Abstract
We consider the following problem: givenk+1 square matrices with rational entries,A 0,A 1,...,A k , decide ifA 0+r 1 A 1+···+r k A k is nonsingular for all possible choices of real numbersr 1, ...,r k in the interval [0, 1]. We show that this question, which is closely related to the robust stability problem, is NP-hard. The proof relies on the new concept ofradius of nonsingularity of a square matrix and on the relationship between computing this radius and a graph-theoretic problem.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
F. Barahona, A solvable case of quadratic 0–1 programming,Discrete Appl. Math.,13 (1986), 24–36.
T. A. Brown and J. Spencer, Minimization of ±1-matrices under line shifts,Colloq. Math. (Poland),23 (1971), 165–171.
A. Deif,Sensitivity Analysis in Linear Systems, Springer-Verlag, Berlin, 1986.
Ch. Delorme and S. Poljak, Laplacian Eigenvalues and the Maximum Cut Problem, Technical Report 599, L.R.I., Université Paris-Sud, 1990.
M. Deza and M. Laurent, Facets for the cut cone, I,Math. Programming,56 (to appear).
J. C. Doyle, Analysis of feedback systems with structured uncertainties,Proc. IEEE,129 (1982), 242–250.
P. Erdös and J. Spencer,Probabilistic Methods in Combinatorics, Akademiai Kiadó, Budapest, 1974.
M. R. Garey and D. S. Johnson,Computers and Intractability: A Guide to the Theory of NP-Completeness, Freeman, San Francisco, 1979.
K. J. Lieberherr and E. Specker, Complexity of partial satisfaction,J. Assoc. Comput. Mach.,28 (1981), 411–421.
M. Mansour, Robust stability of interval matrices,Proceedings of the 28th Conference on Decision and Control, Tampa, FL, 1989, pp. 46–51.
B. Mohar and S. Poljak, Eigenvalues and the max-cut problem,Czech. Math. J.,115 (1990), 343–352.
S. Poljak, V. Rödl, and J. Spencer, Tournament ranking with expected profit in polynomial time,SIAM J. Discrete Math.,1 (1988), 372–376.
S. Poljak and D. Turzik, A polynomial time heuristic for certain subgraph optimization problems with a guaranteed worst case bound,Discrete Math.,58 (1986), 99–104.
J. Rohn, Systems of linear interval equations,Linear Algebra Appl.,126 (1989), 39–78.
A. Schrijver,Theory of Integer and Linear Programming, Wiley, Chichester, 1986.
A. Tarski,A Decision Method for Elementary Algebra and Geometry, University of California Press, Berkeley, CA, 1951.
Author information
Authors and Affiliations
Rights and permissions
About this article
Cite this article
Poljak, S., Rohn, J. Checking robust nonsingularity is NP-hard. Math. Control Signal Systems 6, 1–9 (1993). https://doi.org/10.1007/BF01213466
Received:
Revised:
Issue Date:
DOI: https://doi.org/10.1007/BF01213466