We introduce the notion of an online matroid embedding, which is an algorithm for mapping an unknown matroid that is revealed in an online fashion to a larger-but-known matroid. We establish the existence of such an embedding for binary matroids, and use it to relate variants of the binary matroid secretary problem to each other, showing that seemingly simpler problems are in fact equivalent to seemingly harder ones (up to constant-factors). Specifically, we show this to be the case for the version of the matroid secretary problem in which the matroid is not known in advance, and where it is known in advance. We also show that the version with known matroid structure, is equivalent to the problem where weights are not fully adversarial but drawn from a known pairwise-independent distribution.

翻译：本文引入了在线拟阵嵌入的概念，这是一种将以前馈方式揭示的未知拟阵映射到更大但已知拟阵的算法。我们证明了二进制拟阵在线嵌入的存在性，并利用它将二进制拟阵秘书问题的不同变体相互关联，表明看似更简单的问题实际上等价于看似更难的问题（相差常数因子）。具体而言，我们证明了对于拟阵结构预先未知以及拟阵结构预先已知的秘书问题版本，上述结论均成立。我们还证明，已知拟阵结构的版本等价于权重并非完全对抗性而是从已知的成对独立分布中抽取的问题。