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Matroid Partition Property and the Secretary Problem

Authors Dorna Abdolazimi, Anna R. Karlin, Nathan Klein, Shayan Oveis Gharan

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ACM Subject Classification
  • Theory of computation → Online algorithms
Keywords
  • Online algorithms
  • Matroids
  • Matroid secretary problem

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Abstract

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.

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Dorna Abdolazimi, Anna R. Karlin, Nathan Klein, and Shayan Oveis Gharan. Matroid Partition Property and the Secretary Problem. In 14th Innovations in Theoretical Computer Science Conference (ITCS 2023). Leibniz International Proceedings in Informatics (LIPIcs), Volume 251, pp. 2:1-2:9, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2023) https://doi.org/10.4230/LIPIcs.ITCS.2023.2

Author Details

Dorna Abdolazimi
  • University of Washington, Seattle, WA, USA
Anna R. Karlin
  • University of Washington, Seattle, WA, USA
Nathan Klein
  • University of Washington, Seattle, WA, USA
Shayan Oveis Gharan
  • University of Washington, Seattle, WA, USA

Funding

  • Abdolazimi, Dorna: Research supported by NSF grant CCF-1907845 and Air Force Office of Scientific Research grant FA9550-20-1-0212.
  • Karlin, Anna R.: Research supported by Air Force Office of Scientific Research grant FA9550-20-1-0212 and NSF grant CCF-1813135.
  • Klein, Nathan: Research supported in part by NSF grants DGE-1762114, CCF-1813135, and CCF-1552097.
  • Oveis Gharan, Shayan: Research supported by Air Force Office of Scientific Research grant FA9550-20-1-0212, NSF grant CCF-1907845, and a Sloan fellowship.

References

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