关于强拟阵秘书猜想及其推广
On the Strong Matroid Secretary Conjecture and Beyond
- University of Maryland, College Park（马里兰大学帕克分校）
机构由 AI 辅助整理，请以论文原文为准。
中文总结 AI 辅助
本文通过有限线性规划验证并证明了强拟阵秘书猜想对所有线性拟阵成立，给出1/e竞争算法，并推广了单样本先知算法至任意拟阵。
中文摘要 AI 辅助
强拟阵秘书猜想断言每个拟阵都承认一个1/e竞争的秘书算法，与经典的单选择保证相匹配。我们构造了一个有限线性规划，其值等于任意固定拟阵的最优序数竞争比；对于七个元素上所有正秩拟阵以及八个元素上几乎所有拟阵，该值超过1/e。相同的计算表明，最优比率在拟阵截断下是单调的；我们证明了这一点对于均匀拟阵成立，其中比率随秩严格增加，并反驳了一个图拟阵的情况。在这些证据的指导下，我们证明了对于每个线性拟阵（包括图拟阵、正则拟阵、层状拟阵和伽莫伊德）的猜想，给出了一个1/e竞争的序数秘书算法。该算法维护接受跨度与每个环境子空间的期望交集维数的界限。不交叉和分离表明，在以规定概率接纳每个当前贪心基元素的同时，这些界限可以保持，并且构造使用有限线性规划。对于每个拟阵，我们还给出了一个单样本先知算法，在任意固定到达顺序下具有1/2的竞争比，与样本和值无关。其输出（包括所选值）恰好具有从新乘积抽取中独立公平细化最优值的分布。该算法在n个元素上使用O(n^2)次独立性查询。两个常数在其各自模型中都是紧的。我们还给出了一个自包含的黑盒归约，将单样本先知比率α转换为秘书比率α^2/16，保持多项式运行时间。因此，我们的单样本算法为任意拟阵产生了一个1/64竞争的序数秘书算法。
英文摘要
The strong matroid secretary conjecture asserts that every matroid admits a $1/e$-competitive secretary algorithm, matching the classical single-choice guarantee. We formulate a finite linear program whose value is the optimal ordinal competitive ratio of any fixed matroid; for all matroids of positive rank on seven elements and nearly all on eight, this value exceeds $1/e$. The same computations suggested that the optimal ratio is monotone under truncation of the matroid; we prove this for uniform matroids, where the ratio is strictly increasing in the rank, and refute it for a graphic matroid. Guided by this evidence, we prove the conjecture for every linear matroid, a class that includes graphic matroids, regular matroids, laminar matroids, and gammoids, giving a $1/e$-competitive ordinal secretary algorithm. The algorithm maintains bounds on the expected intersection dimension of the accepted span with every ambient subspace. Uncrossing and separation show that these bounds can be preserved while admitting each current greedy-basis element with a prescribed probability and the construction uses finite linear programs. For every matroid, we also give a single-sample prophet algorithm with competitive ratio $1/2$ in any fixed arrival order independent of the samples and values. Its output, including the selected values, has exactly the law of an independent fair thinning of an optimum from a fresh product draw. The algorithm uses $O(n^2)$ independence queries on $n$ elements. Both constants are tight in their respective models. We also give a self-contained black-box reduction that converts a single-sample prophet ratio $α$ into a secretary ratio $α^2/16$, preserving polynomial running time. Our single-sample algorithm consequently yields a $1/64$-competitive ordinal secretary algorithm for arbitrary matroids.
