## Computing the top betti numbers of semi-algebraic sets defined by quadratic inequalities in polynomial time (2005)

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Venue: | In Proceedings of the ThirtySeventh Annual ACM Symposium on Theory of Computing |

Citations: | 2 - 1 self |

### BibTeX

@INPROCEEDINGS{Basu05computingthe,

author = {Saugata Basu},

title = {Computing the top betti numbers of semi-algebraic sets defined by quadratic inequalities in polynomial time},

booktitle = {In Proceedings of the ThirtySeventh Annual ACM Symposium on Theory of Computing},

year = {2005}

}

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### Abstract

Abstract. For any ℓ> 0, we present an algorithm which takes as input a semi-algebraic set, S, defined by P1 ≤ 0,..., Ps ≤ 0, where each Pi ∈ R[X1,..., Xk] has degree ≤ 2, and computes the top ℓ Betti numbers of S, bk−1(S),..., bk−ℓ(S), in polynomial time. The complexity of the algorithm, stated more precisely, is Pℓ+2 `s ´ i=0 k2 i O(min(ℓ,s)). For fixed ℓ, the complexity of the algorithm can be expressed as sℓ+2k2O(ℓ) , which is polynomial in the input parameters s and k. To our knowledge this is the first polynomial time algorithm for computing non-trivial topological invariants of semi-algebraic sets in Rk defined by polynomial inequalities, where the number of inequalities is not fixed and the polynomials are allowed to have degree greater than one. For fixed s, we obtain by letting ℓ = k, an algorithm for computing all the Betti numbers of S whose complexity is k2O(s). 1.

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Citation Context ...has the homotopy type of the suspension ΣS. Moreover, the description of S ′ can be computed in polynomial time from the description of S. It follows from the basic properties of the suspensions (see =-=[28]-=-) that b1(ΣS) = b0(S), which proves that computing b1(S) is also #P-hard. Iterating the construction, that is taking suspensions of suspensions ℓ times, and noting that bℓ(Σ ℓ (S)) = b0(S), gives the ... |

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Citation Context ...braic set defined by quadratic inequalities in R k is #P-hard. Note that PSPACE-hardness of the problem of counting the number of connected components for general semialgebraic sets were known before =-=[13, 26]-=-, and the proofs of these results extend easily to the quadratic case. In view of these hardness results, it is unlikely that there exist polynomial time algorithms for computing the Betti numbers (or... |

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Citation Context ...rt by an NSF Career Award 0133597 and a Sloan Foundation Fellowship. A preliminary version of this paper appears in the Proceedings of the ACM Symposium on Theory of Computing, 2005. 1s2 SAUGATA BASU =-=[27, 7]-=-, computing the zero-th Betti number (i.e. the number of semi-algebraically connected components of S) [18, 20, 14, 16, 8], as well as the Euler-Poincaré characteristic of S [4]. Very recently a singl... |

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Citation Context ...O(s) . 1. Introduction Let R be a real closed field and S ⊂ R k a semi-algebraic set defined by a Boolean formula with atoms of the form P > 0, P < 0, P = 0 for P ∈ P ⊂ R[X1, . . . , Xk]. It is known =-=[23, 24, 22, 29, 4, 15]-=- that the topological complexity of S (measured by the various Betti numbers of S) is bounded by O(s 2 d) k , where s = #(P) and d = maxP ∈P deg(P ). Note that these bounds are singly exponential in k... |

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Citation Context ...O(s) . 1. Introduction Let R be a real closed field and S ⊂ R k a semi-algebraic set defined by a Boolean formula with atoms of the form P > 0, P < 0, P = 0 for P ∈ P ⊂ R[X1, . . . , Xk]. It is known =-=[23, 24, 22, 29, 4, 15]-=- that the topological complexity of S (measured by the various Betti numbers of S) is bounded by O(s 2 d) k , where s = #(P) and d = maxP ∈P deg(P ). Note that these bounds are singly exponential in k... |

