Reliable Computing

, Volume 3, Issue 1, pp 51–81

NP-Hard Classes of Linear Algebraic Systems with Uncertainties

  • Anatoly V. Lakeyev
  • Vladik Kreinovich

DOI: 10.1023/A:1009938325229

Cite this article as:
Lakeyev, A.V. & Kreinovich, V. Reliable Computing (1997) 3: 51. doi:10.1023/A:1009938325229


For a system of linear equations Ax = b, the following natural questions appear:

• does this system have a solution?

• if it does, what are the possible values of a given objective function f(x1,...,xn) (e.g., of a linear function f(x) = ∑CiXi) over the system's solution set?

We show that for several classes of linear equations with uncertainty (including interval linear equations) these problems are NP-hard. In particular, we show that these problems are NP-hard even if we consider only systems of n+2 equations with n variables, that have integer positive coefficients and finitely many solutions.

Copyright information

© Kluwer Academic Publishers 1997

Authors and Affiliations

  • Anatoly V. Lakeyev
    • 1
  • Vladik Kreinovich
    • 2
  1. 1.Irkutsk Computing Center, Siberian BranchRussian Academy of SciencesIrkutskRussia
  2. 2.Department of Computer ScienceUniversity of Texas atEl Paso, El PasoUSA, e-mail