672 research outputs found
Computing the First Few Betti Numbers of Semi-algebraic Sets in Single Exponential Time
In this paper we describe an algorithm that takes as input a description of a
semi-algebraic set , defined by a Boolean formula with atoms of
the form for
and outputs the first Betti numbers of ,
The complexity of the algorithm is where where s =
#({\mathcal P}) and which is
singly exponential in for any fixed constant. Previously, singly
exponential time algorithms were known only for computing the Euler-Poincar\'e
characteristic, the zero-th and the first Betti numbers
Computing the First Betti Numberand Describing the Connected Components of Semi-algebraic Sets
In this paper we describe a singly exponential algorithm for computing the
first Betti number of a given semi-algebraic set. Singly exponential algorithms
for computing the zero-th Betti number, and the Euler-Poincar\'e
characteristic, were known before. No singly exponential algorithm was known
for computing any of the individual Betti numbers other than the zero-th one.
We also give algorithms for obtaining semi-algebraic descriptions of the
semi-algebraically connected components of any given real algebraic or
semi-algebraic set in single-exponential time improving on previous results
Bounding the number of stable homotopy types of a parametrized family of semi-algebraic sets defined by quadratic inequalities
We prove a nearly optimal bound on the number of stable homotopy types
occurring in a k-parameter semi-algebraic family of sets in , each
defined in terms of m quadratic inequalities. Our bound is exponential in k and
m, but polynomial in . More precisely, we prove the following. Let
be a real closed field and let with . Let be a
semi-algebraic set, defined by a Boolean formula without negations, whose atoms
are of the form, . Let be the projection on the last k co-ordinates. Then, the number of
stable homotopy types amongst the fibers S_{\x} = \pi^{-1}(\x) \cap S is
bounded by Comment: 27 pages, 1 figur
Efficient algorithms for computing the Euler-Poincar\'e characteristic of symmetric semi-algebraic sets
Let be a real closed field and
an ordered domain. We consider the algorithmic problem of computing the
generalized Euler-Poincar\'e characteristic of real algebraic as well as
semi-algebraic subsets of , which are defined by symmetric
polynomials with coefficients in . We give algorithms for computing
the generalized Euler-Poincar\'e characteristic of such sets, whose
complexities measured by the number the number of arithmetic operations in
, are polynomially bounded in terms of and the number of
polynomials in the input, assuming that the degrees of the input polynomials
are bounded by a constant. This is in contrast to the best complexity of the
known algorithms for the same problems in the non-symmetric situation, which
are singly exponential. This singly exponential complexity for the latter
problem is unlikely to be improved because of hardness result
(-hardness) coming from discrete complexity theory.Comment: 29 pages, 1 Figure. arXiv admin note: substantial text overlap with
arXiv:1312.658
Computing the homology of basic semialgebraic sets in weak exponential time
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. All algorithms previously proposed for this problem have a complexity
which is doubly exponential (and this is so for almost all data)
Algorithmic Semi-algebraic Geometry and Topology -- Recent Progress and Open Problems
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.
Aside from describing these results, we discuss briefly the background as well
as the importance of these problems, and also describe the main tools from
algorithmic semi-algebraic geometry, as well as algebraic topology, which make
these advances possible. We end with a list of open problems.Comment: Survey article, 74 pages, 15 figures. Final revision. This version
will appear in the AMS Contemporary Math. Series: Proceedings of the Summer
Research Conference on Discrete and Computational Geometry, Snowbird, Utah
(June, 2006). J.E. Goodman, J. Pach, R. Pollack Ed
A complex analogue of Toda's Theorem
Toda \cite{Toda} proved in 1989 that the (discrete) polynomial time
hierarchy, , is contained in the class \mathbf{P}^{#\mathbf{P}},
namely the class of languages that can be decided by a Turing machine in
polynomial time given access to an oracle with the power to compute a function
in the counting complexity class #\mathbf{P}. This result, which illustrates
the power of counting is considered to be a seminal result in computational
complexity theory. An analogous result (with a compactness hypothesis) in the
complexity theory over the reals (in the sense of Blum-Shub-Smale real machines
\cite{BSS89}) was proved in \cite{BZ09}. Unlike Toda's proof in the discrete
case, which relied on sophisticated combinatorial arguments, the proof in
\cite{BZ09} is topological in nature in which the properties of the topological
join is used in a fundamental way. However, the constructions used in
\cite{BZ09} were semi-algebraic -- they used real inequalities in an essential
way and as such do not extend to the complex case. In this paper, we extend the
techniques developed in \cite{BZ09} to the complex projective case. A key role
is played by the complex join of quasi-projective complex varieties. As a
consequence we obtain a complex analogue of Toda's theorem. The results
contained in this paper, taken together with those contained in \cite{BZ09},
illustrate the central role of the Poincar\'e polynomial in algorithmic
algebraic geometry, as well as, in computational complexity theory over the
complex and real numbers -- namely, the ability to compute it efficiently
enables one to decide in polynomial time all languages in the (compact)
polynomial hierarchy over the appropriate field.Comment: 31 pages. Final version to appear in Foundations of Computational
Mathematic
Computing the Betti numbers of semi-algebraic sets defined by partly quadratic systems of polynomials
Let be a real closed field, with \deg_{Y}(Q) \leq 2, \deg_{X}(Q) \leq
d, Q \in {\mathcal Q}, #({\mathcal Q})=m, and with \deg_{X}(P) \leq d, P \in {\mathcal P}, #({\mathcal
P})=s. Let be a semi-algebraic set defined by a
Boolean formula without negations, with atoms . We describe an algorithm for computing the the
Betti numbers of . The complexity of the algorithm is bounded by . 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 by few
quadratic inequalities known previously. Moreover, for fixed and this
algorithm has polynomial time complexity in the remaining parameters.Comment: 24 pages, 3 figure
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