Partial Implementation of Max Flow and Min Cost Flow in Almost-Linear Time

TL;DR

This paper discusses partial implementation details of a nearly-linear time algorithm for min cost flow problems, clarifying choices and providing insights for future research.

Contribution

It provides a detailed implementation discussion of a recent nearly-linear time min cost flow algorithm, including justifications and partial code stubs.

Findings

Implementation of key algorithm components

Refined analysis of the algorithm's complexity

Suggestions for future research directions

Abstract

In 2022, Chen et al. proposed an algorithm in \cite{main} that solves the min cost flow problem in $m^{1 + o(1)} \log U \log C$ time, where $m$ is the number of edges in the graph, $U$ is an upper bound on capacities and $C$ is an upper bound on costs. However, as far as the authors of \cite{main} know, no one has implemented their algorithm to date. In this paper, we discuss implementations of several key portions of the algorithm given in \cite{main}, including the justifications for specific implementation choices. For the portions of the algorithm that we do not implement, we provide stubs. We then go through the entire algorithm and calculate the $m^{o(1)}$ term more precisely. Finally, we conclude with potential directions for future work in this area.