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

Abstract

The problem of τ -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 τ . 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 τ -synthesis that imposes restrictions for the target net N : we investigate dependency d-restricted τ -synthesis (DRτ S) where each place of N can influence and be influenced by at most d transitions. For a type τ , if τsynthesis is NP-complete then DRτ S is also NP-complete. In this paper, we show that DRτ 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.