# Flow Rounding

@article{Kang2015FlowR, title={Flow Rounding}, author={Donggu Kang and James Payor}, journal={ArXiv}, year={2015}, volume={abs/1507.08139} }

We consider flow rounding: finding an integral flow from a fractional flow. Costed flow rounding asks that we find an integral flow with no worse cost. Randomized flow rounding requires we randomly find an integral flow such that the expected flow along each edge matches the fractional flow. Both problems are reduced to cycle canceling, for which we develop an O(m log n 2 m ) algorithm.

## 8 Citations

### Maximum Integer Flows in Directed Planar Graphs with Multiple Sources and Sinks and Vertex Capacities

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This work considers the problem of finding maximum flows in planar graphs with capacities on both vertices and edges and with multiple sources and sinks and presents three algorithms that improve on the fastest previously known algorithms when the capacities are integers.

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### Tight Bounds for Approximate Carathéodory and Beyond

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- 2017

The result provides a constructive proof for the Approximate Carath-eodory Problem, which states that any point inside a polytope contained in the ball of radius $D$ can be approximated to within $\epsilon$ in $\ell_p$ norm by a convex combination of only $O\left(D^2 p/\ep silon^2\right)$ vertices of the polytopes for $p \geq 2$.

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- 2018

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### Fully Dynamic Electrical Flows: Sparse Maxflow Faster Than Goldberg-Rao

- Computer Science2021 IEEE 62nd Annual Symposium on Foundations of Computer Science (FOCS)
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The algorithm revolves around dynamically maintaining the augmenting electrical flows at the core of the interior point method based algorithm from [Mądry JACM '16]. This entails designing data structures that, in limited settings, return edges with large electric energy in a graph undergoing resistance updates.

### Incremental preference adjustment: a graph-theoretical approach

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The Minimum Dimension Adjustment (MDA) problem is defined, where the preference adjustments are under certain constraints imposed by a specific graph and the goal is to adjust a user’s preference by reversing the personalized rank of two given items while minimizing the number of dimensions with value changed in the preference vector.

### $O(N^3)$ Measurement Cost for Variational Quantum Eigensolver on Molecular Hamiltonians

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A fast, precomputable procedure for creating linearly sized commuting partitions by solving a round-robin scheduling problem via flow networks and the statistical implication of simultaneous measurement is provided.

## References

SHOWING 1-8 OF 8 REFERENCES

### Provably good routing in graphs: regular arrays

- Computer ScienceSTOC '85
- 1985

We examine the problem of routing wires on a VLSI chip where the nodes to be connected are arranged in a two-dimensional array. We develop provably good algorithms that find a solution close to the…

### Randomized rounding: A technique for provably good algorithms and algorithmic proofs

- Computer Science, MathematicsComb.
- 1987

A randomized algorithm for transforming an optimal solution of a relaxed problem into a provably good solution for the 0–1 problem is given and can be extended to provide bounds on the disparity between the rational and 0-1 optima for a given problem instance.

### Optimal Rounding of Instantaneous Fractional Flows Over Time

- MathematicsSIAM J. Discret. Math.
- 2000

This paper improves upon the previous best algorithm to solve the problem without costs by a factor of k and starts with a stationary fractional flow and uses rounding to transform this into an integral flow.

### Approximate Max-Flow on Small Depth Networks

- Computer ScienceSIAM J. Comput.
- 1995

A new deterministic algorithm for solving the relaxed problem of computing an $s$-$t$ flow of value at least $(1-\epsilon)$ of the maximum flow, which is in $\NC$ and uses only $O(m)$ processors, a significant improvement over existing parallel algorithms.

### 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.

### 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.

### 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.

### Finding minimum-cost circulations by canceling negative cycles

- Computer ScienceSTOC '88
- 1988

It is shown that a judicious choice of cycles for canceling leads to a polynomial bound on the number of iterations in this algorithm, which is comparable to those of the fastest previously known algorithms.