# 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…

## 3 Citations

### The complexity of high-dimensional cuts

- Mathematics, Computer ScienceArXiv
- 2021

The algorithmic study of some cut problems in high dimensions, namely, Topological Hitting Set and Boundary Nontrivialization, and randomized (approximation) FPT algorithms for the global variants of THS and BNT are initiated.

### Hardness Results for Laplacians of Simplicial Complexes via Sparse-Linear Equation Complete Gadgets

- MathematicsICALP
- 2022

We study linear equations in combinatorial Laplacians of k -dimensional simplicial complexes ( k -complexes), a natural generalization of graph Laplacians. Combinatorial Laplacians play a crucial…

### Effective Resistance and Capacitance in Simplicial Complexes and a Quantum Algorithm

- Mathematics
- 2021

We investigate generalizations of the graph theoretic notions of effective resistance and capacitance to simplicial complexes and prove analogs of formulas known in the case of graphs. In graphs the…

## References

SHOWING 1-10 OF 30 REFERENCES

### Homology flows, cohomology cuts

- Computer Science, MathematicsSTOC '09
- 2009

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

- MathematicsSODA
- 2021

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

- Mathematics
- 2012

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

- Mathematics
- 2016

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

- Computer ScienceSTOC '11
- 2011

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

- Computer Science, MathematicsSCG '09
- 2009

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

- MathematicsArXiv
- 2012

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

- MathematicsSODA
- 2010

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

- Mathematics, Computer ScienceSODA
- 2020

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

- Computer ScienceSIAM J. Comput.
- 1985

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 $\sqrt n /\log n$.…