Placement of regenerators in optical networks has attracted the attention of recent research works in optical networks. In this problem, we are given a network with an underlying topology of a graph G and with a set of requests that correspond to paths in G. There is a need to put a regenerator every certain distance, because of a decrease in the power of the signal. In this paper, we investigate the problem of minimizing the number of locations to place the regenerators. We present analytical results regarding the complexity of this problem, in four cases, depending on whether or not there is a bound on the number of regenerators at each node, and depending on whether or not the routing is given or only the requests are given (and part of the solution is also to determine the actual routing). These results include polynomial time algorithms, NP-completeness results, approximation algorithms, and inapproximability results.

We give the first algorithm for testing the feasibility of a system of sporadic real-time tasks on a set of identical processors, solving an open problem in the area of multiprocessor real-time scheduling (Baruah and Pruhs in Journal of Scheduling 13(6):577–582, 2009). We also investigate the related notion of schedulability and a notion that we call online feasibility. Finally, we show that discrete-time schedules are as powerful as continuous-time schedules, which answers another open question in the above mentioned survey.

What an organism needs at least from its environment to produce a set of metabolites, e.g. target(s) of interest and/or biomass, has been called a minimal precursor set. Early approaches to enumerate all minimal precursor sets took into account only the topology of the metabolic network (topological precursor sets). Due to cycles and the stoichiometric values of the reactions, it is often not possible to produce the target(s) from a topological precursor set in the sense that there is no feasible flux. Although considering the stoichiometry makes the problem harder, it enables to obtain biologically reasonable precursor sets that we call stoichiometric. Recently a method to enumerate all minimal stoichiometric precursor sets was proposed in the literature. The relationship between topological and stoichiometric precursor sets had however not yet been studied. Such relationship between topological and stoichiometric precursor sets is highlighted. We also present two algorithms that enumerate all minimal stoichiometric precursor sets. The first one is of theoretical interest only and is based on the above mentioned relationship. The second approach solves a series of mixed integer linear programming problems. We compared the computed minimal precursor sets to experimentally obtained growth media of several Escherichia coli strains using genome-scale metabolic networks. The results show that the second approach efficiently enumerates minimal precursor sets taking stoichiometry into account, and allows for broad in silico studies of strains or species interactions that may help to understand e.g. pathotype and niche-specific metabolic capabilities. sasita is written in Java, uses cplex as LP solver and can be downloaded together with all networks and input files used in this paper at http://sasita.gforge.inria.fr/ .