An Fd upper bound for the reachability problem in vector addition systems with states (VASS) in fixed dimension is given, where Fd is the d-th level of the Grzegorczyk hierarchy of complexity classes. The new algorithm combines the idea of the linear path scheme characterization of the reachability in the $2$-dimension VASSes with the general decomposition algorithm by Mayr, Kosaraju and Lambert. The result improves the Fd + 4 upper bound due to Leroux and Schmitz (LICS 2019).
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Fu et al. (2024) studied this question.
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