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.
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Notes
- 1.
Note that \(\mathbb {Z}^n\) is a direct sum of n copies of \(\mathbb {Z}\) while \(\mathbb {Z}_{p_i}\) is a finite cyclic group of order \(p_i\).
- 2.
When k is a part of the input and represented in unary.
- 3.
The notion of \(\textsf{FPT}\) algorithm is defined in the next paragraph.
- 4.
One of the reviewers thankfully pointed out related results about the matrix mortality problem (see, e.g., [10]) which give a shorter proof of this fact and which also give a proof of undecidability of VEST without the initial vector and the parameter.
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Acknowledgments
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C.B. and V.K. were supported by Austrian Science Fund (FWF, project Y1329). C.B. is also funded by the European Union (ERC, CountHom, 101077083). Views and opinions expressed are however those of the author(s) only and do not necessarily reflect those of the European Union or the European Research Council Executive Agency. Neither the European Union nor the granting authority can be held responsible for them.
K.S. was supported by DFG Research Group ADYN via grant DFG 411362735, M.S. acknowledges support by the project “Grant Schemes at CU” (reg. no. CZ.02.2.69/0.0/0.0/19_073/0016935) and GAČR grant 22-19073S.
We also thank the anonymous reviewers for their useful comments.
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Brand, C., Korchemna, V., Simonov, K., Skotnica, M. (2024). Counting Vanishing Matrix-Vector Products. In: Uehara, R., Yamanaka, K., Yen, HC. (eds) WALCOM: Algorithms and Computation. WALCOM 2024. Lecture Notes in Computer Science, vol 14549. Springer, Singapore. https://doi.org/10.1007/978-981-97-0566-5_24
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