The Synthesis Problem for Elementary Net Systems is NP-Complete

The so-called synthesis problem consists in deciding for a class of nets whether a given graph is isomorphic to the case graph of some net and then constructing the net. This problem has been solved for various classes of nets, ranging from elementary nets to Petri nets. The general principle is to compute regions in the graph, i.e. subsets of nodes liable to represent extensions of places of an associated net. The naive method of synthesis which relies on this principle leads to exponential algorithms for an arbitrary class of nets. In an earlier study, we gave algorithms that solve the synthesis problem in polynomial time for the class of bounded Petri nets. We show here that in contrast the synthesis problem is indeed NP-complete for the class of elementary nets. This result is independent from the results of Kunihiko Hiraishi, showing that both problems of separation and inhibition by regions at a given node of the graph are NP-complete

Conditions for duality between fluxes and concentrations in biochemical networks

Mathematical and computational modelling of biochemical networks is often done in terms of either the concentrations of molecular species or the fluxes of biochemical reactions. When is mathematical modelling from either perspective equivalent to the other? Mathematical duality translates concepts, theorems or mathematical structures into other concepts, theorems or structures, in a one-to-one manner. We present a novel stoichiometric condition that is necessary and sufficient for duality between unidirectional fluxes and concentrations. Our numerical experiments, with computational models derived from a range of genome-scale biochemical networks, suggest that this flux-concentration duality is a pervasive property of biochemical networks. We also provide a combinatorial characterisation that is sufficient to ensure flux-concentration duality. That is, for every two disjoint sets of molecular species, there is at least one reaction complex that involves species from only one of the two sets. When unidirectional fluxes and molecular species concentrations are dual vectors, this implies that the behaviour of the corresponding biochemical network can be described entirely in terms of either concentrations or unidirectional fluxes

Narrowing down the Hardness Barrier of Synthesizing Elementary Net Systems

Elementary net system feasibility is the problem to decide for a given automaton A if there is a certain boolean Petri net with a state graph isomorphic to A. This is equivalent to the conjunction of the state separation property (SSP) and the event state separation property (ESSP). Since feasibility, SSP and ESSP are known to be NP-complete in general, there was hope that the restriction of graph parameters for A can lead to tractable and practically relevant subclasses. In this paper, we analyze event manifoldness, the amount of occurrences that an event can have in A, and state degree, the number of allowed successors and predecessors of states in A, as natural input restrictions. Recently, it has been shown that all three decision problems, feasibility, SSP and ESSP, remain NP-complete for linear A where every event occurs at most three times. Here, we show that these problems remain hard even if every event occurs at most twice. Nevertheless, this has to be paid by relaxing the restriction on state degree, allowing every state to have two successor and two predecessor states. As we also show that SSP becomes tractable for linear A where every event occurs at most twice the only open cases left are ESSP and feasibilty for the same input restriction