# Deterministic Decremental SSSP and Approximate Min-Cost Flow in Almost-Linear Time

arXiv: https://arxiv.org/abs/2101.07149 · HTML: https://mlsanity.com/paper/2101.07149/

## Key facts

- Cited by 2 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 18 Jan 2021.

## Built on by (most cited first, 2 of 2)

- [Incremental Approximate Single-Source Shortest Paths with Predictions](https://mlsanity.com/paper/2502.08125/) (2025)
- [Dynamic algorithms for k-center on graphs](https://mlsanity.com/paper/2307.15557/) (2023)

## Abstract (arXiv)

In the decremental single-source shortest paths problem, the goal is to maintain distances from a fixed source s to every vertex v in an m-edge graph undergoing edge deletions. In this paper, we conclude a long line of research on this problem by showing a near-optimal deterministic data structure that maintains (1+ε)-approximate distance estimates and runs in m1+o(1) total update time. Our result, in particular, removes the oblivious adversary assumption required by the previous breakthrough result by Henzinger et al. [FOCS'14], which leads to our second result: the first almost-linear time algorithm for (1-ε)-approximate min-cost flow in undirected graphs where capacities and costs can be taken over edges and vertices. Previously, algorithms for max flow with vertex capacities, or min-cost flow with any capacities required super-linear time. Our result essentially completes the picture for approximate flow in undirected graphs. The key technique of the first result is a novel framework that allows us to treat low-diameter graphs like expanders. This allows us to harness expander properties while bypassing shortcomings of expander decomposition, which almost all previous expander-based algorithms needed to deal with. For the second result, we break the notorious flow-decomposition barrier from the multiplicative-weight-update framework using randomization.