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. 

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