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An efficient heuristic for approximate maximum flow computations

Jingyun Qian, Georg Hahn

TL;DR

This work proposes a new approximation algorithm for the computation of the maximum flow with runtime O(|V||E|^2/k^2) compared to the usual runtime of O(|V||E|^2) for the Edmonds-Karp algorithm.

Abstract

Several concepts borrowed from graph theory are routinely used to better understand the inner workings of the (human) brain. To this end, a connectivity network of the brain is built first, which then allows one to assess quantities such as information flow and information routing via shortest path and maximum flow computations. Since brain networks typically contain several thousand nodes and edges, computational scaling is a key research area. In this contribution, we focus on approximate maximum flow computations in large brain networks. By combining graph partitioning with maximum flow computations, we propose a new approximation algorithm for the computation of the maximum flow with runtime O(|V||E|^2/k^2) compared to the usual runtime of O(|V||E|^2) for the Edmonds-Karp algorithm, where $V$ is the set of vertices, $E$ is the set of edges, and $k$ is the number of partitions. We assess both accuracy and runtime of the proposed algorithm on simulated graphs as well as on graphs downloaded from the Brain Networks Data Repository (https://networkrepository.com).

An efficient heuristic for approximate maximum flow computations

TL;DR

This work proposes a new approximation algorithm for the computation of the maximum flow with runtime O(|V||E|^2/k^2) compared to the usual runtime of O(|V||E|^2) for the Edmonds-Karp algorithm.

Abstract

Several concepts borrowed from graph theory are routinely used to better understand the inner workings of the (human) brain. To this end, a connectivity network of the brain is built first, which then allows one to assess quantities such as information flow and information routing via shortest path and maximum flow computations. Since brain networks typically contain several thousand nodes and edges, computational scaling is a key research area. In this contribution, we focus on approximate maximum flow computations in large brain networks. By combining graph partitioning with maximum flow computations, we propose a new approximation algorithm for the computation of the maximum flow with runtime O(|V||E|^2/k^2) compared to the usual runtime of O(|V||E|^2) for the Edmonds-Karp algorithm, where is the set of vertices, is the set of edges, and is the number of partitions. We assess both accuracy and runtime of the proposed algorithm on simulated graphs as well as on graphs downloaded from the Brain Networks Data Repository (https://networkrepository.com).
Paper Structure (13 sections, 1 equation, 5 figures, 1 table, 1 algorithm)

This paper contains 13 sections, 1 equation, 5 figures, 1 table, 1 algorithm.

Table of Contents

  1. Introduction
  2. Methods
  3. Problem specification
  4. The Edmonds-Karp algorithm
  5. The Kernighan-Lin partitioning algorithm
  6. A heuristic for computing the maximum flow
  7. Runtime
  8. Experimental studies
  9. Setting
  10. The Brain Networks Data Repository
  11. Simulated clustered graphs
  12. Application to brain network graphs
  13. Discussion

Figures (5)

  • Figure 1: Schematic idea of the computation of the maximum flow between and within partitions.
  • Figure 2: An example of clustered graph of 4 cluters each with 10 nodes, inp = 0.5, and outp = 0.05. Log scale on both axes.
  • Figure 3: The log runtime (left) and log max-flow value (right) in sparse and clustered graphs as a function of the number of nodes, where inp = 0.01 and outp = 0.005. The number of assumed clusters is $k=2^3$. Log scale on both axes.
  • Figure 4: The log runtime (left) and log max-flow value (right) in sparse and non-clustered graphs as a function of the number of nodes, where inp = 0.01 and outp = 0.01. The number of assumed clusters is $k=2^3$. Log scale on both axes.
  • Figure 5: The log runtime (left) as a function of the number of partitions $2^l$, where the level $l=\{0,2,4,6,8,10\}$ is given in the legend. The ratio of the runtime for any level $l>0$ to the one at $l=0$ (right). Log scale on both axes.