Given an integer $n\geq 1$ and an irreducible character $\chi_{\lambda}$ of $S_{n}$ for some partition $\lambda$ of $n$, the immanant $\mathrm{imm}_{\lambda}:\mathbb{C}^{n\times n}\to\mathbb{C}$ maps matrices $A\in\mathbb{C}^{n\times n}$ to $\mathrm{imm}_{\lambda}(A)=\sum_{\pi\in S_{n}}\chi_{\lambda}(\pi)\prod_{i=1}^{n}A_{i,\pi(i)}$. 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 $\lambda$ of $n$ with $s$ parts, let $b(\lambda):=n-s$ count the boxes to the right of the first column in the Young diagram of $\lambda$. For a family of partitions $\Lambda$, let $b(\Lambda):=\max_{\lambda\in\Lambda}b(\lambda)$ and write Imm$(\Lambda)$ for the problem of evaluating $\mathrm{imm}_{\lambda}(A)$ on input $A$ and $\lambda\in\Lambda$. If $b(\Lambda)<\infty$, then Imm$(\Lambda)$ is known to be polynomial-time computable. This subsumes the case of the determinant. On the other hand, if $b(\Lambda)=\infty$, then previously known hardness results suggest that Imm$(\Lambda)$ cannot be solved in polynomial time. However, these results only address certain restricted classes of families $\Lambda$. In this paper, we show that the parameterized complexity assumption FPT $\neq$ #W[1] rules out polynomial-time algorithms for Imm$(\Lambda)$ for any computationally reasonable family of partitions $\Lambda$ with $b(\Lambda)=\infty$. We give an analogous result in algebraic complexity under the assumption VFPT $\neq$ VW[1]. Furthermore, if $b(\lambda)$ even grows polynomially in $\Lambda$, we show that Imm$(\Lambda)$ is hard for #P and VNP. This concludes a series of partial results on the complexity of immanants obtained over the last 35 years.

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待婚家庭的完全复杂性二分法

给定一个整数$ n \ geq 1 $和一个不可约字符$ \ chi _ {\ lambda} $的$ S_ {n} $，对于$ n $的某些分区$ \ lambda $，无数的$ \ mathrm {imm} _ { \ lambda}：\ mathbb {C} ^ {n \ times n} \ to \ mathbb {C} $将矩阵$ A \ in \ mathbb {C} ^ {n \ times n} $映射到$ \ mathrm {imm} _ {\ lambda}（A）= \ sum _ {\ pi \ in S_ {n}} \ chi _ {\ lambda}（\ pi）\ prod_ {i = 1} ^ {n} A_ {i，\ pi（i ）} $。重要的特殊情况包括行列式和永久性，它们分别是与符号和琐碎字符相关联的内在性。众所周知，可以在多项式时间内对接近符号字符的字符求出无定值：给定$ n $的分区$ \ lambda $和$ s $部分，让$ b（\ lambda）：= ns $计数$ \ lambda $的Young图中第一列右侧的框。对于一个分区族$ \ Lambda $，让$ b（\ Lambda）：= \ max _ {\ lambda \ in \ Lambda} b（\ lambda）$并写Imm $（\ Lambda）$来解决在输入$ A上评估$ \ mathrm {imm} _ {\ lambda}（A）$的问题$和$ \ lambda \ in \ Lambda $。如果$ b（\ Lambda）<\ infty $，则已知Imm $（\ Lambda）$是多项式时间可计算的。这包含了行列式的情况。另一方面，如果$ b（\ Lambda）= \ infty $，则先前已知的硬度结果表明Imm $（\ Lambda）$无法在多项式时间内求解。但是，这些结果仅针对某些受限类别的家庭$ \ Lambda $。在本文中，我们证明了参数化复杂度假设FPT $ \ neq $ #W [1]排除了任何计算合理的分区$ \ Lambda $和$ b（$）的Imm $（\ Lambda）$多项式时间算法。 \ Lambda）= \ infty $。在假设VFPT $ \ neq $ VW [1]的情况下，我们给出了代数复杂度的类似结果。此外，如果$ b（\ lambda）$甚至在$ \ Lambda $中呈多项式增长，则表明Imm $（\ Lambda）$对于#P和VNP来说很难。这得出了有关过去35年中获得的无名氏的复杂性的一系列部分结果。