Problem
Research questions and friction points this paper is trying to address.
Matroid Secretary Problem
Online Algorithm
Competitive Ratio
A 3.7321-Competitive Algorithm for Matroid Secretary
🤖 AI Summary
该研究针对拟阵秘书问题，提出了一种竞争比为3.7321的在线算法，通过保留随机样本部分并平衡剩余元素，改进了Singla的4-竞争算法。
📝 Abstract
The matroid secretary problem asks an online algorithm to select a high-weight independent set from elements arriving in uniformly random order, with immediate and irrevocable decisions. Singla (2026) recently gave a $4$-competitive algorithm for arbitrary matroids using only the number of elements and independence queries on already-arrived elements. Following his approach, we obtain an improved competitive ratio of $2+\sqrt3\approx3.7321$ in the same information model. Our algorithm accepts every element of a fixed canonical optimum with probability at least $2-\sqrt3$ and uses $O(n^2)$ independence queries. The algorithm modifies Singla's reversible reference process by retaining a randomly chosen part of the sample as a reserve whose membership in the reference greedy solution is not frozen. Balancing the remaining sample and post-sample elements preserves reversibility and allows an exact calculation of the probability that an exchange partner blocks a target element. The resulting guarantee has a direct analytic proof.
Innovation
Methods, ideas, or system contributions that make the work stand out.
competitive ratio
matroid secretary problem
independence queries
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2024-07-14arXiv.orgCitations: 0
