# Max flows in O(nm) time, or better

@inproceedings{Orlin2013MaxFI, title={Max flows in O(nm) time, or better}, author={James B. Orlin}, booktitle={STOC '13}, year={2013} }

In this paper, we present improved polynomial time algorithms for the max flow problem defined on sparse networks with n nodes and m arcs. We show how to solve the max flow problem in O(nm + m31/16 log2 n) time. In the case that m = O(n1.06), this improves upon the best previous algorithm due to King, Rao, and Tarjan, who solved the max flow problem in O(nm logm/(n log n)n) time. This establishes that the max flow problem is solvable in O(nm) time for all values of n and m. In the case that m… Expand

#### Tables and Topics from this paper

#### 349 Citations

O(nm)-TIME ALGORITHM FOR MAX FLOW

- 2013

In this paper, we describe the recent work by Orlin[1] which gives a strongly-polynomial O(nm)-time algorithm for max flow, where m is the number of edges in our graph and n is the number of nodes.… Expand

A Fast Max Flow Algorithm

- Mathematics, Computer Science
- ArXiv
- 2019

A new variant of the excess scaling algorithm for the max flow problem whose running time strictly dominates the running time of the algorithm by King et al., and for graphs in which $m = O(n \log n)$, the runningTime dominates that of King etAl. Expand

Unit Capacity Maxflow in Almost O(m) Time

- 2020

We present an algorithm, which given any m-edge n-vertex directed graph with positive integer capacities at most U computes a maximum s-t flow for any vertices s and t in O(mU) time. This improves… Expand

Faster energy maximization for faster maximum flow

- Mathematics, Computer Science
- STOC
- 2020

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. Expand

A New Push-Relabel Algorithm for the Maximum Flow Problem

- Computer Science
- ArXiv
- 2013

A faster push-relabel algorithm for the maximum flow problem on bounded-degree networks with n vertices and m arcs is presented and an algorithm incorporating some or all of the techniques may be a promising avenue towards an O(mn)-time algorithm for all edge densities. Expand

Computing Maximum Flow with Augmenting Electrical Flows

- Mathematics, Computer Science
- 2016 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. Expand

A Faster Algorithm for Maximum Flow in Directed Planar Graphs with Vertex Capacities

- Computer Science
- ISAAC
- 2021

11 We give an O(k3n log n min(k, log2 n) log2(nC))-time algorithm for computing maximum integer 12 flows in planar graphs with integer arc and vertex capacities bounded by C, and k sources and sinks.… Expand

A faster strongly polynomial time algorithm to solve the minimum cost tension problem

- Mathematics, Computer Science
- J. Comb. Optim.
- 2017

This paper presents a strongly polynomial time algorithm for the minimum cost tension problem, which runs in O(max{m3n, m2logn(m+nlogn)}) time, where n and m denote the number of nodes and number of arcs, respectively. Expand

An $$O(mn \log U)$$O(mnlogU) time algorithm for estimating the maximum cost of adjusting an infeasible network

- Computer Science
- Telecommun. Syst.
- 2016

An algorithm to solve the feasibility problem, where n, m and U are the number of nodes, arcs, and the value of maximum upper bound, respectively, which improves upon the previous method due to Ghiyasvand. Expand

Review of Generic Preflow Push Algorithm

- 2018

In 2013, Orlin proved that the max flow problem could be solved in O(nm) time. His algorithm ran in O(nm + m) time, which was the fastest for very sparse graphs. If the graph was not sufficiently… Expand

#### References

SHOWING 1-10 OF 34 REFERENCES

Improved Time Bounds for the Maximum Flow Problem

- Mathematics, Computer Science
- SIAM J. Comput.
- 1989

Possible improvements to the Ahuja-Orlin algorithm are explored and it is shown that the use of dynamic trees in the latter algorithm reduces the running time to $O(nm\log (({n / m})(\log U)^{{1 / 2}} + 2))$. Expand

A Fast and Simple Algorithm for the Maximum Flow Problem

- Mathematics, Computer Science
- Oper. Res.
- 1989

This work presents a simple sequential algorithm for the maximum flow problem on a network with n nodes, m arcs, and integer arc capacities bounded by U and describes a parallel implementation that runs in On2 log U log p time in the PRAM model with EREW and uses only p processors. Expand

An o(n³)-Time Algorithm Maximum-Flow Algorithm

- Mathematics, Computer Science
- SIAM J. Comput.
- 1996

It is argued that several of the results yield parallel algorithms with optimal speedup that are based on O(n^3/\log n) time on a uniform-cost RAM. Expand

Theoretical Improvements in Algorithmic Efficiency for Network Flow Problems

- Mathematics, Computer Science
- J. ACM
- 1972

New algorithms for the maximum flow problem, the Hitchcock transportation problem, and the general minimum-cost flow problem are presented, and Dinic shows that, in a network with n nodes and p arcs, a maximum flow can be computed in 0 (n2p) primitive operations by an algorithm which augments along shortest augmenting paths. Expand

A linear-time algorithm for a special case of disjoint set union

- Mathematics, Computer Science
- STOC
- 1983

A linear-time algorithm for the special case of the disjoint set union problem in which the structure of the unions (defined by a “union tree”) is known in advance, which gives similar improvements in the efficiency of algorithms for solving a number of other problems. Expand

A new approach to the maximum-flow problem

- Mathematics, Computer Science
- JACM
- 1988

An alternative method based on the preflow concept of Karzanov, which runs as fast as any other known method on dense graphs, achieving an O(n) time bound on an n-vertex graph and faster on graphs of moderate density. Expand

An O(V5/3E2/3) algorithm for the maximal flow problem

- Mathematics, Computer Science
- Acta Informatica
- 2004

A new algorithm for finding a maximal flow in a given network is presented that runs in time O(V5/3E2/3), where V and E are the number of the vertices and edges in the network. Expand

A data structure for dynamic trees

- Computer Science, Mathematics
- STOC '81
- 1981

An O(mn log n)-time algorithm is obtained to find a maximum flow in a network of n vertices and m edges, beating by a factor of log n the fastest algorithm previously known for sparse graphs. Expand

Maximal Flow Through a Network

- Mathematics
- 1956

Introduction. The problem discussed in this paper was formulated by T. Harris as follows: "Consider a rail network connecting two cities by way of a number of intermediate cities, where each link of… Expand

A Computational Study of the Pseudoflow and Push-Relabel Algorithms for the Maximum Flow Problem

- Mathematics, Computer Science
- Oper. Res.
- 2009

The results show that the implementation of the pseudoflow algorithm is faster than the best-known implementation of push-relabel on most of the problem instances within this computational study. Expand