Almost-Linear Time Algorithms for Decremental Graphs: Min-Cost Flow and More via Duality

• Published in: Proceedings / annual Symposium on Foundations of Computer Science pp. 2010 - 2032
• Main Authors: Van Den Brand, Jan, Chen, Li, Kyng, Rasmus, Liu, Yang P., Meierhans, Simon, Gutenberg, Maximilian Probst, Sachdeva, Sushant
• Format: Conference Proceeding
• Language: English
• Published: IEEE 27.10.2024
• Subjects: Approximation algorithms; Computer science; Convex optimization; Costs; Data structures; Directed graphs; Heuristic algorithms; Interior point methods; Maximum flow; Minimum cost flow; Optimization; Runtime
• ISSN: 2575-8454
• DOI: 10.1109/FOCS61266.2024.00120

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 [ \text{Goranci-Racke-} SaranurakTan, SODA 2021] that also works in capacitated graphs. All our algorithms are deterministc, though they can be sped up further using randomized techniques while still working against an adaptive adversary.