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  "path": "/abs/2605.03979v1",
  "publishedAt": "2026-05-06T00:00:00.000Z",
  "site": "https://arxiv.org",
  "tags": [
    "Sanjeev Khanna",
    "Aaron Putterman",
    "Junkai Song"
  ],
  "textContent": "**Authors:** Sanjeev Khanna, Aaron Putterman, Junkai Song\n\nWe 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.",
  "title": "An $\\widetilde{O} (n^{3/7})$ Round Parallel Algorithm for Matroid Bases"
}