Polynomial time algorithm for computing the top Betti numbers of semi-algebraic sets defined by quadratic inequalities

Authors:
Saugata Basu
Affiliations:
Georgia Institute of Technology, Atlanta, GA
Venue:
Proceedings of the thirty-seventh annual ACM symposium on Theory of computing
Year:
2005

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Abstract

For any fixed 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 Σi=0l+2 (si k2O((l,s)). For fixed l, the complexity of the algorithm can be expressed as sl+2 k2O(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)