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

The \textit{Unsplittable Flow on a Path} (UFP) problem has sparked remarkable attention as a challenging combinatorial problem with profound practical implications. Steered by its prominent application in \textit{power engineering}, the present work formulates a novel generalization of UFP, wherein demands and capacities in the input instance are monotone step functions over the set of edges. As an initial step towards tackling this generalization, we draw on and extend ideas from prior research to devise a a quasi-polynomial time approximation scheme (QPTAS) under the premise that the demands and capacities lie in a quasi-polynomial range. Second, retaining the same assumption, an efficient logarithmic approximation is introduced for the single-source variant of the problem. Finally, we round up the contributions by designing a (kind of) black-box reduction that, under some mild conditions, allows to translate LP-based approximation algorithms for the studied problem into their counterparts for the \textit{Alternating Current Optimal Power Flow} (AC OPF) problem -- a fundamental workflow in operation and control of power systems.