Parameterized Complexity and Approximability of Coverability Problems in Weighted Petri Nets

Many databases have been filled with the chemical reactions found in scientific publications and the associated information (efficiency, chemical products involved...). They can be used to define functions representing costs such as the ecoligical impact of the reactions. A major challenge is to use computer driven optimization in order to improve synthesis process and to provide algorithms to help determining a minimum cost pathway (series of reactions) for the synthesis of a molecule. As, the classical Petri nets do not allows us to consider the optimization component, a weighted model has to be defined and the complexity of the associated problems studied. In this paper we introduce the weighted Petri nets in which each transition is associated with a weight. We define the Minimum Weight Synthesis Problem: find a minimum weight series of transitions to fire to produce a given target component. It mainly differ from classical coverability as it is an optimization problem. We prove that this problem is EXPSPACE-Complete and that there is no polynomial approximation even when both in and outdegree are fixed to two and the target state is a single component. We also consider a more constraint version of the problem limiting the number of fired transitions. We prove this problem falls into PSPACE and the parametrized versions into XP but it remains not approximable.