TY - JOUR
T1 - Computing the Homology of Semialgebraic Sets. I
T2 - Lax Formulas
AU - Bürgisser, Peter
AU - Cucker, Felipe
AU - Tonelli-Cueto, Josué
PY - 2020/2
Y1 - 2020/2
N2 - 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.
AB - 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.
KW - Homology groups
KW - Numerical algorithms
KW - Weak complexity
UR - http://www.scopus.com/inward/record.url?scp=85065338522&partnerID=8YFLogxK
UR - https://www.scopus.com/record/pubmetrics.uri?eid=2-s2.0-85065338522&origin=recordpage
U2 - 10.1007/s10208-019-09418-y
DO - 10.1007/s10208-019-09418-y
M3 - 21_Publication in refereed journal
VL - 20
SP - 71
EP - 118
JO - Foundations of Computational Mathematics
JF - Foundations of Computational Mathematics
SN - 1615-3375
IS - 1
ER -