3 research outputs found
A full complexity dichotomy for immanant families
Given an integer and an irreducible character of
for some partition of , the immanant
maps matrices
to . Important special cases
include the determinant and permanent, which are the immanants associated with
the sign and trivial character, respectively.
It is known that immanants can be evaluated in polynomial time for characters
that are close to the sign character: Given a partition of with
parts, let count the boxes to the right of the first
column in the Young diagram of . For a family of partitions ,
let and write Imm
for the problem of evaluating on input and
. If , then Imm is known to be
polynomial-time computable. This subsumes the case of the determinant. On the
other hand, if , then previously known hardness results
suggest that Imm cannot be solved in polynomial time. However, these
results only address certain restricted classes of families .
In this paper, we show that the parameterized complexity assumption FPT
#W[1] rules out polynomial-time algorithms for Imm for any
computationally reasonable family of partitions with
. We give an analogous result in algebraic complexity under
the assumption VFPT VW[1]. Furthermore, if even grows
polynomially in , we show that Imm is hard for #P and VNP.
This concludes a series of partial results on the complexity of immanants
obtained over the last 35 years.Comment: 28 pages, to appear at STOC'2