NDLI: On Topological Lower Bounds for Algebraic Computation Trees

Content Provider	Springer Nature Link
Author	Vorobjov, Nicolai Gabrielov, Andrei
Copyright Year	2015
Abstract	We prove that the height of any algebraic computation tree for deciding membership in a semialgebraic set $$\Sigma \subset {\mathbb R}^n$$ is bounded from below by $$\begin{aligned} \frac{c_1\log (\mathrm{b}_m(\Sigma ))}{m+1} -c_2n, \end{aligned}$$ where $$\mathrm{b}_m(\Sigma )$$ is the mth Betti number of $$\Sigma $$ with respect to “ordinary” (singular) homology and $$c_1,\ c_2$$ are some (absolute) positive constants. This result complements the well-known lower bound by Yao (J Comput Syst Sci 55:36–43, 1997) for locally closed semialgebraic sets in terms of the total Borel–Moore Betti number. We also prove that if $$\rho :\> {\mathbb R}^n \rightarrow {\mathbb R}^{n-r}$$ is the projection map, then the height of any tree deciding membership in $$\Sigma $$ is bounded from below by $$\begin{aligned} \frac{c_1\log (\mathrm{b}_m(\rho (\Sigma )))}{(m+1)^2} -\frac{c_2n}{m+1} \end{aligned}$$ for some positive constants $$c_1,\ c_2$$ . We illustrate these general results by examples of lower complexity bounds for some specific computational problems.
Ending Page	72
Page Count	12
Starting Page	61
File Format	PDF
ISSN	16153375
e-ISSN	16153383
Journal	Foundations of Computational Mathematics
Issue Number	1
Volume Number	17
Language	English
Publisher	Springer US
Publisher Date	2015-08-27
Publisher Place	New York
Access Restriction	One Nation One Subscription (ONOS)
Subject Keyword	Economics Complexity lower bounds Semialgebraic sets Numerical Analysis Algebraic computation trees Computer Science Linear and Multilinear Algebras, Matrix Theory Computational difficulty of problems (lower bounds, completeness, difficulty of approximation, etc.) Applications of Mathematics Math Applications in Computer Science Topology of real algebraic varieties
Content Type	Text
Resource Type	Article
Subject	Applied Mathematics Analysis Computational Theory and Mathematics Computational Mathematics

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