Counting vanishing matrix-vector products

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 v ∈ Q d be a rational vector, ( T 1 , T 2 … , T m ) a list of d × d rational matrices, S ∈ Q h × 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 , … , T i k from the list such that S T i k ⋯ T i 1 v = 0 ∈ 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 by the matrix dimensions and the order of the field.