The secretary problem on an unknown poset. Issue 4 (5th November 2012)

• Record Type: Journal Article
• Authors: Garrod, Bryn; Morris, Robert
• Is Part Of: Random structures & algorithms. Volume 43:Issue 4(2013)
• Page Start: 429
• Page End: 451
• Publication Date: 2012-11-05
• Subjects: Random graphs -- Periodicals; Mathematical analysis -- Periodicals; 519
• DOI: 10.1002/rsa.20466
• Languages: English
• ISSNs: 1042-9832

Abstract

<abstract abstract-type="main" xml:lang="en"> <title>Abstract</title> <p>We consider generalizations of the classical secretary problem, also known as the problem of optimal choice, to posets where the only information we have is the size of the poset and the number of maximal elements. We show that, given this information, there is an algorithm that is successful with probability at least <tex-math notation="LaTeX"><![CDATA[\documentclass{article} \usepackage{mathrsfs} \usepackage{amsmath} \pagestyle{empty} \begin{document} \begin{align*}\frac{1}{e}\end{align*} \end{document}]]></tex-math><inline-graphic xlink:href="ark:/27927/pgg3qnzz5b1" mimetype="image" xlink:type="simple" xmlns:xlink="http://www.w3.org/1999/xlink" />. We conjecture that if there are <italic>k</italic> maximal elements and <italic>k</italic> ≥ 2 then this can be improved to <tex-math notation="LaTeX"><![CDATA[\documentclass{article} \usepackage{mathrsfs} \usepackage{amsmath} \pagestyle{empty} \begin{document} \begin{align*}\sqrt\lbrack k-1\rbrack {\frac{1}{k}}\end{align*} \end{document}]]></tex-math><inline-graphic xlink:href="ark:/27927/pgg3qnzz57c" mimetype="image" xlink:type="simple" xmlns:xlink="http://www.w3.org/1999/xlink" />, and prove this conjecture for posets of width <italic>k</italic>. We also show that no better bound is possible. © 2012 Wiley Periodicals, Inc. Random Struct. Alg., 43, 429–451, 2013</p> </abstract>