This paper was not presented at any IFAC meeting. This paper was recommended for publication in revised form by Associate Editor S. Hara under the direction of Editor Roberto Tempo. This work was carried out while both authors were in their previous places of employ- ment. M. Vidyasagar was with the Centre for Arti"cial Intelligence and Robotics, Bangalore, India. Vincent Blondel was with the Universite H de Lie`ge, Belgium. * Corresponding author. E-mail addresses: sagar@hydbad.tcs.co.in (M. Vidyasagar), blondel@inma.ucl.ac.be (V. Blondel). See Garey and Johnson (1979) for a dated but still highly readable account of NP-completeness and NP-hardness, and Papadimitrou (1994) for a more up-to-date treatment. Automatica 37 (2001) 1397}1405 Brief Paper Probabilistic solutions to some NP-hard matrix problems M. Vidyasagar*, Vincent D. Blondel Tata Consultancy Services, Coromandel House, 1-2-10 S.P. Road, Hyderabad 500 003, India Department of Mathematical Engineering, Center for Systems Engineering and Applied Mechanics, Universite & catholique de Louvain, Avenue George Lemai K tre, 4 B-1348 Louvain-la-Neuve, Belgium Received 16 July 1999; revised 24 October 2000; received in "nal form 4 March 2001 Abstract During recent years, it has been shown that a number of problems in matrix theory are NP-hard, including robust nonsingularity, robust stability, robust positive semide"niteness, robust bounded norm, state feedback stabilization with structural and norm constraints, etc. In this paper, we use standard bounds on empirical probabilities as well as recent results from statistical learning theoryontheVC-dimensionoffamiliesofsetsde"nedbya "nitenumberofpolynomialinequalities,toshowthatforeachoftheabove problems, as well as for still more general and more di$cult problems, there exists a polynomial-time randomized algorithm that can provide a yes or no answer to arbitrarily small levels of accuracy and con"dence. 2001 Elsevier Science Ltd. All rights reserved. Keywords: NP-hard; Matrix stability; VC-dimension; Interval matrices; Static output feedback 1. Introduction During recent years, several researchers have explored the computational complexity of various problems aris- ing in robust control theory and in matrix theory. Owing to these e!orts, it is now known that several problems in matrix theory are NP-hard. A survey of computational complexity results in systems and control can be found in Blondel and Tsitsiklis (2000). We give below a partial catalog of some such NP-hard problems. These problems can be grouped naturally into two categories: Problems of analysis, and problems of synthesis. Both types of problemsarestatedintermsof `interval matricesa,which are de"ned next. Given an integer n, let > denote the subset of R de"ned by > : "( , ), i, j"1, 2 , n: , 3Q ∀i, j, where Q denotes the set of rational numbers. Let y3> be a typical element. The corresponding set A y is de"ned by A y : "A3R: )a ) ∀i, j. Now let > : "y3>: " and " ∀i, j. For a typical element y3> , de"ne A y : "A3R: A is symmetric and )a ) ∀i, j. The set A y is referred to as an `interval matrixa while A y is a `symmetric interval matrixa. With this notation we can now state several NP-hard problems, all of which pertain to analysis. 1. Robust stability: Given an element y3>, determine whether every matrix in the set A y is stable, in the sense that all of its eigenvalues have negative real parts. 2. Robust positive semidexniteness: Given a vector y3> , determine whether every symmetric matrix in A y is positive semide"nite. 0005-1098/01/$-see front matter 2001 Elsevier Science Ltd. All rights reserved. PII:S0005-1098(01)00089-9