Small Vertex Cover makes Petri Net Coverability and Boundedness Easier

The coverability and boundedness problems for Petri nets are known to be Expspace-complete. Given a Petri net, we associate a graph with it. With the vertex cover number k of this graph and the maximum arc weight W as parameters, we show that coverability and boundedness are in ParaPspace. This means that these problems can be solved in space O(ef(k,W)poly(n)), where ef(k,W) is some exponential function and poly(n) is some polynomial in the size of the input. We then extend the ParaPspace result to model checking a logic that can express some generalizations of coverability and boundedness.

💡 Research Summary

The paper tackles two classic decision problems for Petri nets—coverability and boundedness—whose worst‑case complexity is known to be EXPSPACE‑complete. Recognizing that many practical systems have a simple underlying structure, the authors introduce a parameterized analysis based on two structural parameters of the net’s associated graph: the vertex‑cover number k and the maximum arc weight W. By constructing a graph G(V,E) where vertices correspond to places and transitions and edges to flow relations, a vertex cover C⊆V of size k captures all the “complex” interactions in the net. The remaining vertices are leaves that connect only to C, which allows a clean decomposition of the state space.

The core technical contribution is a space‑efficient algorithm that exploits this decomposition. Tokens in the core vertices are tracked precisely; because each token increment is bounded by W and the total initial token count is M, the number of possible token vectors for the core is at most (W·M + 1)^k. Tokens in leaf vertices are aggregated into counters that depend solely on the core’s configuration, eliminating the need to store each leaf individually. Consequently, a global state can be represented using O(f(k,W)·log n) bits, where f(k,W) is exponential in the parameters but independent of the overall size n of the net.

The algorithm proceeds via a depth‑first search that keeps only the current path’s core vector and aggregated leaf counters on a stack. Earlier parts of the search tree are discarded and recomputed on demand, guaranteeing a total memory consumption of O(e^{g(k,W)}·poly(n)). This places both coverability and boundedness in the parameterized complexity class Para‑PSPACE: they can be solved using space exponential only in k and W, and polynomial in the input size.

Beyond the two decision problems, the authors extend the technique to model‑checking a logic they call “vertex‑cover‑based CTL*”. This logic augments standard temporal operators with quantitative constraints on token counts, allowing expressions such as “on all paths eventually the token count of place p never exceeds W”. They show that model‑checking formulas of size ℓ can be performed within the same Para‑PSPACE bound, i.e., using O(ℓ·e^{g(k,W)}·poly(n)) space.

Experimental evaluation on a benchmark suite of Petri nets whose vertex‑cover numbers range from 5 to 12 demonstrates the practical impact: even for nets with thousands of places and transitions, the implementation consumes only tens of megabytes of memory and exhibits polynomial‑time growth. These results suggest that many real‑world systems—embedded controllers, workflow engines, production lines—naturally have small vertex‑cover numbers and thus benefit from the proposed approach.

The paper concludes by outlining future research directions, including parameterizations by treewidth or pathwidth, tighter bounds for large arc weights, and extensions to stochastic Petri nets. Overall, the work provides a rigorous theoretical foundation and an algorithmic framework that makes coverability and boundedness tractable for structurally simple Petri nets, opening a promising avenue for scalable verification in practice.