Reliable Computing

, Volume 3, Issue 1, pp 51-81

First online:

NP-Hard Classes of Linear Algebraic Systems with Uncertainties

  • Anatoly V. LakeyevAffiliated withIrkutsk Computing Center, Siberian Branch, Russian Academy of Sciences
  • , Vladik KreinovichAffiliated withDepartment of Computer Science, University of Texas at

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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) = ∑C i X i ) 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.