  The Unsplittable Flow on a Path (UFP) problem has garnered considerable
attention as a challenging combinatorial optimization problem with notable
practical implications. Steered by its pivotal applications in 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 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 Alternating Current Optimal Power Flow (AC OPF) problem --
a fundamental workflow in operation and control of power systems.
