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arXiv · 2111.04114

Formal Barriers to Simple Algorithms for the Matroid Secretary Problem

Maryam Bahrani·Hedyeh Beyhaghi·Sahil Singla·S. Matthew Weinberg
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Abstract

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.

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Maryam Bahrani, Hedyeh Beyhaghi, Sahil Singla, S. Matthew Weinberg. 2021-11-07. Formal Barriers to Simple Algorithms for the Matroid Secretary Problem. https://arxiv.org/abs/2111.04114

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