Generalized max-flows and min-cuts in simplicial complexes

@inproceedings{Maxwell2021GeneralizedMA,
  title={Generalized max-flows and min-cuts in simplicial complexes},
  author={William Maxwell and Amir Nayyeri},
  booktitle={ESA},
  year={2021}
}
We consider high dimensional variants of the maximum flow and minimum cut problems in the setting of simplicial complexes and provide both algorithmic and hardness results. By viewing flows and cuts topologically in terms of the simplicial (co)boundary operator we can state these problems as linear programs and show that they are dual to one another. Unlike graphs, complexes with integral capacity constraints may have fractional max-flows. We show that computing a maximum integral flow is NP… 
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References

SHOWING 1-10 OF 30 REFERENCES

Homology flows, cohomology cuts

TLDR
This work describes the first algorithms to compute maximum flows in surface-embedded graphs in near-linear time, and key insight is to optimize the relative homology class of the flow, rather than directly optimizing the flow itself.

Minimum-cost integer circulations in given homology classes

TLDR
It is shown that the convex hull of feasible solutions has a very simple polyhedral description and a pseudo-polynomial time algorithm is described for the case in which the surface has fixed genus and the circulation is only restricted to be non-negative.

Cuts and flows of cell complexes

We study the vector spaces and integer lattices of cuts and flows associated with an arbitrary finite CW complex, and their relationships to group invariants including the critical group of a

Simplicial and Cellular Trees

Much information about a graph can be obtained by studying its spanning trees. On the other hand, a graph can be regarded as a 1-dimensional cell complex, raising the question of developing a theory

Improved algorithms for min cut and max flow in undirected planar graphs

TLDR
These are the first algorithms breaking the O(n log n) barrier for those two problems, which has been standing for more than 25 years, and the first known non-trivial dynamic algorithm for min st-cut and max st-flow.

Minimum cuts and shortest homologous cycles

TLDR
The first algorithms to compute minimum cuts in surface-embedded graphs in near-linear time are described and it is proved that finding a minimum-cost subgraph homologous to a single input cycle is {NP}-hard.

The Maxflow problem and a generalization to simplicial complexes

TLDR
In the last section, this work suggests a generalization in the context of simplicial complexes, that reduces to the problem of Maxflow in graphs, when the authors consider a graph as a simplicial complex of dimension 1.

Hardness Results for Homology Localization

TLDR
This paper addresses the problem of localizing homology classes, namely, finding the cycle representing a given class with the most concise geometric measure, and proves the problem is NP-hard to approximate within any constant factor.

Computing Minimal Persistent Cycles: Polynomial and Hard Cases

TLDR
This paper proves that it is NP-hard to compute minimal persistent d-cycles (d>1) for both types of intervals given arbitrary simplicial complexes and identifies two interesting cases which are polynomially tractable.

An O(n log2 n) Algorithm for Maximum Flow in Undirected Planar Networks

A new algorithm is given to find a maximum flow in an undirected planar flow network in $O(n\log ^2 n)$ time, which is faster than the best method previously known by a factor of $\sqrt n /\log n$.