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Computing the Homology of Basic Semialgebraic Sets in Weak Exponential Time

2018
*
Journal of the ACM
*

We describe and analyze an algorithm for

doi:10.1145/3275242
fatcat:fxzmiqobkzcavdmkdeudaufiqa
*computing**the**homology*(Betti numbers and torsion coefficients)*of**basic**semialgebraic**sets*which works*in**weak**exponential**time*. ... That is, out*of*a*set**of**exponentially*small measure*in**the*space*of*data*the*cost*of**the*algorithm is*exponential**in**the*size*of**the*data. ... This work has been supported by*the*Einstein Foundation, Berlin, by*the*DFG research grant no. BU 1371/2-2, and by*the*Research Grants Council*of**the*Hong Kong SAR project no. CityU-11202017. ...##
###
Computing the Homology of Semialgebraic Sets I: Lax Formulas
[article]

2019
*
arXiv
*
pre-print

*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. ... 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 ... (ii) Part (iii)

*of*Theorem 1.1 shows,

*in*

*the*terminology introduced by [1] , that

*Homology*works

*in*

*weak*

*exponential*

*time*. ...

##
###
Computing the Homology of Semialgebraic Sets. II: General formulas
[article]

2020
*
arXiv
*
pre-print

*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. ... We describe and analyze a numerical algorithm for

*computing*

*the*

*homology*(Betti numbers and torsion coefficients)

*of*

*semialgebraic*

*sets*given by Boolean formulas. ... Details can also be found

*in*[17, 4 §3 -2] . (ii) Part (iii)

*of*Theorem 1.1 shows that

*Homology*works

*in*

*weak*

*exponential*

*time*. ...

##
###
Persistent homology of semi-algebraic sets
[article]

2022
*
arXiv
*
pre-print

We give an algorithm with singly

arXiv:2202.09591v2
fatcat:mpujh5pspjdonazhxqcbxwgli4
*exponential*complexity for*computing**the*barcodes up to dimension ℓ (for any fixed ℓ≥ 0)*of**the*filtration*of*a given semi-algebraic*set*by*the*sub-level*sets**of*a given ... Our algorithm is*the*first algorithm for this problem with singly*exponential*complexity, and generalizes*the*corresponding results for*computing**the*Betti numbers up to dimension ℓ*of*semi-algebraic*sets*... Acknowledgements*The*authors are grateful to Ezra Miller for his comments on a previous version*of*this paper and for pointing out several related prior works. ...##
###
On bounding the Betti numbers and computing the Euler characteristic of semi-algebraic sets

1996
*
Proceedings of the twenty-eighth annual ACM symposium on Theory of computing - STOC '96
*

*In*this paper we prove new bounds on

*the*sum

*of*

*the*Betti numbers

*of*closed semi-algebraic

*sets*and also give

*the*first single

*exponential*

*time*algorithm for

*computing*

*the*Euler characteristic

*of*arbitrary ... Using

*the*tools developed for

*the*above results, as well as some additional techniques, we give

*the*first single

*exponential*

*time*algorithm for

*computing*

*the*Euler characteristic

*of*arbitrary closed semi-algebraic ... We prove

*the*following theorem. We remark that

*computing*

*the*

*homology*groups

*of*semi-algebraic

*sets*

*in*single

*exponential*

*time*is a central open problem

*of*

*computational*real algebraic geometry. ...

##
###
On Bounding the Betti Numbers and Computing the Euler Characteristic of Semi-Algebraic Sets

1999
*
Discrete & Computational Geometry
*

*In*this paper we prove new bounds on

*the*sum

*of*

*the*Betti numbers

*of*closed semi-algebraic

*sets*and also give

*the*first single

*exponential*

*time*algorithm for

*computing*

*the*Euler characteristic

*of*arbitrary ... Using

*the*tools developed for

*the*above results, as well as some additional techniques, we give

*the*first single

*exponential*

*time*algorithm for

*computing*

*the*Euler characteristic

*of*arbitrary closed semi-algebraic ... We prove

*the*following theorem. We remark that

*computing*

*the*

*homology*groups

*of*semi-algebraic

*sets*

*in*single

*exponential*

*time*is a central open problem

*of*

*computational*real algebraic geometry. ...

##
###
An Adaptive Grid Algorithm for Computing the Homology Group of Semialgebraic Set
[article]

2019
*
arXiv
*
pre-print

Looking for an efficient algorithm for

arXiv:1903.02388v1
fatcat:mw4brytrazayba2qemqoomn6ee
*the**computation**of**the**homology*groups*of*an algebraic*set*or even a semi-algebraic*set*is an important problem*in**the*effective real algebraic geometry. ... Recently, Peter Burgisser, Felipe Cucker and Pierre Lairez wrote a paper [1], which made a step forward by giving an algorithm*of**weak**exponential**time*. However,*the*algorithm is not not practical. ... Notice 3.4 Algorithm for*the**computation**of**homology*on a*semialgebraic**set*. Now let us design a more adaptive algorithms than*the*main algorithm*in**the*proposition 5.1*of*[1] . ...##
###
Efficient simplicial replacement of semi-algebraic sets
[article]

2022
*
arXiv
*
pre-print

*computing*

*the*first ℓ homotopy groups

*of*S to

*the*combinatorial problem

*of*

*computing*

*the*first ℓ homotopy groups

*of*a finite simplicial complex

*of*size bounded by (sd)^k^O(ℓ). ...

