the solution to a similar equation. To this end, we first analyze the sparse property of coefficient matrix of linear equations and find that the matrices occurring in iteration are very sparse. Then, we present an algorithm locSolver (for localized system solver) for linear equations with sparse coefficient matrices and right-hand-sides. This algorithm can reduce the number of variables. After that, we present the algorithm incSolver (for incremental system solver) which utilizes similarity in the iterations of the program for a linear equation system. All three techniques can be used in column generation of multi-commodity problems. Preliminary numerical experiments show that the incSolver is significantly faster than the existing algorithms. For example, random test cases show that incSolver is at least 37 times and up to 341 times faster than popular solver LAPACK.