Complexity of Some Linear Problems with Interval Data

During the recent years, a number of linear problems with interval data have been proved to be NP-hard. These results may seem rather obscure as regards the ways in which they were obtained. This survey paper is aimed at demonstrating that in fact it is not so, since many of these results follow easily from the recently established fact that for the subordinate matrix norm · ,1 it is NP-hard to decide whether A,1 1 holds, even in the class of symmetric positive definite rational matrices. After a brief introduction into the basic topics of the complexity theory in Section 1 and formulation of the underlying norm complexity result in Section 2, we present NP-hardness results for checking properties of interval matrices (Section 3), computing enclosures (Section 4), solvability of rectangular linear interval systems (Section 5), and linear and quadratic programming (Section 6). Due to space limitations, proofs are mostly only ketched to reveal the unifying role of the norm complexity result; technical details are omitted.