Deterministic Minimum Steiner Cut in Maximum Flow Time

We devise a deterministic algorithm for minimum Steiner cut, which uses (log n)^O(1) maximum flow calls and additional near-linear time. This algorithm improves on Li and Panigrahi's (FOCS 2020) algorithm, which uses (log n)^O(1/ϵ^4) maximum flow calls and additional O(m^1+ϵ) time, for ϵ > 0. Our algorithm thus shows that deterministic minimum Steiner cut can be solved in maximum flow time up to polylogarithmic factors, given any black-box deterministic maximum flow algorithm. Our main technical contribution is a novel deterministic graph decomposition method for terminal vertices that generalizes all existing s-strong partitioning methods, which we believe may have future applications.