# The Polynomial Complexity of Vector Addition Systems with States

## TL;DR

This paper characterizes the asymptotic computational complexity of vector addition systems with states, showing it is either polynomial with a computable degree or exponential, and provides methods to determine this degree efficiently.

## Contribution

It establishes a precise complexity classification for vector addition systems with states, including a polynomial-time method to compute the degree of polynomial complexity.

## Key findings

- Complexity is either polynomial with a computable degree or exponential.
- The degree of polynomial complexity can be computed in polynomial time.
- The degree k satisfies 1 ≤ k ≤ 2^n, where n is the system's dimension.

## Abstract

Vector addition systems are an important model in theoretical computer science and have been used in a variety of areas. In this paper, we consider vector addition systems with states over a parameterized initial configuration. For these systems, we are interested in the standard notion of computational complexity, i.e., we want to understand the length of the longest trace for a fixed vector addition system with states depending on the size of the initial configuration. We show that the asymptotic complexity of a given vector addition system with states is either $\Theta(N^k)$ for some computable integer $k$, where $N$ is the size of the initial configuration, or at least exponential. We further show that $k$ can be computed in polynomial time in the size of the considered vector addition system. Finally, we show that $1 \le k \le 2^n$, where $n$ is the dimension of the considered vector addition system.