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We give the first almost-linear total time algorithm for deciding if a flow of cost at most $$F$$ still exists in a directed graph, with edge costs and capacities, undergoing decremental updates, i.e., edge deletions, capacity decreases, and cost increases. This implies almost-linear time algorithms for approximating the minimum-cost flow value and s-t distance on such decremental graphs. Our framework additionally allows us to maintain decremental strongly connected components in almost-linear time deterministically. These algorithms also improve over the current best known runtimes for statically computing minimum-cost flow, in both the randomized and deterministic settings. We obtain our algorithms by taking the dual perspective, which yields cut-based algorithms. More precisely, our algorithm computes the flow via a sequence of $$m^{1+o(1)}$$-dynamic min-ratio cut problems, the dual analog of the dynamic min-ratio cycle problem that underlies recent fast algorithms for minimum-cost flow. Our main technical contribution is a new data structure that returns an approximately optimal min-ratio cut in amortized $$m^{o(1)}$$ time by maintaining a tree-cut sparsifier. This is achieved by devising a new algorithm to maintain the dynamic expander hierarchy of [Goranci-Racke-Saranurak-Tan, SODA 2021] that also works in capacitated graphs. All our algorithms are deterministic, though they can be sped up further using randomized techniques while still working against an adaptive adversary. 

With the wide adoption of deep neural network (DNN) models for various applications, enterprises, and cloud providers have built deep learning clusters and increasingly deployed specialized accelerators, such as GPUs and TPUs, for DNN training jobs. To arbitrate cluster resources among multi-user jobs, existing schedulers fall short, either lacking fine-grained heterogeneity awareness or hardly generalizable to various scheduling policies. To fill this gap, we propose a novel design of a task-level heterogeneity-aware scheduler, Hadar, based on an online optimization framework that can express other scheduling algorithms. Hadar leverages the performance traits of DNN jobs on a heterogeneous cluster, characterizes the task-level performance heterogeneity in the optimization problem, and makes scheduling decisions across both spatial and temporal dimensions. The primal-dual framework is employed, with our design of a dual subroutine, to solve the optimization problem and guide the scheduling design. Extensive trace-driven simulations with representative DNN models have been conducted to demonstrate that Hadar improves the average job completion time (JCT) by 3× over an Apache YARN-based resource manager used in production. Moreover, Hadar outperforms Gavel[1], the state-of-the-art heterogeneity-aware scheduler, by 2.5× for the average JCT, and shortens the queuing delay by 13% and improve FTF (Finish-Time-Fairness) by 1.5%.