Abstract
A symmetric matrix A is said to be sign-nonsingular if every symmetric matrix with the same sign pattern as A is nonsingular. Hall, Li and Wang showed that the inertia of a sign-nonsingular symmetric matrix is determined uniquely by its sign pattern. The purpose of this paper is to present an efficient algorithm for computing the inertia of such matrices. The algorithm runs in O(nm) time for a symmetric matrix of order n with m nonzero entries. The correctness of the algorithm provides an alternative proof of the result by Hall et al. In addition, for a symmetric matrix in general, it is shown to be NP-complete to decide whether the inertia of the matrix is not determined by the sign pattern.
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References
Ahuja, R.K., Magnanti, T.L., Orlin, J.B.: Network Flows: Theory, Algorithms, and Applications. Prentice Hall, Englewood Cliffs (1993)
Brualdi, R.A., Shader, B.L.: Matrices of Sign-solvable Linear Systems. Cambridge University Press, Cambridge (1995)
Hall, F.J., Li, Z., Wang, D.: Symmetric sign pattern matrices that require unique inertia. Linear Algebra and Its Applications 338, 153–169 (2001)
Horn, R.A., Johnson, C.R.: Matrix Analysis. Cambridge University Press, Cambridge (1985)
Klee, V., Ladner, R., Manber, R.: Sign-solvability revisited. Linear Algebra and Its Applications 59, 131–158 (1984)
Lovász, L., Plummer, M.D.: Matching Theory. Annals of Discrete Math., vol. 29. North-Holland, Amsterdam (1986)
McCuaig, W.: Pólya’s permanent problem. The Electronic Journal of Combinatorics 1, # R79 (2004)
Murota, K.: An identity for bipartite matching and symmetric determinant. Linear Algebra and Its Applications 222, 261–274 (1995)
Robertson, N., Seymour, P.D., Thomas, R.: Permanents, Pfaffian orientations, and even directed circuits. Annals of Mathematics 150(3), 929–975 (1999)
Vazirani, V.V., Yannakakis, M.: Pfaffian orientations, 0-1 permanents, and even cycles in directed graphs. Discrete Applied Mathematics 25, 179–190 (1989)
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© 2005 Springer-Verlag Berlin Heidelberg
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Kakimura, N., Iwata, S. (2005). Computing the Inertia from Sign Patterns. In: Jünger, M., Kaibel, V. (eds) Integer Programming and Combinatorial Optimization. IPCO 2005. Lecture Notes in Computer Science, vol 3509. Springer, Berlin, Heidelberg. https://doi.org/10.1007/11496915_18
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DOI: https://doi.org/10.1007/11496915_18
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-540-26199-5
Online ISBN: 978-3-540-32102-6
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