English

Complexity and Completeness of Immanants

Computational Complexity 2007-05-23 v2 Combinatorics

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(nδ)\Omega(n^\delta) for some δ>0\delta>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.

Keywords

Cite

@article{arxiv.cs/0301024,
  title  = {Complexity and Completeness of Immanants},
  author = {Jean-Luc Brylinski and Ranee Brylinski},
  journal= {arXiv preprint arXiv:cs/0301024},
  year   = {2007}
}

Comments

10 pages, Latex