Improved Algorithm for Reachability in $d$-VASS

TL;DR

This paper presents an improved algorithm for the reachability problem in fixed-dimension vector addition systems with states, achieving a tighter complexity bound by combining linear path schemes with a decomposition approach.

Contribution

It introduces a new algorithm that refines the upper bound for reachability in d-dimensional VASS, improving upon previous complexity bounds.

Findings

Achieves an $ extsf{F}_d$ upper bound for reachability in fixed dimension.

Combines linear path scheme characterization with a decomposition algorithm.

Improves previous upper bounds from $ extsf{F}_{d+4}$ to $ extsf{F}_d$.

Abstract

An $\mathsf{F}_{d}$ upper bound for the reachability problem in vector addition systems with states (VASS) in fixed dimension is given, where $\mathsf{F}_d$ 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 $\mathsf{F}_{d + 4}$ upper bound due to Leroux and Schmitz (LICS 2019).