A matroid M on a set E of elements has the α-partition property, for some α > 0, if it is possible to (randomly) construct a partition matroid 𝒫 on (a subset of) elements of M such that every independent set of 𝒫 is independent in M and for any weight function w:E → ℝ_{≥0}, the expected value of the optimum of the matroid secretary problem on 𝒫 is at least an α-fraction of the optimum on M. We show that the complete binary matroid, B_d on 𝔽₂^d does not satisfy the α-partition property for any constant α > 0 (independent of d). Furthermore, we refute a recent conjecture of [Kristóf Bérczi et al., 2021] by showing the same matroid is 2^d/d-colorable but cannot be reduced to an α 2^d/d-colorable partition matroid for any α that is sublinear in d.
@InProceedings{abdolazimi_et_al:LIPIcs.ITCS.2023.2, author = {Abdolazimi, Dorna and Karlin, Anna R. and Klein, Nathan and Oveis Gharan, Shayan}, title = {{Matroid Partition Property and the Secretary Problem}}, booktitle = {14th Innovations in Theoretical Computer Science Conference (ITCS 2023)}, pages = {2:1--2:9}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-263-1}, ISSN = {1868-8969}, year = {2023}, volume = {251}, editor = {Tauman Kalai, Yael}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ITCS.2023.2}, URN = {urn:nbn:de:0030-drops-175051}, doi = {10.4230/LIPIcs.ITCS.2023.2}, annote = {Keywords: Online algorithms, Matroids, Matroid secretary problem} }
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