Computing the Homology of Semialgebraic Sets. I: Lax Formulas

We describe and analyze an algorithm for computing the homology (Betti numbers and torsion coefficients) of closed semialgebraic sets given by Boolean formulas without negations over lax polynomial inequalities. The algorithm works in weak exponential time. This means that outside a subset of data h...

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Bibliographic Details
Published in:Foundations of computational mathematics Vol. 20; no. 1; pp. 71 - 118
Main Authors: Bürgisser, Peter, Cucker, Felipe, Tonelli-Cueto, Josué
Format: Journal Article
Language:English
Published: New York Springer US 01.02.2020
Springer Nature B.V
Subjects:
Acceleration
Algorithms
Applications of Mathematics
Boolean algebra
Computation
Computer Science
Economics
Homology
Linear and Multilinear Algebras
Math Applications in Computer Science
Mathematics
Mathematics and Statistics
Matrix Theory
Numerical Analysis
Polynomials
68Q25
Homology groups
Numerical algorithms
14P10
65Y20
Weak complexity
65D18
ISSN:1615-3375, 1615-3383
Online Access:Get full text
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Summary:We describe and analyze an algorithm for computing the homology (Betti numbers and torsion coefficients) of closed semialgebraic sets given by Boolean formulas without negations over lax polynomial inequalities. 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. All previous algorithms solving this problem have doubly exponential complexity. Our algorithm thus represents an exponential acceleration over state-of-the-art algorithms for all input data outside a set that vanishes exponentially fast.
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ISSN:1615-3375
1615-3383
DOI:10.1007/s10208-019-09418-y