Counting Vanishing Matrix-Vector Products

Abstract

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 $\textbf{v} \in \mathbb {Q}^d$ 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 $ST_{i_k} \cdots T_{i_1}\textbf{v} = \textbf{0} \in \mathbb {Q}^h$.

In this paper, we show that this problem is $\# \textsf{W}[2]$-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 $\# \textsf{W}[2]$-hard for parameter k. We also discuss a decision version of the problem and its several modifications for which we show $\textsf{W}[1]/\textsf{W}[2]$-hardness. This is in contrast to the parameterized k-sum problem, which is only $\textsf{W}[1]$-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 by the matrix dimensions and the order of the field.