On complexity of mixed discriminants of rank 2 matrices and related problems

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We prove that it is {number_sign}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 [30] and in [32]. We also present a deterministic poly-time algorithm computing the mixed discriminant D(A{sub 1},..,A{sub N}) provided that the linear (matrix) subspace generated by {l_brace}A{sub 1},..,A{sub N}{r_brace} is small and discuss randomized algorithms approximating mixed discriminants within absolute error.

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12 p.

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Gurvits, L. (Leonid) January 1, 2003.

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We prove that it is {number_sign}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 [30] and in [32]. We also present a deterministic poly-time algorithm computing the mixed discriminant D(A{sub 1},..,A{sub N}) provided that the linear (matrix) subspace generated by {l_brace}A{sub 1},..,A{sub N}{r_brace} is small and discuss randomized algorithms approximating mixed discriminants within absolute error.

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12 p.

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  • Submitted to: 28th International Symposium on Mathematical Foundations of Computer Science, August 25-29, 2003, Bratislave, Slovar Republic

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  • Report No.: LA-UR-03-2248
  • Grant Number: none
  • Office of Scientific & Technical Information Report Number: 976587
  • Archival Resource Key: ark:/67531/metadc930054

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  • January 1, 2003

Added to The UNT Digital Library

  • Nov. 13, 2016, 7:26 p.m.

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  • Dec. 12, 2016, 4:31 p.m.

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Gurvits, L. (Leonid). On complexity of mixed discriminants of rank 2 matrices and related problems, article, January 1, 2003; United States. (digital.library.unt.edu/ark:/67531/metadc930054/: accessed October 23, 2018), University of North Texas Libraries, Digital Library, digital.library.unt.edu; crediting UNT Libraries Government Documents Department.