• Corpus ID: 244798799

Gomory-Hu Trees in Quadratic Time

@article{Zhang2021GomoryHuTI,
  title={Gomory-Hu Trees in Quadratic Time},
  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

SHOWING 1-10 OF 18 REFERENCES
Gomory-Hu Tree in Subcubic Time
TLDR
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*
TLDR
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.
An Õ(mn) Gomory-Hu tree construction algorithm for unweighted graphs
TLDR
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.
Faster Cut-Equivalent Trees in Simple Graphs
TLDR
This paper improves the running time of the proposed first subcubic time algorithm for constructing a cut-equivalent tree to Ô(n) if almost-linear time max-flow algorithms exist, and also uses the currently fastestmax-flow algorithm by van den Brand.
Friendly Cut Sparsifiers and Faster Gomory-Hu Trees
TLDR
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.
A matroid approach to finding edge connectivity and packing arborescences
  • H. Gabow
  • Computer Science, Mathematics
    STOC '91
  • 1991
TLDR
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.
New Algorithms and Lower Bounds for All-Pairs Max-Flow in Undirected Graphs
TLDR
The first hardness reductions for All-Pairs Max-Flow in undirected graphs are designed, giving an essentially optimal lower bound for the node-capacities setting and explaining the lack of lower bounds by proving a $\textit{non-reducibility}$ result.
Very Simple Methods for All Pairs Network Flow Analysis
TLDR
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
TLDR
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.
Computing Maximum Flow with Augmenting Electrical Flows
  • A. Madry
  • Computer Science, Mathematics
    2016 IEEE 57th Annual Symposium on Foundations of Computer Science (FOCS)
  • 2016
TLDR
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.
...
1
2
...