A Note on the Parameterised Complexity of Coverability in Vector Addition Systems

We investigate the parameterised complexity of the classic coverability problem for vector addition systems (VAS): given a finite set of vectors $V \subseteq\mathbb{Z}^d$, an initial configuration $s\in\mathbb{N}^d$, and a target configuration $t\in\mathbb{N}^d$, decide whether starting from $s$, one can iteratively add vectors from $V$ to ultimately arrive at a configuration that is larger than or equal to $t$ on every coordinate, while not observing any negative value on any coordinate along the way. We consider two natural parameters for the problem: the dimension $d$ and the size of $V$, defined as the total bitsize of its encoding. We present several results charting the complexity of those two parameterisations, among which the highlight is that coverability for VAS parameterised by the dimension and with all the numbers in the input encoded in unary is complete for the class XNL under PL-reductions. We also discuss open problems in the topic, most notably the question about fixed-parameter tractability for the parameterisation by the size of $V$.

💡 Research Summary

This paper investigates the parameterised complexity of the classic coverability problem for vector addition systems (VAS). A VAS consists of a finite set of integer vectors $V\subseteq\mathbb Z^d$; starting from an initial configuration $s\in\mathbb N^d$, one may repeatedly add vectors from $V$ as long as no coordinate becomes negative. The coverability question asks whether there exists a configuration $t’\ge t$ (coordinate‑wise) reachable from $s$. Historically, this problem is EXPSPACE‑hard and belongs to EXPSPACE.

The authors focus on two natural parameters: the dimension $d$ of the system, and the total bitsize $|V|$ of the vector set (the sum of absolute values in unary encoding, or the binary bit length in binary encoding). They define two parameterised decision problems: $p\text{-dim‑COVERABILITY}$ (parameter $d$) and $p\text{-size‑COVERABILITY}$ (parameter $|V|$).

For $p\text{-dim‑COVERABILITY}$ with a unary encoding of all numbers, the paper shows that the problem lies in XNL, the class of problems solvable in non‑deterministic space $f(d)\cdot\log n$. This follows from known non‑deterministic space bounds $2^{O(d)}\cdot\log n$ for coverability runs (Rackoff, Künnemann et al.). The authors then prove XNL‑hardness by a PL‑reduction from the finite‑automata intersection non‑emptiness problem studied by Wehar. The reduction maps automaton states and transition labels to VAS dimensions and vectors while preserving the parameter $d$, establishing XNL‑completeness.

When the input is binary‑encoded, the same problem becomes para‑PSPACE‑complete. The upper bound again follows from the $2^{O(d)}\cdot\log n$ space algorithm, which now runs in polynomial space after a pre‑computation that depends only on $d$. For the lower bound, the authors leverage the PSPACE‑hardness of coverability for a fixed VAS (Draghici, Haase, Ryzhikov) and show that this hardness lifts to the parameterised setting via an FPT reduction that fixes the dimension. Consequently, $p\text{-dim‑COVERABILITY}$ is para‑PSPACE‑hard, and together with the upper bound it is para‑PSPACE‑complete.

The paper also analyses $p\text{-size‑COVERABILITY}$, where the parameter is the bitsize of $V$. A simple PL‑reduction maps an instance $(\langle V,s,t\rangle,|V|)$ to $(\langle V,s,t\rangle,d)$, showing that the complexity results for the dimension parameter transfer directly: with unary encoding the problem is XNL‑complete, and with binary encoding it is para‑PSPACE‑complete.

These results are significant because XNL and para‑PSPACE are relatively obscure parameterised complexity classes with few natural complete problems. By showing that the intuitive parameter “dimension” yields XNL‑complete and para‑PSPACE‑complete problems, the paper enriches the library of natural complete problems and demonstrates the usefulness of parameterised analysis for VAS and Petri nets.

Finally, the authors discuss open questions. The most prominent is whether coverability becomes fixed‑parameter tractable when parameterised by the size of $V$ (i.e., $|V|$). Existing results show PSPACE‑hardness for a fixed $V$, but it remains unknown whether an FPT algorithm exists when $|V|$ is treated as the parameter. The paper positions this as a key direction for future research, suggesting that deeper structural insights into VAS may be required to resolve the FPT status.