Reliable Computing 3: 315–323, 1997.
1997 Kluwer Academic Publishers. Printed in the Netherlands.
Complexity of Some Linear Problems with
Faculty of Mathematics and Physics, Charles University, Prague, Czech Republic, and Institute of
Computer Science, Academy of Sciences, Prague, Czech Republic, e-mail: email@example.com
(Received: 1 November 1996; accepted: 24 January 1997)
Abstract. 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
NP-hard to decide whether A
≥ 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.
An algorithm is called a polynomial-time algorithm if there exists a polynomial
p such that for each instance (input data) of length the number of steps of the
algorithm is ≤ p(). Length: number of bits of the input. Consequence: only rational
data allowed (usually represented by pairs of integers). Example: modiﬁed Gaussian
Decision (“yes or no”) problems are considered in complexity theory. A problem
belongs to the class P if it is solvable by a polynomial-time algorithm, and to the
class NP if a guessed candidate for a solution can be veriﬁed by a polynomial-time
A problem I can be reduced in polynomial time to problem J, which we denote
by I → J, if there exists a polynomial-time algorithm
which transforms each
instance i of I to an instance
(i)ofJso that the answer to i is “yes” if and only if
the answer to
(i) is “yes” (or, the answer to i is “yes” if and only if the answer to
(i) is “no”). Hence, if I → J, then each algorithm for solving J may be employed
for solving I; consequently, J is “at least as difﬁcult” as I.
A problem J is called NP-hard if I → J for each I
NP. An NP-hard problem
exists (Cook ; hundreds of them have been found since). Method for proving
NP-hardness: if J is NP-hard and J → K,thenKis NP-hard.