Strong Algorithms for the Ordinal Matroid Secretary Problem

José A. Soto (), Abner Turkieltaub () and Victor Verdugo ()
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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

Mathematics of Operations Research, 2021, vol. 46, issue 2, 642-673

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.

Keywords: Primary: 68W27, secondary: 68R05, Primary: Mathematics: combinatorics; secondary: decision analysis: theory, secretary problem, matroids, online algorithms (search for similar items in EconPapers)
Date: 2021
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Persistent link: https://EconPapers.repec.org/RePEc:inm:ormoor:v:46:y:2021:i:2:p:642-673

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