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

TL;DR

This paper introduces the first almost-linear time algorithms for decremental graph problems like min-cost flow and s-t distance, using duality and a new data structure for approximate min-ratio cuts.

Contribution

It presents a novel duality-based framework and a data structure for maintaining approximate min-ratio cuts in decremental graphs, improving runtimes for several problems.

Findings

Achieved almost-linear time algorithms for decremental min-cost flow and s-t distance.

Developed a new data structure for maintaining approximate min-ratio cuts.

Improved static min-cost flow algorithms in both randomized and deterministic settings.

Abstract

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-R\"{a}cke-Saranurak-Tan, 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.