# Electrical flows, laplacian systems, and faster approximation of maximum flow in undirected graphs

@inproceedings{Christiano2011ElectricalFL, title={Electrical flows, laplacian systems, and faster approximation of maximum flow in undirected graphs}, author={Paul Christiano and Jonathan A. Kelner and Aleksander Madry and Daniel A. Spielman and Shang-Hua Teng}, booktitle={STOC '11}, year={2011} }

We introduce a new approach to computing an approximately maximum s-t flow in a capacitated, undirected graph. This flow is computed by solving a sequence of electrical flow problems. Each electrical flow is given by the solution of a system of linear equations in a Laplacian matrix, and thus may be approximately computed in nearly-linear time. Using this approach, we develop the fastest known algorithm for computing approximately maximum s-t flows. For a graph having n vertices and m edges…

## 301 Citations

A new approach to computing maximum flows using electrical flows

- Computer ScienceSTOC '13
- 2013

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.

Computing Maximum Flow with Augmenting Electrical Flows

- Computer Science, Mathematics2016 IEEE 57th Annual Symposium on Foundations of Computer Science (FOCS)
- 2016

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.

An Almost-Linear-Time Algorithm for Approximate Max Flow in Undirected Graphs, and its Multicommodity Generalizations

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- 2014

A new framework for approximately solving flow problems in capacitated, undirected graphs is introduced and it is applied to provide asymptotically faster algorithms for the maximum s-t flow and maximum concurrent multicommodity flow problems.

Fast Approximation of Maximum Flow using Electrical Flows

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- 2011

A proof that the algorithm can be run without the binary search technique described in the original paper is presented, and how the slightly modified algorithm may help better understand the overall behavior of the technique is discussed.

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Lecture 23 1 Fast Max-flow 1.2 Experts for Max Flow

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Given undirected graph G(V, E) where each edge has capacity 1, the objective is to find the maximum flow from s to t, such that the flow on an edge does not exceed its capacity. The running time for…

Almost Linear Time Algorithms for Flows in Graphs

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- 2017

An algorithm to find an approximately maximum flow in undirected graphs in almost-linear time with the help of nearly-linear Laplacian solvers and the multiplicative weights update method is described.

Approximate Maximum Flow on Separable Undirected Graphs

- MathematicsSODA
- 2013

Key to the algorithm is an intermediate problem that is term grouped L2 flow, which exists between maximum flows and electrical flows, and makes use of spectral vertex sparsifiers in order to remove vertices while preserving the energy dissipation of electrical flows.

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- Computer ScienceSTOC '12
- 2012

This paper gives algorithms that find 1-ε approximate solutions to the maximum concurrent flow problem and maximum weighted multicommodity flow problem in time O(m4/3poly(k,ε-1)).

Navigating Central Path with Electrical Flows: From Flows to Matchings, and Back

- Computer Science2013 IEEE 54th Annual Symposium on Foundations of Computer Science
- 2013

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.

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