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We consider an optimization problem arising in the design of controllers for OLED displays. Our objective is to minimize amplitude of the electrical current through the diodes which has a direct impact on the lifetime of such a display. Modeling the problem in mathematical terms yields a class of network flow problems where we group the arcs and pay in each group only for the arc carrying the maximum flow. We develop (fully) combinatorial approximation heuristics suitable for being implemented in the hardware of a control device that drives an OLED display