← 研踪 PaperTrailarXiv:2611.05021 · v1
Nearly-Linear Query Algorithm for Submodular Maximization subject to a Matroid Constraint
首次提交 2026-11-03 23:00
版本更新 2026-11-03 23:00
雷达观测 2026-11-04 23:39
摘要
We consider monotone submodular maximization subject to a matroid constraint. The seminal continuous greedy algorithm of Calinescu, Chekuri, Pál, and Vondrák~[SICOMP'11] achieves the tight approximation of $(1-1/e)$. Since then, a line of works by Badanidiyuru and Vondrák~[SODA'14], Buchbinder, Feldman, and Schwartz~[SODA'15 \& MOR'17], Kobayashi and Terao~[ICALP'24], Buchbinder and Feldman~[FOCS'24 \& SICOMP'25], and Buchbinder and Feldman~[STOC'25] has developed efficient $(1-1/e-\varepsilon)$-approximation algorithms for this problem. We present a combinatorial randomized $(1-1/e-\varepsilon)$-approximation algorithm that requires $\tilde{O}_{\varepsilon}(n)$ independence-oracle and value-oracle queries, where $n$ is the number of elements in the matroid. This is the first $(1-1/e-\varepsilon)$-approximation algorithm with both independence-oracle and value-oracle query complexities nearly-linear in $n$ and polynomial in $1/\varepsilon$. To achieve this, we develop a new local search algorithm, which we call \emph{protected local search}, based on the potential function of Buchbinder and Feldman~[FOCS'24 \& SICOMP'25].
作者备注：21 pages
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v1 · 2026-11-03 23:00 · 最新观测版本
Nearly-Linear Query Algorithm for Submodular Maximization subject to a Matroid Constraint
We consider monotone submodular maximization subject to a matroid constraint. The seminal continuous greedy algorithm of Calinescu, Chekuri, Pál, and Vondrák~[SICOMP'11] achieves the tight approximation of $(1-1/e)$. Since then, a line of works by Badanidiyuru and Vondrák~[SODA'14], Buchbinder, Feldman, and Schwartz~[SODA'15 \& MOR'17], Kobayashi and Terao~[ICALP'24], Buchbinder and Feldman~[FOCS'24 \& SICOMP'25], and Buchbinder and Feldman~[STOC'25] has developed efficient $(1-1/e-\varepsilon)$-approximation algorithms for this problem. We present a combinatorial randomized $(1-1/e-\varepsilon)$-approximation algorithm that requires $\tilde{O}_{\varepsilon}(n)$ independence-oracle and value-oracle queries, where $n$ is the number of elements in the matroid. This is the first $(1-1/e-\varepsilon)$-approximation algorithm with both independence-oracle and value-oracle query complexities nearly-linear in $n$ and polynomial in $1/\varepsilon$. To achieve this, we develop a new local search algorithm, which we call \emph{protected local search}, based on the potential function of Buchbinder and Feldman~[FOCS'24 \& SICOMP'25].
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