• Corpus ID: 244798799

# Gomory-Hu Trees in Quadratic Time

@article{Zhang2021GomoryHuTI,
author={Tianyi Zhang},
journal={ArXiv},
year={2021},
volume={abs/2112.01042}
}
Gomory-Hu tree [Gomory and Hu, 1961] is a succinct representation of pairwise minimum cuts in an undirected graph. When the input graph has general edge weights, classic algorithms need at least cubic running time to compute a Gomory-Hu tree. Very recently, the authors of [AKL+, arXiv v1, 2021] have improved the running time to Õ(n) which breaks the cubic barrier for the first time. In this paper, we refine their approach and improve the running time to Õ(n). This quadratic upper bound is also…

## References

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• Computer Science
ArXiv
• 2021
For unweighted graphs, the techniques show equivalence (up to poly-logarithmic factors in running time) between Gomory-Hu tree (i.e., all-pairs max-flow values) and a singlepair max- flow.
APMF < APSP? Gomory-Hu Tree for Unweighted Graphs in Almost-Quadratic Time*
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2021 IEEE 62nd Annual Symposium on Foundations of Computer Science (FOCS)
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The designed algorithm is the first subcubic deterministic algorithm for Gomory-Hu Tree in simple graphs, showing that randomness is not necessary for beating the $n-1$ times max-flow bound from 1961.
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The first Õ(mn) algorithm for constructing a Gomory-Hu tree for simple unweighted graphs is presented, which is an efficient tree packing algorithm for computing Steiner edge connectivity and uses this algorithm as the main subroutine.
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• Computer Science
SODA
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Under the hypothesis that an Õ(n)-edge sparsifier that preserves all friendly minimum st-cuts can be computed efficiently, the upper bound improves to Õ (m + n) which is the best possible without breaking the cubic barrier for constructing Gomory-Hu trees in non-simple graphs.
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An algorithm that finds k edge-disjoint arborescences on a directed graph in time O(kmn + k3n2)2 is presented, based on two theorems of Edmonds that link these two problems and show how they can be solved.
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SODA
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Very Simple Methods for All Pairs Network Flow Analysis
A very simple version of the Gomory–Hu cut tree method that finds one minimum cut for every pair of nodes is derived, and it is shown that the seemingly fundamental operation of that method, node contraction, is not needed, nor must crossing cuts be avoided.
Faster energy maximization for faster maximum flow
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STOC
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Computing Maximum Flow with Augmenting Electrical Flows