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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. 

The authors present an algorithm that, given an n-vertex m-edge Eulerian graph with polynomially bounded weights, computes an 𝑂 ~ ( 𝑛 log ⁡ 2 𝑛 ⋅ 𝜀 − 2 ) O ~ (nlog 2 n⋅ε −2 )-edge 𝜀 ε-approximate Eulerian sparsifier with high probability in 𝑂 ~ ( 𝑚 log ⁡ 3 𝑛 ) O ~ (mlog 3 n) time (where 𝑂 ~ ( ⋅ ) O ~ (⋅) hides polyloglog(n) factors). By a reduction from Peng-Song (STOC ’22), this yields an 𝑂 ~ ( 𝑚 log ⁡ 3 𝑛 + 𝑛 log ⁡ 6 𝑛 ) O ~ (mlog 3 n+nlog 6 n)-time algorithm for solving n-vertex m-edge Eulerian Laplacian systems with polynomially bounded weights with high probability, improving on the previous state-of-the-art runtime of Ω ( 𝑚 log ⁡ 8 𝑛 + 𝑛 log ⁡ 23 𝑛 ) Ω(mlog 8 n+nlog 23 n). They also provide a polynomial-time algorithm that computes sparsifiers with 𝑂 ( min ⁡ ( 𝑛 log ⁡ 𝑛 ⋅ 𝜀 − 2 + 𝑛 log ⁡ 5 / 3 𝑛 ⋅ 𝜀 − 4 / 3 , 𝑛 log ⁡ 3 / 2 𝑛 ⋅ 𝜀 − 2 ) ) O(min(nlogn⋅ε −2 +nlog 5/3 n⋅ε −4/3 ,nlog 3/2 n⋅ε −2 )) edges, improving the previous best bounds. Furthermore, they extend their techniques to yield the first 𝑂 ( 𝑚 ⋅ polylog ( 𝑛 ) ) O(m⋅polylog(n))-time algorithm for computing 𝑂 ( 𝑛 𝜀 − 1 ⋅ polylog ( 𝑛 ) ) O(nε −1 ⋅polylog(n))-edge graphical spectral sketches, along with a natural Eulerian generalization. Unlike prior approaches using short cycle or expander decompositions, their algorithms leverage a new effective resistance decomposition scheme, combined with natural sampling and electrical routing for degree balance. The analysis applies asymmetric variance bounds specialized to Eulerian Laplacians and tools from discrepancy theory.