On the Parameterized Complexity of Synthesizing Boolean Petri Nets With Restricted Dependency (Technical Report)

TL;DR

This paper studies the computational complexity of synthesizing Boolean Petri nets with restricted dependencies, showing that the problem is in XP and W[2]-hard for certain net types, highlighting the challenges in constrained synthesis.

Contribution

It introduces dependency-restricted $ au$-synthesis, analyzing its complexity and establishing XP membership and W[2]-hardness for various Boolean Petri net types.

Findings

DR$ au$S is in XP when parameterized by dependency d.

DR$ au$S is W[2]-hard for net types with set and reset interactions.

NP-completeness persists for many types even with dependency restrictions.

Abstract

The problem of $\tau$-synthesis consists in deciding whether a given directed labeled graph $A$ is isomorphic to the reachability graph of a Boolean Petri net $N$ of type $\tau$. In case of a positive decision, $N$ should be constructed. For many Boolean types of nets, the problem is NP-complete. This paper deals with a special variant of $\tau$-synthesis that imposes restrictions for the target net $N$: we investigate \emph{dependency $d$-restricted $\tau$-synthesis (DR$\tau$S)} where each place of $N$ can influence and be influenced by at most $d$ transitions. For a type $\tau$, if $\tau$-synthesis is NP-complete then DR$\tau$S is also NP-complete. In this paper, we show that DR$\tau$S parameterized by $d$ is in XP. Furthermore, we prove that it is $W[2]$-hard, for many Boolean types that allow unconditional interactions $set$ and $reset$.