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Citation Context ...xes of vector spaces, spectral sequences and triangulations of semi-algebraic sets. We do not prove any results since all of them are quite classical and we refer the reader to appropriate references =-=[12, 21, 9]-=- for the proofs. In Section 4, we recall some basic algorithms in semi-algebraic geometry that we will need later. We state the inputs, outputs and complexities of these algorithms. For classical resu... |

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Citation Context ...rt by an NSF Career Award 0133597 and a Sloan Foundation Fellowship. A preliminary version of this paper appears in the Proceedings of the ACM Symposium on Theory of Computing, 2005. 1s2 SAUGATA BASU =-=[27, 7]-=-, computing the zero-th Betti number (i.e. the number of semi-algebraically connected components of S) [18, 20, 14, 16, 8], as well as the Euler-Poincaré characteristic of S [4]. Very recently a singl... |

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Citation Context ...rs in the Proceedings of the ACM Symposium on Theory of Computing, 2005. 1s2 SAUGATA BASU [27, 7], computing the zero-th Betti number (i.e. the number of semi-algebraically connected components of S) =-=[18, 20, 14, 16, 8]-=-, as well as the Euler-Poincaré characteristic of S [4]. Very recently a singly exponential time algorithm has been given for computing the first few Betti numbers of semi-algebraic sets [3] (see also... |

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Citation Context ...O(s) . 1. Introduction Let R be a real closed field and S ⊂ R k a semi-algebraic set defined by a Boolean formula with atoms of the form P > 0, P < 0, P = 0 for P ∈ P ⊂ R[X1, . . . , Xk]. It is known =-=[23, 24, 22, 29, 4, 15]-=- that the topological complexity of S (measured by the various Betti numbers of S) is bounded by O(s 2 d) k , where s = #(P) and d = maxP ∈P deg(P ). Note that these bounds are singly exponential in k... |

37 | Computing roadmaps of semi-algebraic sets on a variety
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Citation Context ...rs in the Proceedings of the ACM Symposium on Theory of Computing, 2005. 1s2 SAUGATA BASU [27, 7], computing the zero-th Betti number (i.e. the number of semi-algebraically connected components of S) =-=[18, 20, 14, 16, 8]-=-, as well as the Euler-Poincaré characteristic of S [4]. Very recently a singly exponential time algorithm has been given for computing the first few Betti numbers of semi-algebraic sets [3] (see also... |

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Citation Context ... linear programming. In this case the set S is either a convex polyhedron or empty, and (weakly) polynomial time algorithms are known to decide emptiness of such a set. In another direction, Barvinok =-=[2]-=- designed a polynomial time algorithm for deciding feasibility of systems of quadratic inequalities, but under the condition that the number of inequalities is bounded by a constant (see also [17] for... |

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Citation Context ...xes of vector spaces, spectral sequences and triangulations of semi-algebraic sets. We do not prove any results since all of them are quite classical and we refer the reader to appropriate references =-=[12, 21, 9]-=- for the proofs. In Section 4, we recall some basic algorithms in semi-algebraic geometry that we will need later. We state the inputs, outputs and complexities of these algorithms. For classical resu... |

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Citation Context ...rs in the Proceedings of the ACM Symposium on Theory of Computing, 2005. 1s2 SAUGATA BASU [27, 7], computing the zero-th Betti number (i.e. the number of semi-algebraically connected components of S) =-=[18, 20, 14, 16, 8]-=-, as well as the Euler-Poincaré characteristic of S [4]. Very recently a singly exponential time algorithm has been given for computing the first few Betti numbers of semi-algebraic sets [3] (see also... |

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Citation Context |

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Citation Context ...R〈ε〉 k defined by S ′ = Ext(S, R〈ε〉) ∩ ¯ Bk(0, 1/ε), where ¯Bk(0, r) denotes the closed ball of radius r centered at the origin. It follows from Hardt’s triviality theorem for semi-algebraic mappings =-=[19]-=- that bi(S) = bi(S ′ ) for all i ≥ 0. We then replace S ′ by the set S ′′ ⊂ R〈ε, δ〉 k defined by, � P ∈P1 P ≤ 0∧−P ≤ 0 � P ∈P2 −P +δ ≤ 0 � P ∈P3 −P −δ ≤ 0 � ε 2 (X 2 1 +· · ·+X2 k )−1 ≤ 0. It follows ... |

