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arXiv · 2602.15483

Exploring VASS Parameterised by Geometric Dimension

Wojciech Czerwiński·Roland Guttenberg·Łukasz Orlikowski·Henry Sinclair-Banks·Yangluo Zheng
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Abstract

The geometric dimension $g$ of a Vector Addition System with States (VASS) is the dimension of the vector space generated by cycles in the VASS; this parameter refines the standard dimension $d$, the number of counters. Recently, it was discovered that the fastest-known algorithm for solving the reachability problem for VASS has the same complexity in terms of $g$ as in terms of $d$. This suggests that the geometric dimension may in fact be a more adequate parameter for measuring the complexity of VASS reachability problems. We initiate a more systematic study of the geometric dimension. We discuss differences between two parameters: the geometric dimension and the SCC dimension. Our main technical result states that classical results about the coverability and boundedness problems can be improved from dimension $d$ to geometric dimension $g$. Namely, coverability is witnessed by runs of length $n^{2^{\mathcal{O}(g)}}$ instead of $n^{2^{\mathcal{O}(d)}}$, and unboundedness can be witnessed by runs of length $n^{2^{\mathcal{O}(g\log g)}}$ instead of $n^{2^{\mathcal{O}(d\log d )}}$, where $n$ is the size of the instance. We also study integer reachability and simultaneous unboundedness in VASS parameterised by the geometric dimension.

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Wojciech Czerwiński, Roland Guttenberg, Łukasz Orlikowski, Henry Sinclair-Banks, Yangluo Zheng. 2026-02-17. Exploring VASS Parameterised by Geometric Dimension. https://arxiv.org/abs/2602.15483

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