Complexity and Completeness of Immanants

Jean-Luc Brylinski; Ranee Brylinski

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arXiv · cs/0301024

Complexity and Completeness of Immanants

Jean-Luc Brylinski·Ranee Brylinski

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Abstract

Immanants are polynomial functions of n by n matrices attached to irreducible characters of the symmetric group S_n, or equivalently to Young diagrams of size n. Immanants include determinants and permanents as extreme cases. Valiant proved that computation of permanents is a complete problem in his algebraic model of NP theory, i.e., it is VNP-complete. We prove that computation of immanants is VNP-complete if the immanants are attached to a family of diagrams whose separation is $Ω(n^δ)$ for some $δ>0$. We define the separation of a diagram to be the largest number of overhanging boxes contained in a single row. Our theorem proves a conjecture of Buergisser for a large variety of families, and in particular we recover with new proofs his VNP-completeness results for hooks and rectangles.

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Jean-Luc Brylinski, Ranee Brylinski. 2003-01-23. Complexity and Completeness of Immanants. https://arxiv.org/abs/cs/0301024

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