# On the Computational Complexity of Betti Numbers: Reductions from Matrix Rank

@inproceedings{Edelsbrunner2014OnTC, title={On the Computational Complexity of Betti Numbers: Reductions from Matrix Rank}, author={Herbert Edelsbrunner and Salman Parsa}, booktitle={SODA}, year={2014} }

We give evidence for the difficulty of computing Betti numbers of simplicial complexes over a finite field. We do this by reducing the rank computation for sparse matrices with m non-zero entries to computing Betti numbers of simplicial complexes consisting of at most a constant times m simplices. Together with the known reduction in the other direction, this implies that the two problems have the same computational complexity.

## 27 Citations

Complexity of simplicial homology and independence complexes of chordal graphs

- MathematicsComput. Geom.
- 2016

Computing Height Persistence and Homology Generators in ℝ3 Efficiently

- Computer ScienceSODA
- 2019

It is shown that the persistence for height functions on general simplicial complexes K linearly embedded in R, hence called height persistence, can be computed in O(n log n) time, which improves significantly the current best bound of O( n), ω being the exponent of matrix multiplication.

Height Persistence and Homology Generators in R 3 Efficiently

- Computer Science
- 2019

It is shown that the persistence for height functions on general simplicial complexes K linearly embedded in R, hence called height persistence, can be computed in O(n log n) time, which improves significantly the current best bound of O( n), ω being the exponent of matrix multiplication.

Condition and homology in semialgebraic geometry

- Mathematics, Computer Science
- 2019

This PhD thesis shows how to obtain a numerical algorithm running in single exponential time with very high probability, which improves the state-of-the-art.

On Complexity of Computing Bottleneck and Lexicographic Optimal Cycles in a Homology Class

- Mathematics, Computer ScienceSoCG
- 2022

The main result is that, in the case of 3-manifolds of size n2 in the Euclidean 3-space, the problem of finding a bottleneck optimal cycle cannot be solved more efficiently than solving a system of linear equations with an n × n sparse matrix.

Solving 1-Laplacians in Nearly Linear Time: Collapsing and Expanding a Topological Ball

- Computer Science, MathematicsSODA
- 2014

An efficient algorithm for solving a linear system arising from the 1-Laplacian corresponding to a collapsible simplicial complex with a known collapsing sequence and is applicable to convex simplicial complexes embedded in R3.

Barcodes of Towers and a Streaming Algorithm for Persistent Homology

- Computer ScienceSoCG
- 2017

This work shows how to compute a filtration, a sequence of nested simplicial complexes, with the same persistent barcode as the tower, based on the coning strategy by Dey et al. (SoCG, 2014).

Small Model $2$-Complexes in $4$-space and Applications

- MathematicsArXiv
- 2015

It is shown that computing first homology of $2-complexes is equivalent in computational complexity to matrix diagonalization, which means that the usual procedures for computing homology cannot be improved other than by matrix methods.

Persistent Homology and Nested Dissection

- MathematicsSODA
- 2016

It is shown that one can improve the computation of persistent homology if the underlying space has some additional structure and reasonable geometric conditions under which one can beat the matrix multiplication bound for persistent homological.

Computing Height Persistence and Homology Generators in $\mathbb{R}^3$ Efficiently

- Computer Science
- 2018

The persistence for height functions on simplicial complexes, hence called em height persistence, can be computed in O(n\log n) time, which improves significantly the current best bound of $O(n^{\omega})$, $\omega$ being the matrix multiplication exponent.

## References

SHOWING 1-10 OF 21 REFERENCES

An incremental algorithm for Betti numbers of simplicial complexes on the 3-sphere

- MathematicsComput. Aided Geom. Des.
- 1995

A deterministic o(m log m) time algorithm for the reeb graph

- Computer ScienceSoCG '12
- 2012

We present a deterministic algorithm to compute the Reeb graph of a PL real-valued function on a simplicial complex in O(m log m) time, where m is the size of the 2-skeleton. The problem reduces to…

A randomized O(m log m) time algorithm for computing Reeb graphs of arbitrary simplicial complexes

- Computer Science, MathematicsSCG
- 2010

This paper presents the first sub-quadratic algorithm to compute the Reeb graph for a function on an arbitrary simplicial complex K, and is faster than current algorithms for any other special cases (e.g, 3-manifolds).

Matrix Sparsification for Rank and Determinant Computations via Nested Dissection

- Computer Science2008 49th Annual IEEE Symposium on Foundations of Computer Science
- 2008

The main results of this paper show that one can remove the three restrictions of being "symmetric", being "real", and being "positive definite" and still be able to compute the rank and, when relevant, also the absolute determinant, while keeping the running time of nested dissection.

Algebraic Topology

- Mathematics

The focus of this paper is a proof of the Nielsen-Schreier Theorem, stating that every subgroup of a free group is free, using tools from algebraic topology.

Extending Persistence Using Poincaré and Lefschetz Duality

- MathematicsFound. Comput. Math.
- 2009

An algebraic formulation is given that extends persistence to essential homology for any filtered space, an algorithm is presented to calculate it, and how it aids the ability to recognize shape features for codimension 1 submanifolds of Euclidean space is described.

Multiplying matrices faster than coppersmith-winograd

- Computer ScienceSTOC '12
- 2012

An automated approach for designing matrix multiplication algorithms based on constructions similar to the Coppersmith-Winograd construction is developed and a new improved bound on the matrix multiplication exponent ω<2.3727 is obtained.

Solving sparse linear equations over finite fields

- Computer Science, MathematicsIEEE Trans. Inf. Theory
- 1986

A "coordinate recurrence" method for solving sparse systems of linear equations over finite fields is described and a probabilistic algorithm is shown to exist for finding the determinant of a square matrix.

Computational topology: ambient isotopic approximation of 2-manifolds

- Computer ScienceTheor. Comput. Sci.
- 2003

Faster inversion and other black box matrix computations using efficient block projections

- Mathematics, Computer ScienceISSAC '07
- 2007

The correctness of the algorithm to find rational solutions for sparse systems of linear equations is established by proving the existence of efficient block projections for arbitrary non-singular matrices over sufficiently large fields by incorporating them into existing black-box matrix algorithms to derive improved bounds for the cost of several matrix problems.