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

Nithin Kavi

TL;DR

This work translates the near-linear-time max-flow/min-cost-flow framework of Chen et al. into a concrete implementation effort, assembling and partially implementing core components such as dynamic trees, low-stretch decomposition, and a rebuilding game. It provides detailed rationale for design choices, reports empirical validations for LSST and HLD, and offers heuristic IPM experiments to gauge practicality. While several subparts remain stubbed (notably expander decomposition and a fully correct IPM), the study clarifies the practical challenges and lays a roadmap for completing the implementation, with a clear analysis of the theoretical time bounds and where they manifest in practice. Overall, the paper contributes both a partial realizing of the algorithmic blueprint and a critical assessment of its near-linear-time promises, informing future work toward a fully operational, scalable solver.

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.

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

TL;DR

This work translates the near-linear-time max-flow/min-cost-flow framework of Chen et al. into a concrete implementation effort, assembling and partially implementing core components such as dynamic trees, low-stretch decomposition, and a rebuilding game. It provides detailed rationale for design choices, reports empirical validations for LSST and HLD, and offers heuristic IPM experiments to gauge practicality. While several subparts remain stubbed (notably expander decomposition and a fully correct IPM), the study clarifies the practical challenges and lays a roadmap for completing the implementation, with a clear analysis of the theoretical time bounds and where they manifest in practice. Overall, the paper contributes both a partial realizing of the algorithmic blueprint and a critical assessment of its near-linear-time promises, informing future work toward a fully operational, scalable solver.

Abstract

In 2022, Chen et al. proposed an algorithm in \cite{main} that solves the min cost flow problem in time, where is the number of edges in the graph, is an upper bound on capacities and 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 term more precisely. Finally, we conclude with potential directions for future work in this area.
Paper Structure (28 sections, 10 theorems, 18 equations, 6 figures)

This paper contains 28 sections, 10 theorems, 18 equations, 6 figures.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Definitions
  4. Layout of Algorithm
  5. Dynamic Trees
  6. Low Stretch Decomposition
  7. Low-stretch spanning trees
  8. create_petal
  9. petal_decomposition
  10. hierarchical-petal-decomposition
  11. LSST Implementation Validation
  12. Heavy Light Decomposition
  13. Implementation Details
  14. Heavy-Light Decomposition Implementation Validation
  15. Rebuilding Game
  16. ...and 13 more sections

Key Result

Lemma 3.1

(main) There is a deterministic data structure $\mathcal{D}^{(T)}$ that maintains a dynamic tree $T \subseteq G = (V, E)$ under insertion/deletion of edges with gradients $\mathbf{g}$ and lengths $\mathbf{\ell}$ and supports the following operations:

Figures (6)

  • Figure 1: For different random graphs, we graph the average stretch vs $m \log n \log \log n.$ Empirically, we can see that the stretch is bounded above by $0.045 m \log n \log \log n$ and that the pattern of growth appears to suggest that this growth rate will continue. The purple line is the best fit line which minimizes squared L2 error.
  • Figure 2: For different random graphs, we graph the average duration vs $m \log n \log \log n.$ Empirically, we can see that the duration is bounded above by $5.7 \cdot 10^{-6} m \log n \log \log n$ and that the pattern of growth appears to suggest that this growth rate will continue. The purple line is the best fit line which minimizes squared L2 error.
  • Figure 3: The average number of intersections appears to be bounded by $O(\log n)$. The purple line is the best fit line.
  • Figure 4: The time to compute the heavy light decomposition increases linearly with the number of vertices, which matches our desired $O(n)$ time complexity. The purple line is the best fit line.
  • Figure 5: The runtime of the rebuilding game appears to be $O\left(\frac{C_r K d}{\gamma_g}(m + T)\right)$ as stated in main. The purple line is the best fit line.
  • ...and 1 more figures

Theorems & Definitions (42)

  • Definition 2.1
  • Definition 2.2
  • Definition 2.4
  • Definition 2.6
  • Definition 2.7
  • Definition 2.8
  • Definition 2.9
  • Definition 2.10
  • Definition 2.11
  • Remark 2.12
  • ...and 32 more