Abstract
We prove that it is #P-hard to compute the mixed discriminant of rank 2 positive semidefinite matrices. We present poly-time algorithms to approximate the ”beast”. We also prove NP-hardness of two problems related to mixed discriminants of rank 2 positive semidefinite matrices. One of them, the so called Full Rank Avoidance problem, had been conjectured to be NP-Complete in [23] and in [25]. We also present a deterministic poly-time algorithm computing the mixed discriminant D(A 1,..,A N ) provided that the linear (matrix) subspace generated by {A 1,..,A N } is small and discuss randomized algorithms approximating mixed discriminants within absolute error.
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Gurvits, L. (2005). On the Complexity of Mixed Discriminants and Related Problems. In: Jȩdrzejowicz, J., Szepietowski, A. (eds) Mathematical Foundations of Computer Science 2005. MFCS 2005. Lecture Notes in Computer Science, vol 3618. Springer, Berlin, Heidelberg. https://doi.org/10.1007/11549345_39
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DOI: https://doi.org/10.1007/11549345_39
Publisher Name: Springer, Berlin, Heidelberg
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