Robust satisfiability of systems of equations

P. Franek, M. Krcál, Journal of the ACM 62 (2015).


Journal Article | Published | English
Author
;
Abstract
We study the problem of robust satisfiability of systems of nonlinear equations, namely, whether for a given continuous function f:K→ ℝn on a finite simplicial complex K and α > 0, it holds that each function g: K → ℝn such that ||g - f || ∞ < α, has a root in K. Via a reduction to the extension problem of maps into a sphere, we particularly show that this problem is decidable in polynomial time for every fixed n, assuming dimK ≤ 2n - 3. This is a substantial extension of previous computational applications of topological degree and related concepts in numerical and interval analysis. Via a reverse reduction, we prove that the problem is undecidable when dim K > 2n - 2, where the threshold comes from the stable range in homotopy theory. For the lucidity of our exposition, we focus on the setting when f is simplexwise linear. Such functions can approximate general continuous functions, and thus we get approximation schemes and undecidability of the robust satisfiability in other possible settings.
Publishing Year
Date Published
2015-08-01
Journal Title
Journal of the ACM
Volume
62
Issue
4
Article Number
26
IST-REx-ID

Cite this

Franek P, Krcál M. Robust satisfiability of systems of equations. Journal of the ACM. 2015;62(4). doi:10.1145/2751524
Franek, P., & Krcál, M. (2015). Robust satisfiability of systems of equations. Journal of the ACM, 62(4). https://doi.org/10.1145/2751524
Franek, Peter, and Marek Krcál. “Robust Satisfiability of Systems of Equations.” Journal of the ACM 62, no. 4 (2015). https://doi.org/10.1145/2751524.
P. Franek and M. Krcál, “Robust satisfiability of systems of equations,” Journal of the ACM, vol. 62, no. 4, 2015.
Franek P, Krcál M. 2015. Robust satisfiability of systems of equations. Journal of the ACM. 62(4).
Franek, Peter, and Marek Krcál. “Robust Satisfiability of Systems of Equations.” Journal of the ACM, vol. 62, no. 4, 26, ACM, 2015, doi:10.1145/2751524.

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