Robust satisfiability of systems of equations Journal Article

Author(s): Franek, Peter; Krčál, Marek
Article Title: Robust satisfiability of systems of equations
Affiliation IST Austria
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.
Keywords: satisfiability; Nonlinear equations; Topological extension; Undecidability
Journal Title: Journal of the ACM
Volume: 62
Issue 4
ISSN: 0004-5411
Publisher: ACM  
Date Published: 2015-08-01
Start Page: Article number: 26
Sponsor: This research was supported by the Center of Excellence – Institute for Theoretical Computer Science, Prague (project P202/12/G061 of GA ˇ CR), by the Project LL1201 ERCCZ CORES and by institutional support RVO:67985807.
DOI: 10.1145/2751524
Open access: yes (repository)
IST Austria Authors
  1. Marek Krčál
    10 Krčál
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