The Matroid Secretary Conjecture is a notorious open problem in online optimization. It claims the existence of an $O(1)$-competitive algorithm for the Matroid Secretary Problem (MSP). Here, the elements of a weighted matroid appear one-by-one, revealing their weight at appearance, and the task is to select elements online with the goal to get an independent set of largest possible weight. $O(1)$-competitive MSP algorithms have so far only been obtained for restricted matroid classes and for MSP variations, including Random-Assignment MSP (RA-MSP), where an adversary fixes a number of weights equal to the ground set size of the matroid, which then get assigned randomly to the elements of the ground set. Unfortunately, these approaches heavily rely on knowing the full matroid upfront. This is an arguably undesirable requirement, and there are good reasons to believe that an approach towards resolving the MSP Conjecture should not rely on it. Thus, both Soto [SIAM Journal on Computing 2013] and Oveis Gharan & Vondrak [Algorithmica 2013] raised as an open question whether RA-MSP admits an $O(1)$-competitive algorithm even without knowing the matroid upfront. In this work, we answer this question affirmatively. Our result makes RA-MSP the first well-known MSP variant with an $O(1)$-competitive algorithm that does not need to know the underlying matroid upfront and without any restriction on the underlying matroid. Our approach is based on first approximately learning the rank-density curve of the matroid, which we then exploit algorithmically.

翻译：拟阵秘书猜想是在线优化领域一个著名的未解决问题。该猜想声称存在一种$O(1)$-竞争比的算法可解决拟阵秘书问题（MSP）：加权拟阵的元素逐一出现，并在出现时揭示其权重，目标是在线选择元素，使得所选元素构成独立集且总权重尽可能大。迄今为止，仅针对受限拟阵类及MSP变体获得了$O(1)$-竞争比的算法，其中包括随机分配MSP（RA-MSP）：对手预先固定与拟阵基集大小相等的权重集合，随后将这些权重随机分配给基集中的元素。遗憾的是，这些方法严重依赖预先获知整个拟阵的结构信息。这一要求显然不尽合理，且我们有充分理由相信，解决MSP猜想的可行方案不应依赖此类先验知识。因此，Soto [SIAM Journal on Computing 2013] 与 Oveis Gharan & Vondrak [Algorithmica 2013] 均提出开放性问题：RA-MSP能否在不预先获知拟阵的情况下实现$O(1)$-竞争比算法。本文对此问题给出肯定回答。该结果使RA-MSP成为首个已知的MSP变体，它在不预先获知底层拟阵且不对拟阵类型施加任何限制的条件下，即可实现$O(1)$-竞争比算法。我们的方法基于首先近似学习拟阵的秩密度曲线，进而将其应用于算法设计。