Babaioff et al. [BIK2007] introduced the matroid secretary problem in 2007, a natural extension of the classic single-choice secretary problem to matroids, and conjectured that a constant-competitive online algorithm exists. The conjecture still remains open despite substantial partial progress, including constant-competitive algorithms for numerous special cases of matroids, and an $O(\log \log \text{rank})$-competitive algorithm in the general case. Many of these algorithms follow principled frameworks. The limits of these frameworks are previously unstudied, and prior work establishes only that a handful of particular algorithms cannot resolve the matroid secretary conjecture. We initiate the study of impossibility results for frameworks to resolve this conjecture. We establish impossibility results for a natural class of greedy algorithms and for randomized partition algorithms, both of which contain known algorithms that resolve special cases.
翻译:Babaioff等人[BIK2007]在2007年引入了机器人秘书问题,这是典型的单选秘书问题自然延伸至机器人的延伸,并推测存在着一个持续竞争的在线算法。 尽管出现了实质性的部分进展,这种推测仍然没有定论,包括许多机器人特殊案例的不断竞争算法和一般案例中的美元(log\log\log\ text{rank})美元竞争算法。许多这些算法遵循了原则性框架。这些算法的局限性以前是未经研究的,而先前的工作仅确定少数特定的算法无法解决机器人秘书的推测。我们开始研究解决这一推测的框架不可能的结果。我们为自然的贪婪算法和随机分割算法确定了不可能的结果,这两种算法都包含解决特殊案例的已知算法。