On the complexity of mixed discriminants and related problems

Authors:
Leonid Gurvits
Affiliations:
Los Alamos National Laboratory, Los Alamos, NM
Venue:
MFCS'05 Proceedings of the 30th international conference on Mathematical Foundations of Computer Science
Year:
2005

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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(A1,..,AN) provided that the linear (matrix) subspace generated by {A1,..,AN } is small and discuss randomized algorithms approximating mixed discriminants within absolute error.