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The Unsplittable Flow on a Path (UFP) problem has garnered considerableattention as a challenging combinatorial optimization problem with notablepractical implications. Steered by its pivotal applications in powerengineering, the present work formulates a novel generalization of UFP, whereindemands and capacities in the input instance are monotone step functions overthe set of edges. As an initial step towards tackling this generalization, wedraw on and extend ideas from prior research to devise a quasi-polynomial timeapproximation scheme (QPTAS) under the premise that the demands and capacitieslie in a quasi-polynomial range. Second, retaining the same assumption, anefficient logarithmic approximation is introduced for the single-source variantof the problem. Finally, we round up the contributions by designing a (kind of)black-box reduction that, under some mild conditions, allows to translateLP-based approximation algorithms for the studied problem into theircounterparts for the Alternating Current Optimal Power Flow (AC OPF) problem --a fundamental workflow in operation and control of power systems.

Approximations for Generalized Unsplittable Flow on Paths with Application to Power Systems Optimization