*In*particular, since ℓ-equivalence implies that

*the*homotopy groups up to dimension ℓ

*of*|Δ| are isomorphic to those

*of*S, we obtain a reduction (having singly

*exponential*complexity)

*of*

*the*problem

*of*...

*In*order to be able to associate a "multiplicity" to

*the*

*set*

*of*

*homology*classes which are born at

*time*s and dies at

*time*t we interpret them as classes

*in*certain subquotients

*of*H * (X s )

*in*what follows ...

##
###
Algorithmic Semi-algebraic Geometry and Topology -- Recent Progress and Open Problems
[article]

2007
*
arXiv
*
pre-print

We give a survey

arXiv:0708.2854v2
fatcat:eqquvlr2rvcw3oekjjm6f2ug2q
*of*algorithms for*computing*topological invariants*of*semi-algebraic*sets*with special emphasis on*the*more recent developments*in*designing algorithms for*computing**the*Betti numbers ...*of*semi-algebraic*sets*. ... Acknowledgment Acknowledgment*The*author thanks Richard Pollack for his careful reading and*the*anonymous referees for many helpful comments which helped to substantially improve*the*article. ...##
###
Algorithms in Real Algebraic Geometry: A Survey
[article]

2014
*
arXiv
*
pre-print

We also describe some recent results linking

arXiv:1409.1534v1
fatcat:nyprfglktvdtnmhu3zwqrb547y
*the**computational*hardness*of*decision problems*in**the*first order theory*of**the*reals, with that*of**computing*certain topological invariants*of*semi-algebraic ... , to more recent algorithms for*computing*topological invariants*of*semi-algebraic*sets*. ... semi-algebraically connected components*of*semi-algebraic*sets**in*singly*exponential**time*, that we can*compute**the*first Betti number*of*closed and bounded semi-algebraic*sets**in*singly*exponential**time*...##
###
Real algebraic geometry

2014
*
ChoiceReviews
*

*the*pool

*of*

*the*Real Algebraic Geometry

*of*tomorrow. ... We also wish to thank

*the*numerous sponsors who allowed us to welcome so many participants under proper conditions,

*in*particular a large number

*of*PhD students and post-PhD students called to constitute ... connected components

*of*semi-algebraic

*sets*

*in*singly

*exponential*

*time*, that we can

*compute*

*the*first Betti number

*of*closed and bounded semi-algebraic

*sets*

*in*singly

*exponential*

*time*(see Remark 3.16 ...

##
###
Computing the Betti numbers of semi-algebraic sets defined by partly quadratic systems of polynomials

2009
*
Journal of Algebra
*

*The*complexity

*of*

*the*algorithm interpolates between

*the*doubly

*exponential*

*time*bounds for

*the*known algorithms

*in*

*the*general case, and

*the*polynomial complexity

*in*case

*of*semi-algebraic

*sets*defined ... Let S ⊂^ℓ+k be a semi-algebraic

*set*defined by a Boolean formula without negations, with atoms P=0, P ≥ 0, P ≤ 0, P ∈ P∪ Q. We describe an algorithm for

*computing*

*the*

*the*Betti numbers

*of*S. ... numbers

*of*a given semi-algebraic

*set*

*in*singly

*exponential*

*time*. ...

##
###
Efficient computation of a semi-algebraic basis of the first homology group of a semi-algebraic set
[article]

2021
*
arXiv
*
pre-print

*The*complexity

*of*

*the*algorithm is bounded singly

*exponentially*. It is not known how to

*compute*such a basis for

*the*higher

*homology*groups with singly

*exponential*complexity. ... We give an algorithm for

*computing*a semi-algebraic basis for

*the*first

*homology*group, H_1(S,𝔽), with coefficients

*in*a field 𝔽,

*of*any given semi-algebraic

*set*S ⊂R^k defined by a closed formula. ... Finally,

*the*

*set*

*of*

*weak*sign conditions on Q j whose realizations are contained

*in*clos(π i+1,j (ass(γ)))

*computed*using Algorithm 13.1 (

*Computing*realizable sign conditions)

*in*[6, pp. 511]

*in*Line 5 ...

##
###
Computing the first Betti number and the connected components of semi-algebraic sets

2005
*
Proceedings of the thirty-seventh annual ACM symposium on Theory of computing - STOC '05
*

*In*this paper we describe

*the*first singly

*exponential*algorithm for

*computing*

*the*first Betti number

*of*a given semi-algebraic

*set*. ... No singly

*exponential*algorithm was known for

*computing*any

*of*

*the*individual Betti numbers other than

*the*zero-th one. ...

*In*

*the*absence

*of*a singly

*exponential*

*time*algorithm for

*computing*triangulations

*of*semi-algebraic

*sets*, algorithms with single

*exponential*complexity are known only for

*the*problems

*of*testing emptiness ...

##
###
Algorithmic and topological aspects of semi-algebraic sets defined by quadratic polynomial
[article]

2007
*
arXiv
*
pre-print

*The*first algorithm

*computes*

*the*number

*of*connected components and

*the*first Betti number

*of*a semi-algebraic

*set*defined by compact objects

*in*R^k which are simply connected. ...

*The*second algorithm

*computes*efficiently

*the*real intersection

*of*three quadratic surfaces

*in*R^3 using a semi-numerical approach. ... Acknowledgements

*The*writing

*of*this thesis has been one

*of*

*the*most significant academic challenges I have had to face. ...

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