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arXiv · 2609.17782

A 3.1462-Competitive Algorithm for Matroid Secretary

Hau Chan·Jianan Lin·Chenhao Wang
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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 of maintaining a dynamically updated reference set, we introduce random deletions and a time-dependent acceptance rule, improving the competitive ratio to $C_*\approx3.1462$ in the same information model, where $C_*-\log C_*=2$. Our ordinal algorithm accepts every element of a fixed canonical optimum with probability exactly $1/C_*$ and uses $O(n^2)$ independence queries. The algorithm maintains a greedy reference solution, protects only accepted elements, and randomly deletes unaccepted reference-basis elements. A time-dependent acceptance rule makes the accepted set, conditional on the reference set, an independent thinning of its greedy basis. The resulting guarantee has a direct analytic proof.

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Hau Chan, Jianan Lin, Chenhao Wang. 2026-09-20. A 3.1462-Competitive Algorithm for Matroid Secretary. https://arxiv.org/abs/2609.17782

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