Consider the following parameterized counting variation of the classic subset sum problem, which arises notably in the context of higher homotopy groups of topological spaces: Let $\mathbf{v} \in \mathbb{Q}^d$ be a rational vector, $(T_{1}, T_{2} \ldots T_{m})$ a list of $d \times d$ rational matrices, $S \in \mathbb{Q}^{h \times d}$ a rational matrix not necessarily square and $k$ a parameter. The goal is to compute the number of ways one can choose $k$ matrices $T_{i_1}, T_{i_2}, \ldots, T_{i_k}$ from the list such that $ST_{i_k} \cdots T_{i_1}\mathbf{v} = \mathbf{0} \in \mathbb{Q}^h$. In this paper, we show that this problem is $\# W[2]$-hard for parameter $k$. %This strengthens a result of Matou\v{s}ek, who showed $\# W[1]$-hardness of that problem. As a consequence, computing the $k$-th homotopy group of a $d$-dimensional topological space for $d > 3$ is $\# W[2]$-hard for parameter $k$. We also discuss a decision version of the problem and its several modifications for which we show $W[1]/W[2]$-hardness. This is in contrast to the parameterized $k$-sum problem, which is only $W[1]$-hard (Abboud-Lewi-Williams, ESA'14). In addition, we show that the decision version of the problem without parameter is an undecidable problem, and we give a fixed-parameter tractable algorithm for matrices of bounded size over finite fields, parameterized the matrix dimensions and the order of the field.

翻译：考虑经典子集和问题的一种参数化计数变体，该变体在拓扑空间的高阶同伦群背景下尤为突出：设 $\mathbf{v} \in \mathbb{Q}^d$ 为有理向量，$(T_{1}, T_{2} \ldots T_{m})$ 为 $d \times d$ 有理矩阵列表，$S \in \mathbb{Q}^{h \times d}$ 为未必是方阵的有理矩阵，$k$ 为参数。目标为计算从该列表中选择 $k$ 个矩阵 $T_{i_1}, T_{i_2}, \ldots, T_{i_k}$ 使得 $ST_{i_k} \cdots T_{i_1}\mathbf{v} = \mathbf{0} \in \mathbb{Q}^h$ 的方案总数。本文证明该问题对参数 $k$ 是 $\# W[2]$-困难的。%这强化了 Matou\v{s}ek 关于该问题为 $\# W[1]$-困难的结果。由此推论，对于 $d > 3$ 维拓扑空间的第 $k$ 阶同伦群计算问题，对参数 $k$ 是 $\# W[2]$-困难的。我们还讨论了该问题的判定版本及其若干变体，并证明其具有 $W[1]/W[2]$-困难性。这与参数化 $k$-和问题形成对比——后者仅为 $W[1]$-困难（Abboud-Lewi-Williams, ESA'14）。此外，我们证明无参数版本的判定问题是不可判定的，并给出一个针对有限域上有界尺寸矩阵的固定参数可解算法，该算法以矩阵维度和域阶数为参数。