Solving Interval Linear Systems is NP-Hard

Abstract

In the previous chapter, we showed how to compute the range of a fractionally linear function f(x₁,..., x_n) on given intervals x₁,..., x_n in polynomial time. A fractionally linear function f(x₁,...,x_n) can be described as a solution of a linear equation (b₀+Σb_ix_i) · f = a₀ + Σa_ix_i. The problem of finding the range of f is thus equivalent to finding the set of all possible solutions of this equation when x_i take the values in their respective intervals. A natural generalization is, therefore, the solution of a system of linear equations Σ a_ij · f_j = b_i, where the coefficients a_ij- and b_i are linear functions of the variables that are defined with interval uncertainty.

In this chapter and in the next Chapter 12, we analyze the computational complexity and feasibility of solving such interval linear equations. In most formulations, this problem is NP-hard, but some particular cases of this problem turn out to be feasible.