# 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…

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## References

SHOWING 1-10 OF 34 REFERENCES

### Improved Time Bounds for the Maximum Flow Problem

- Computer ScienceSIAM 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 $O(nm\log (({n / m})(\log U)^{{1 / 2}} + 2))$.

### A Fast and Simple Algorithm for the Maximum Flow Problem

- Computer ScienceOper. 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.

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

- Computer ScienceSIAM 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.

### Beyond the flow decomposition barrier

- MathematicsProceedings 38th Annual Symposium on Foundations of Computer Science
- 1997

Borders are improved for the Gomory-Hu tree problem, the parametric flow problem, and the approximate s-t cut problem by introducing a new approach to the maximum flow problem.

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

- Computer Science, MathematicsJ. Comput. Syst. Sci.
- 1985

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.

### A new approach to the maximum-flow problem

- Computer ScienceJACM
- 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.

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

- BusinessActa 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.

### A data structure for dynamic trees

- Computer ScienceSTOC '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.

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

- Computer ScienceOper. 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.

### A New Combinatorial Approach for Sparse Graph Problems

- Computer Science, MathematicsICALP
- 2008

A new combinatorial data structure for representing arbitrary Boolean matrices that can perform fast vector multiplications with a given matrix, where the runtime depends on the sparsity of the input vector.