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We study the classic problem of fairly allocating a set of indivisible goods among a group of agents, and focus on the notion of approximate proportionality known as PROPm. Prior work showed that there exists an allocation that satisfies this notion of fairness for instances involving up to five agents, but fell short of proving that this is true in general. We extend this result to show that a PROPm allocation is guaranteed to exist for all instances, independent of the number of agents or goods. Our proof is constructive, providing an algorithm that computes such an allocation and, unlike prior work, the running time of this algorithm is polynomial in both the number of agents and the number of goods.

Machine vision solutions can perform within a wide range of applications and are commonly used to verify the operation of production systems. They offer the potential to automatically record assembly states and derive information, but simultaneously require a high effort of planning, configuration and implementation. This generally leads to an iterative, expert based implementation with long process times and sets major barriers for many companies. Furthermore the implementation is task specific and needs to be repeated with every variation of product, environment or process. Therefore a novel concept of a simulation-based process chain for both—configuration and enablement—of machine vision systems is presented in this paper. It combines related work of sensor planning algorithms with new methods of training data generation and detailed task specific analysis for assembly applications.