eng
Schloss Dagstuhl – Leibniz-Zentrum für Informatik
Leibniz International Proceedings in Informatics
1868-8969
2023-02-01
2:1
2:9
10.4230/LIPIcs.ITCS.2023.2
article
Matroid Partition Property and the Secretary Problem
Abdolazimi, Dorna
1
Karlin, Anna R.
1
Klein, Nathan
1
Oveis Gharan, Shayan
1
University of Washington, Seattle, WA, USA
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.
https://drops.dagstuhl.de/storage/00lipics/lipics-vol251-itcs2023/LIPIcs.ITCS.2023.2/LIPIcs.ITCS.2023.2.pdf
Online algorithms
Matroids
Matroid secretary problem