Approximate linear algebra is intractable Original Research Article

It is shown that the problem of computing the optimal solutions of several versions of imprecise linear systems of equations is NP-hard. An imprecise linear system is a linear system Ax = b where A = A(0) + ∑pμ A(μ), b = b(0) + ∑qv b(v), with unknown coefficients pμ, qv constrained by one of the five relations imagedouble vertical barpdouble vertical bar∞ less-than-or-equals, slant α, double vertical barqdouble vertical bar∞ less-than-or-equals, slant β, imagedouble vertical barpdouble vertical bar2 less-than-or-equals, slant α, double vertical barqdouble vertical bar∞ less-than-or-equals, slant β, imagedouble vertical barpdouble vertical bar∞ less-than-or-equals, slant α, double vertical barqdouble vertical bar2 less-than-or-equals, slant β, imagedouble vertical bar(p,q)double vertical bar2 less-than-or-equals, slant α, where (p, q) is a vector formed by all values pμ and qv. Given such a system, we would like to find its optimal solution, i.e., the largest possible (x+j) and the smallest possible (x−j) values of xj. Our main result is that this problem is NP-hard in any of the five versions. This holds even when the A(μ), b(v) (μ ≠ 0, v ≠ 0) are restricted to have only a single nonzero entry each, in disjoint positions, and even if instead of the exact values x+j, x−j, we want to compute their δ-approximations (for a given accuracy δ > 0). So (unless NP = P), algorithms that find the optimal solution of imprecise linear systems require (in the worst case) exponential time.