A Polynomial-Time Algorithm for Reachability Problem of a Subclass of Petri Net and Chip Firing Games

Reachability is a fundamental problem in the study of many dynamical systems; especially, the complexity of this problem for Petri nets has been open for many years. In this paper, we investigate to the reachability of state machine - a special class of Petri nets, in which every transition has exactly one input place and one output place by giving a correspondence between this problem and flow networks problem. We first study the bijection between state machine of n places and Conflicting Chip Firing Game (CCFG) on a graph of n vertices. Then we use Push-Relabel algorithm on flow networks to solve the reachability problem of CCFG with complexity O(n³), which implies a corresponding algorithm for state machines.