Junkai Song

Sanjeev Khanna, Aaron Putterman, Junkai Song

We study the parallel (adaptive) complexity of the classic problem of finding a basis in an $n$-element matroid, given access via an \emph{independence oracle}. In this model, the algorithm may submit polynomially many independence queries in each round, and the central question is: how many rounds are necessary and sufficient to find a basis? Karp, Upfal, and Wigderson (FOCS~1985, JCSS~1988; hereafter KUW) initiated this study, showing that $O(\sqrt{n})$ adaptive rounds suffice for any matroid, and that $\widetildeΩ(n^{1/3})$ rounds are necessary even for partition matroids. This left a substantial gap that persisted for nearly four decades, until Khanna, Putterman, and Song (FOCS~2025; hereafter KPS) achieved $\widetilde O(n^{7/15})$ rounds, the first improvement since~KUW. In this work, we make another conceptual advance beyond KPS, giving a new algorithm that finds a matroid basis in $\widetilde O(n^{3/7})$ rounds. We develop a structural and algorithmic framework that brings a new lens to the analysis of random circuits, moving from reasoning about individual elements to understanding how dependencies span multiple elements simultaneously.

Julia Chuzhoy, Sanjeev Khanna, Junkai Song

In the fully dynamic maximal matching problem, the goal is to maintain a maximal matching in a graph undergoing an online sequence of edge insertions and deletions. The problem has been studied extensively in the oblivious-adversary setting, where randomized algorithms with polylogarithmic worst-case and constant amortized update time have been known for some time. A major challenge in this area has been designing an algorithm with non-trivial update time against an adaptive adversary. In a recent breakthrough, Bernstein, Bhattacharya, Kiss, and Saranurak (STOC 2025; hereafter, BBKS25) obtained the first algorithms with sublinear update time for this setting: namely, a randomized algorithm with $\tilde{O}(n^{3/4})$ amortized update time, and a deterministic algorithm with $\tilde{O}(n^{8/9})$ amortized update time. Our main result is a deterministic algorithm for fully dynamic maximal matching with amortized update time $n^{1/2+o(1)}$. A powerful tool in dynamic matching is the use of matching sparsifiers: sparse subgraphs that preserve enough information to recover matchings with desired properties. Sparsifiers, such as the EDCS data structure, have been successfully used for approximate maximum matching. For maximal matching, however, this paradigm is not as natural, since maximality must hold with respect to the entire graph. Nevertheless, BBKS25 showed that EDCS can be repurposed as a verification-and-repair mechanism for fully dynamic maximal matching against adaptive adversaries. We introduce a new deterministic framework, referred to as the subgraph system, which, in contrast to EDCS, is purpose-built for verification and maintenance of maximality. It is also designed to allow efficient recursive refinements leading to stronger and stronger parameters, that yield our deterministic algorithm with $n^{1/2+o(1)}$ amortized update time.