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 $k$. As a consequence, computing the $k$-th homotopy group of a $d$-dimensional 1-connected 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以下参数化计数变体，该变体在拓扑空间的高阶同伦群背景下尤为重要：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，$S \in \mathbb{Q}^{h \times d}$ 为不一定为方阵的有理矩阵，$k$ 为参数。目标是计算从该列表中选取 $k$ 个矩阵 $T_{i_1}, T_{i_2}, \ldots, T_{i_k}$ 的方式数，使得 $ST_{i_k} \cdots T_{i_1}\mathbf{v} = \mathbf{0} \in \mathbb{Q}^h$。在本文中，我们证明该问题对于参数 $k$ 是 $\# W[2]$-困难的。由此可知，对于 $d > 3$ 的 $d$ 维 1-连通拓扑空间，计算其第 $k$ 个同伦群对于参数 $k$ 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，并证明了它们的 $W[1]/W[2]$-困难性。这与参数化 $k$-和问题（仅 $W[1]$-困难，见 Abboud-Lewi-Williams, ESA'14）形成对比。此外，我们证明无参数版本的判定问题是不可判定的，并针对有限域上有界尺寸的矩阵，给出一个以矩阵维数和域阶为参数的固定参数可解算法。