Faster Sparse Minimum Cost Flow by Electrical Flow Localization

@article{Axiotis2022FasterSM,
  title={Faster Sparse Minimum Cost Flow by Electrical Flow Localization},
  author={Kyriakos Axiotis and Aleksander Mkadry and Adrian Vladu},
  journal={2021 IEEE 62nd Annual Symposium on Foundations of Computer Science (FOCS)},
  year={2022},
  pages={528-539}
}
We give an $\tilde{O}(m^{3/2-1/762}\log(U+W))$ time algorithm for minimum cost flow with capacities bounded by $U$ and costs bounded by $W$. For sparse graphs with general capacities, this is the first algorithm to improve over the $\tilde{O}(m^{3/2}\log^{O(1)}(U+W))$ running time obtained by an appropriate instantiation of an interior point method [Daitch-Spielman, 2008]. Our approach is extending the framework put forth in [Gao-Liu-Peng, 2021] for computing the maximum flow in graphs with… 
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References

SHOWING 1-10 OF 26 REFERENCES
Circulation Control for Faster Minimum Cost Flow in Unit-Capacity Graphs
TLDR
This work obtains a faster algorithm for solving the minimum cost flow problem in graphs with unit capacity by combining a regularized version of the standard Newton step with a customized preconditioning method which aims to ensure that the graph on which these circulations are computed has sufficiently large conductance.
Computing Maximum Flow with Augmenting Electrical Flows
  • A. Madry
  • Computer Science, Mathematics
    2016 IEEE 57th Annual Symposium on Foundations of Computer Science (FOCS)
  • 2016
TLDR
The presented algorithm takes a primal dual approach in which each iteration uses electrical flows computations both to find an augmenting s-t flow in the current residual graph and to update the dual solution, and shows that by maintain certain careful coupling of these primal and dual solutions the authors are always guaranteed to make significant progress.
Navigating Central Path with Electrical Flows: From Flows to Matchings, and Back
  • A. Madry
  • Computer Science
    2013 IEEE 54th Annual Symposium on Foundations of Computer Science
  • 2013
TLDR
A deeper understanding of interior-point methods is acquired - a powerful tool in convex optimization - in the context of flow problems, as well as, utilizing certain interplay between maximum flows and bipartite matchings.
A new approach to computing maximum flows using electrical flows
TLDR
An algorithm which computes a (1-ε)-approximately maximum st-flow in an undirected uncapacitated graph in time O(1/ε√m/F⋅ m log2 n) where F is the flow value and the minimizer is related to an approximate blocking flow is shown.
Negative-Weight Shortest Paths and Unit Capacity Minimum Cost Flow in Õ(m 10/7 log W) Time.
TLDR
This paper studies a set of combinatorial optimization problems on weighted graphs, and shows that each one of these four problems can be solved in O(m) time, which gives the first in over 25 years polynomial improvement in their sparse-graph time complexity.
Bipartite Matching in Nearly-linear Time on Moderately Dense Graphs
  • J. V. D. Brand, Y. Lee, Di Wang
  • Computer Science
    2020 IEEE 61st Annual Symposium on Foundations of Computer Science (FOCS)
  • 2020
TLDR
A simple sublinear-time algorithm for detecting and sampling high-energy edges in electric flows on expanders and show that when combined with recent advances in dynamic expander decompositions, this yields efficient data structures for maintaining the iterates of both [v.d.Brand-Lee-Sidford-Song 2020] and the authors' new IPMs.
Faster energy maximization for faster maximum flow
TLDR
An algorithm which given any m-edge n-vertex directed graph with integer capacities at most U computes a maximum s-t flow for any vertices s and t in m 11/8+o(1) U 1/4 time with high probability.
Faster approximate lossy generalized flow via interior point algorithms
TLDR
These algorithms reduce the problem of solving a linear system in a symmetric M-matrix to that of solving O{log n} linear systems in symmetric diagonally-dominant matrices, which they can do in time Om using the algorithm of Spielman and Teng.
Electrical flows, laplacian systems, and faster approximation of maximum flow in undirected graphs
TLDR
This work introduces a new approach to computing an approximately maximum s-t flow in a capacitated, undirected graph and develops the fastest known algorithm for computing approximately maximums-t flows.
Nearly Maximum Flows in Nearly Linear Time
  • Jonah Sherman
  • Computer Science
    2013 IEEE 54th Annual Symposium on Foundations of Computer Science
  • 2013
We introduce a new approach to the maximum flow problem in undirected, capacitated graphs using congestion-approximators: easy-to-compute functions that approximate the congestion required to route
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