Computing the Homology of Semialgebraic Sets. I: Lax Formulas

Abstract

We describe and analyze an algorithm for computing the homology (Betti numbers and torsion coefficients) of semialgebraic sets given by Boolean formulas. The algorithm works in weak exponential time. This means that outside a subset of data having exponentially small measure, the cost of the algorithm is single exponential in the size of the data. This extends the work in [2] to arbitrary semialgebraic sets.All previous algorithms proposed for this problem have doubly exponential complexity (and this is so for almost all input data).

Plantinga-Vegter Algorithm takes Average Polynomial Time

“…We observe that in contrast with the complexity analyses (of condition numbers closely related to κ aff ) in the literature (see, e.g., [2,3,[8][9][10][11]), the bounds in eorem 6.7 depend on E x ∈[−a, a] n (κ aff (f , x) n ) and not on max x ∈[−a, a] κ aff (f , x) n . Whereas the former has finite expectation (over f ), the la er has not.…”

Computing the Homology of Semialgebraic Sets. II: General Formulas

“…This means that outside a subset of data having exponentially small measure, the cost of the algorithm is single exponential in the size of the data. This extends the work in [2] to arbitrary semialgebraic sets.All previous algorithms proposed for this problem have doubly exponential complexity.…”

“…This paper is a continuation of [2]. In the latter, we exhibited a numerical algorithm computing the topology of a closed semialgebraic set described by a monotone Boolean combination of polynomial equalities and lax inequalities.…”

“…Our goal in this paper is to exhibit a numerical algorithm, with similar complexity bounds to that in [2], but working for arbitrary semialgebraic sets. That is, complements are allowed in the Boolean combinations (or, equivalently, negations in the Boolean formulas) describing semialgebraic sets and so are strict inequalities.…”

“…Outside this exceptional set then, our algorithm works exponentially faster than state-of-the-art algorithms (which are doubly exponential). For a background on numerical algorithms, condition, and weak cost -and to avoid being repetitious-we refer the reader to the introduction of [2]. We thus proceed with a description of the problem and our main result.…”