Deterministic Minimum Steiner Cut in Maximum Flow Time

Abstract: We devise a deterministic algorithm for minimum Steiner cut, which uses $(\log n)<sup>{O(1)}$ maximum flow calls and additional near-linear time. This algorithm improves on Li and Panigrahi's (FOCS 2020) algorithm, which uses $(\log n)<sup>{O(1/\epsilon<sup>4)}$ maximum flow calls and additional $O(m<sup>{1+\epsilon})$ time, for $\epsilon &gt; 0$. 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.