Computing the top betti numbers of semi-algebraic sets defined by quadratic inequalities in polynomial time (2005)
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| Venue: | In Proceedings of the ThirtySeventh Annual ACM Symposium on Theory of Computing |
| Citations: | 1 - 1 self |
BibTeX
@INPROCEEDINGS{Basu05computingthe,
author = {Saugata Basu},
title = {Computing the top betti numbers of semi-algebraic sets defined by quadratic inequalities in polynomial time},
booktitle = {In Proceedings of the ThirtySeventh Annual ACM Symposium on Theory of Computing},
year = {2005}
}
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Abstract
Abstract. For any ℓ> 0, we present an algorithm which takes as input a semi-algebraic set, S, defined by P1 ≤ 0,..., Ps ≤ 0, where each Pi ∈ R[X1,..., Xk] has degree ≤ 2, and computes the top ℓ Betti numbers of S, bk−1(S),..., bk−ℓ(S), in polynomial time. The complexity of the algorithm, stated more precisely, is Pℓ+2 `s ´ i=0 k2 i O(min(ℓ,s)). For fixed ℓ, the complexity of the algorithm can be expressed as sℓ+2k2O(ℓ) , which is polynomial in the input parameters s and k. To our knowledge this is the first polynomial time algorithm for computing non-trivial topological invariants of semi-algebraic sets in Rk defined by polynomial inequalities, where the number of inequalities is not fixed and the polynomials are allowed to have degree greater than one. For fixed s, we obtain by letting ℓ = k, an algorithm for computing all the Betti numbers of S whose complexity is k2O(s). 1.