20 |
Construction of roadmaps of semi-algebraic sets
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Citation Context |

19 | Computing the first few Betti numbers of semi-algebraic sets in single exponential time
- Basu
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Citation Context ...0, 14, 16, 8], as well as the Euler-Poincaré characteristic of S [4]. Very recently a singly exponential time algorithm has been given for computing the first few Betti numbers of semi-algebraic sets =-=[3]-=- (see also [6]). In this paper, we consider a restricted class of semi-algebraic sets – namely, semi-algebraic sets defined by a conjunction of quadratic inequalities. Since sets defined by linear ine... |

18 | Counting complexity classes for numeric computations II: Algebraic and semialgebraic sets
- Bürgisser, Cucker
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Citation Context ...braic set defined by quadratic inequalities in R k is #P-hard. Note that PSPACE-hardness of the problem of counting the number of connected components for general semialgebraic sets were known before =-=[13, 26]-=-, and the proofs of these results extend easily to the quadratic case. In view of these hardness results, it is unlikely that there exist polynomial time algorithms for computing the Betti numbers (or... |

17 |
The topology of quadratic mappings and Hessians of smooth mappings
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Citation Context ...tion 5 we describe certain topological properties of semi-algebraic sets defined by quadratic inequalities which are crucial for our algorithm. Most of the results in this section are due to Agrachev =-=[1]-=-. In Section 6 we prove the main mathematical results necessary for our algorithm. In Section 7 we describe our algorithm for computing the top Betti numbers of semi-algebraic sets defined by quadrati... |

17 |
Geometrie algebrique reelle. Springer-Verlag
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Citation Context ... of ∆ and h| h −1 (S1) : h −1 (S1) → S1 is a semi-algebraic triangulation of S1. We will refer to this sub-complex by ∆|S1. We will need the following theorem which can be deduced from Section 9.2 in =-=[11]-=- (see also [9]). Theorem 3.2. Let S1 ⊂ S2 ⊂ R k be closed and bounded semi-algebraic sets, and let hi : ∆i → Si, i = 1, 2 be semi-algebraic triangulations of S1, S2. Then, there exists a semi-algebrai... |

16 |
Computing the first Betti number and the connected components of semi-algebraic sets
- Basu, Pollack, et al.
- 2005
(Show Context)
Citation Context ... as well as the Euler-Poincaré characteristic of S [4]. Very recently a singly exponential time algorithm has been given for computing the first few Betti numbers of semi-algebraic sets [3] (see also =-=[6]-=-). In this paper, we consider a restricted class of semi-algebraic sets – namely, semi-algebraic sets defined by a conjunction of quadratic inequalities. Since sets defined by linear inequalities have... |

15 |
On Bounding the Betti Numbers and Computing the Euler
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10 |
Vorobjov Betti Numbers for Quantifier-free Formulae
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Pasechnik Polynomial time computing over quadratic maps I. Sampling in real algebraic sets, Computational Complexity
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Citation Context ...inok [2] designed a polynomial time algorithm for deciding feasibility of systems of quadratic inequalities, but under the condition that the number of inequalities is bounded by a constant (see also =-=[17]-=- for an interesting generalization as well as a constructive version of this result). 2. Brief Outline Given any compact semi-algebraic set S, we will denote by bi(S) the rank of Hi (S, Q) (the i-th s... |

2 |
On different bounds on different Betti numbers, Discrete and Computational Geometry
- Basu
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Citation Context ...numbers of S) is bounded by O(s 2 d) k , where s = #(P) and d = maxP ∈P deg(P ). Note that these bounds are singly exponential in k. More precise bounds on the individual Betti numbers of S appear in =-=[5]-=-. Designing efficient algorithms for computing the homology groups, and in particular the Betti numbers, of semi-algebraic sets are considered amongst the most important problems in algorithmic semi-a... |