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. Partially Observable Markov Decision Process (POMDP) is a fundamental model for probabilistic planning in stochastic domains. More recent extensions are constrained and chance-constrained POMDPs, allowing constraints to be specified on some aspects of the policy in addition to the objective function. Despite their expressive power, these models assume all actions take a fixed duration, which poses a limitation in modeling real-world planning problems. In this work, we propose a unified model for durative POMDP and its constrained extensions. First, we convert these extensions into an Integer Linear Programming (ILP) formulation, which can be solved using existing solvers in the ILP literature. Second, a heuristic search approach is provided that can efficiently prune the search space, guided by solving successive partial ILP programs. Finally, evaluation results show that our approach is empirically superior to the state-of-the-art fixed-horizon chance-constrained POMDP solver.

Areg Karapetyan, Khaled Elbassioni, Majid Khonji, Sid Chi-Kin Chau (2022).“Approximations for Generalized Unsplittable Flow on Paths with Application to Power Systems Optimization.”