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Computing the Homology of Basic Semialgebraic Sets in Weak Exponential Time

Peter Bürgisser, Felipe Cucker, Pierre Lairez
2018 Journal of the ACM  
We describe and analyze an algorithm for 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.  ... 
doi:10.1145/3275242 fatcat:fxzmiqobkzcavdmkdeudaufiqa

Computing the Homology of Semialgebraic Sets I: Lax Formulas [article]

Peter Bürgisser and Felipe Cucker and Josué Tonelli-Cueto
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.  ... 
arXiv:1807.06435v3 fatcat:cpn6gnkhfndovf6v64w25qgiae

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

Peter Bürgisser, Felipe Cucker, Josué Tonelli-Cueto
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.  ... 
arXiv:1903.10710v2 fatcat:yuqqr33z3fb33g2ikazgvuxmpa

Persistent homology of semi-algebraic sets [article]

Saugata Basu, Negin Karisani
2022 arXiv   pre-print
We give an algorithm with singly 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.  ... 
arXiv:2202.09591v2 fatcat:mpujh5pspjdonazhxqcbxwgli4

On bounding the Betti numbers and computing the Euler characteristic of semi-algebraic sets

Saugata Basu
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.  ... 
doi:10.1145/237814.237988 dblp:conf/stoc/Basu96 fatcat:k6fzrfqswfgh7ek6rk4op3aaum

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

S. Basu
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.  ... 
doi:10.1007/pl00009443 fatcat:odogbws56vb6tmffbdtlj6mb5u

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

Han Jiadong
2019 arXiv   pre-print
Looking for an efficient algorithm for 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] .  ... 
arXiv:1903.02388v1 fatcat:mw4brytrazayba2qemqoomn6ee

Efficient simplicial replacement of semi-algebraic sets [article]

Saugata Basu, Negin Karisani
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  ... 
arXiv:2009.13365v2 fatcat:3vvuo42gcndipiaoxmkaeqpyxy

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

Saugata Basu
2007 arXiv   pre-print
We give a survey 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.  ... 
arXiv:0708.2854v2 fatcat:eqquvlr2rvcw3oekjjm6f2ug2q

Algorithms in Real Algebraic Geometry: A Survey [article]

Saugata Basu
2014 arXiv   pre-print
We also describe some recent results linking 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  ... 
arXiv:1409.1534v1 fatcat:nyprfglktvdtnmhu3zwqrb547y

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  ... 
doi:10.5860/choice.51-3296 fatcat:xm5l6ieafzewtfpuegkxsbtzhy

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

Saugata Basu, Dmitrii V. Pasechnik, Marie-Françoise Roy
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.  ... 
doi:10.1016/j.jalgebra.2008.09.043 fatcat:w55xgh4a35h7jnnqdfxgfmyu5e

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

Saugata Basu, Sarah Percival
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  ... 
arXiv:2107.08947v1 fatcat:34lidw35svbplahcecvrr2hdee

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

Saugata Basu, Richard Pollack, Marie-Françoise Roy
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  ... 
doi:10.1145/1060590.1060636 dblp:conf/stoc/BasuPR05 fatcat:gi4itaphczek3gzakd3xofweom

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

Michael Kettner
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.  ... 
arXiv:0709.3283v1 fatcat:fe2h4yewmjcarmcqxaotgr5vny
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