AI 中文总结
本研究针对未知图拟阵的拟阵秘书问题，开发了仅依赖独立性神谕、运行时间为多项式的算法，其生成独立集权重至少为最大权重独立集的1/36，是首个此类常数竞争比算法。
AI 中文摘要
拟阵秘书问题（MSP）是在线算法领域中最简洁且最具吸引力的开放问题之一。著名的MSP猜想指出存在一个具有常数竞争比的算法，但迄今为止已知的最优算法的竞争比为O(log log(秩))。人们普遍认为，MSP算法应使用的所有信息都可通过已到达元素的独立性神谕获取。尽管如此，对于一些自然类别的拟阵，若预先提供关于该拟阵的额外信息，就存在常数竞争比算法；而若算法仅能使用已到达元素的独立性神谕，则不存在此类算法。本研究针对其中最具吸引力的拟阵类别——图拟阵展开研究，开发了一个仅能访问独立性神谕的MSP算法。该算法运行时间为多项式时间，若基础拟阵为图拟阵，其生成的独立集权重至少为最大权重独立集权重的1/36。本研究是首个针对未知图拟阵的MSP的常数竞争比算法。
英文摘要
The matroid secretary problem (MSP) is one of the cleanest, and most captivating open problems in online algorithms. The famous MSP conjecture stipulates that there exists a constant-competitive algorithm, yet to date the best known algorithms are $O(\log \log (\text{rank}))$ competitive. It is widely believed that all information that an algorithm for the MSP should use is information that is available through an independence oracle on the already arrived elements. Despite this, there are natural classes of matroids where a constant-competitive algorithm is known if we are given additional upfront information about the matroid; while no such algorithm is known if all the algorithm can use is an independence oracle on the arrived elements. In this work, we tackle the perhaps most appealing such class of matroids, graphic matroids. We develop an algorithm for the MSP that has access to the independence oracle only. Our algorithm runs in polynomial time, and if the underlying matroid is graphic, it produces an independent set whose weight is at least $1/36$ of the maximum-weight independent set. Ours is the first constant-competitive algorithm for MSP on unknown graphic matroids.
