Fachbereich Maschinenbau und Mechatronik
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An equitable graph coloring is a proper vertex coloring of a graph G where the sizes of the color classes differ by at most one. The equitable chromatic number is the smallest number k such that G admits such equitable k-coloring. We focus on enumerative algorithms for the computation of the equitable coloring number and propose a general scheme to derive pruning rules for them: We show how the extendability of a partial coloring into an equitable coloring can be modeled via network flows. Thus, we obtain pruning rules which can be checked via flow algorithms. Computational experiments show that the search tree of enumerative algorithms can be significantly reduced in size by these rules and, in most instances, such naive approach even yields a faster algorithm. Moreover, the stability, i.e., the number of solved instances within a given time limit, is greatly improved.
Since the execution of flow algorithms at each node of a search tree is time consuming, we derive arithmetic pruning rules (generalized Hall-conditions) from the network model. Adding these rules to an enumerative algorithm yields an even larger runtime improvement.
Assistance systems have been widely adopted in the manufacturing sector to facilitate various processes and tasks in production environments. However, existing systems are mostly equipped with rigid functional logic and do not provide individual user experiences or adapt to their capabilities. This work integrates human factors in assistance systems by adjusting the hardware and instruction presented to the workers’ cognitive and physical demands. A modular system architecture is designed accordingly, which allows a flexible component exchange according to the user and the work task. Gamification, the use of game elements in non-gaming contexts, has been further adopted in this work to provide level-based instructions and personalised feedback. The developed framework is validated by applying it to a manual workstation for industrial assembly routines.
We present a new Min-Max theorem for an optimization problem closely connected to matchings and vertex covers in balanced hypergraphs. The result generalizes Kőnig’s Theorem (Berge and Las Vergnas in Ann N Y Acad Sci 175:32–40, 1970; Fulkerson et al. in Math Progr Study 1:120–132, 1974) and Hall’s Theorem (Conforti et al. in Combinatorica 16:325–329, 1996) for balanced hypergraphs.