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Strong Algorithms for the Ordinal Matroid Secretary Problem

Author

Listed:
  • José A. Soto

    (Department of Mathematical Engineering and CMM, Universidad de Chile. UMI 2807 CNRS, Santiago, Chile)

  • Abner Turkieltaub

    (Department of Computer Science, University of British Columbia, Vancouver, Canada)

  • Victor Verdugo

    (Institute of Engineering Sciences, Universidad de O’Higgins, Rancagua, Chile)

Abstract

In the ordinal matroid secretary problem (MSP), candidates do not reveal numerical weights, but the decision maker can still discern if a candidate is better than another. An algorithm α is probability-competitive if every element from the optimum appears with probability 1 / α in the output. This measure is stronger than the standard utility competitiveness. Our main result is the introduction of a technique based on forbidden sets to design algorithms with strong probability-competitive ratios on many matroid classes. We improve upon the guarantees for almost every matroid class considered in the MSP literature. In particular, we achieve probability-competitive ratios of 4 for graphic matroids and of 3 3 ≈ 5.19 for laminar matroids. Additionally, we modify Kleinberg’s 1 + O ( 1 / ρ ) utility-competitive algorithm for uniform matroids of rank ρ in order to obtain a 1 + O ( log ρ / ρ ) probability-competitive algorithm. We also contribute algorithms for the ordinal MSP on arbitrary matroids.

Suggested Citation

  • José A. Soto & Abner Turkieltaub & Victor Verdugo, 2021. "Strong Algorithms for the Ordinal Matroid Secretary Problem," Mathematics of Operations Research, INFORMS, vol. 46(2), pages 642-673, May.
    • Handle: RePEc:inm:ormoor:v:46:y:2021:i:2:p:642-673
    DOI: 10.1287/moor.2020.1083
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    References listed on IDEAS

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    1. D. V. Lindley, 1961. "Dynamic Programming and Decision Theory," Journal of the Royal Statistical Society Series C, Royal Statistical Society, vol. 10(1), pages 39-51, March.
    2. Niv Buchbinder & Kamal Jain & Mohit Singh, 2014. "Secretary Problems via Linear Programming," Mathematics of Operations Research, INFORMS, vol. 39(1), pages 190-206, February.
    3. Moran Feldman & Ola Svensson & Rico Zenklusen, 2018. "A Simple O (log log(rank))-Competitive Algorithm for the Matroid Secretary Problem," Mathematics of Operations Research, INFORMS, vol. 43(2), pages 638-650, May.
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