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Abstract. We have studied the network geometry of the endoplasmic reticulum by means of graph theoretical and integer programming models. The purpose is to represent this structure as close as possible by a class of finite, undirected and connected graphs the nodes of which have to be either of degree three or at most of degree three. We determine plane graphs of minimal total edge length satisfying degree and angle constraints, and we show that the optimal graphs are close to the ER network geometry. Basically, two procedures are formulated to solve the optimization problem: a binary linear program, that iteratively constructs an optimal solution, and a linear program, that iteratively exploits additional cutting planes from different families to accelerate the solution process. All formulations have been implemented and tested on a series of real-life and randomly generated cases. The cutting plane approach turns out to be particularly efficient for the real-life testcases, since it outperforms the pure integer programming approach by a factor of at least 10.Keywords: endoplasmic reticulum, plane graph, 0-1 programming, separation procedure ProblemThe endoplasmic reticulum (ER) is a membrane-bound organelle that forms a highly complicated interconnected network of tubules and flattened sacs (known as cisternae) [10,6]. As the cortical ER in plant cells occupies a very thin, almost two-dimensional, layer of cytoplasm beneath the plasma membrane, our study of the ER network will be based on 2D approximations of the ER network in Tobacco leaf epidermal cells. Figure 1 (a) shows an instance of live cell images of an ER network [13]. Transition between tubules and cisternae can be highly dynamic and tubules can also dynamically change their polygonal network [10]. The dynamic ER shape is suggested to be adaptable to the cells requirements for ER function; for example, ER cisternae may be the preferred site of protein translocation while tubules might be the preferred site for ER vesicle budding. As an expanding number of proteins have been identified that mediate the generation and shape of the ER network, the stage is now set for investigations into the mechanisms regulating ER morphology within the cell. To be able to carry out such investigations, better tools are required to quantify the morphology and dynamic rearrangements that the ER undergoes. It is important to consider that whilst the proteins and stresses on the system driving formation and changes in the remodelling may differ between eukaryotic systems, the network geometry of the ER appears to be fairly well conserved in terms of it consisting of a polygonal network of interconnected tubules and cisternae. Therefore, the problem considered here has universal appeal towards understanding the constraints placed on ER network formation in all systems.A quantitative analysis [11] of the ER network in tobacco leaf epidermal cells suggests that it is a perturbed Euclidean Steiner network between terminals, where terminals are persistent nodes (static elemen...

Cloud federation enables service providers to collaborate to provide better services to customers. For cloud storage services, optimizing customer object placement for a member of a federation is a real challenge. Storage, migration, and latency costs need to be considered. These costs are contradictory in some cases. In this article, we modeled object placement as a multi-objective optimization problem. The proposed model takes into account parameters related to the local infrastructure, the federated environment, customer workloads, and their SLAs. For resolving this problem, we propose CDP-NSGAII IR , a Constraint Data Placement matheuristic based on NSGAII with Injection and Repair functions. The injection function aims to enhance the solutions’ quality. It consists to calculate some solutions using an exact method then inject them into the initial population of NSGAII. The repair function ensures that the solutions obey the problem constraints and so prevents from exploring large sets of unfeasible solutions. It reduces drastically the execution time of NSGAII. Experimental results show that the injection function improves the HV of NSGAII and the exact method by up to 94% and 60%, respectively, while the repair function reduces the execution time by an average of 68%.