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Algorithms for Secretary Problems on Graphs and Hypergraphs

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Automata, Languages and Programming (ICALP 2009)

Part of the book series: Lecture Notes in Computer Science ((LNTCS,volume 5556))

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Abstract

We examine online matching problems with applications to Internet advertising reservation systems. Consider an edge-weighted bipartite graph G(L ∪ R, E). We develop an 8-competitive algorithm for the following secretary problem: Initially given R, and the size of L, the algorithm receives the vertices of L sequentially, in a random order. When a vertex l ∈ L is seen, all edges incident to l are revealed, together with their weights. The algorithm must immediately either match l to an available vertex of R, or decide that l will remain unmatched.

In [5], the authors show a 16-competitive algorithm for the transversal matroid secretary problem, which is the special case with weights on vertices, not edges. (Equivalently, one may assume that for each l ∈ L, the weights on all edges incident to l are identical.) We use a very similar algorithm, but simplify and improve the analysis to obtain a better competitive ratio for the more general problem. Our analysis is easily extended to obtain competitive algorithms for a class of similar problems, such as to find disjoint sets of edges in hypergraphs where edges arrive online. We also introduce secretary problems with adversarially chosen groups.

Finally, we give a 2e-competitive algorithm for the secretary problem on graphic matroids, where, with edges appearing online, the goal is to find a maximum-weight acyclic subgraph of a given graph.

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Authors and Affiliations

  1. Dept. of Computer Science, University of Illinois, Urbana, IL, 61801, USA

    Nitish Korula

  2. Google Inc., 76 9th Avenue, New York, NY, 10011, USA

    Martin Pál

Authors
  1. Nitish Korula
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  2. Martin Pál
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Editor information

Editors and Affiliations

  1. Department of Computer Science, University of Freiburg, Georges Köhler Allee 79, 79110, Freiburg, Germany

    Susanne Albers

  2. Department of Computer and Systems Sciences, Sapienza University of Rome, Via Ariosto 25, 00184, Roma, Italy

    Alberto Marchetti-Spaccamela

  3. Tel Aviv University Google R&D Center, School of Computer Science, Tel Aviv University, 69978, Tel Aviv, Israel

    Yossi Matias

  4. University of Patras and CTI, N. Kazantzaki Street 1, 26504, Rion, Patras, Greece

    Sotiris Nikoletseas

  5. RWTH Aachen, Lehrstuhl Informatik 7, Ahornstraße 55, 52056, Aachen, Germany

    Wolfgang Thomas

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© 2009 Springer-Verlag Berlin Heidelberg

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Korula, N., Pál, M. (2009). Algorithms for Secretary Problems on Graphs and Hypergraphs. In: Albers, S., Marchetti-Spaccamela, A., Matias, Y., Nikoletseas, S., Thomas, W. (eds) Automata, Languages and Programming. ICALP 2009. Lecture Notes in Computer Science, vol 5556. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-02930-1_42

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  • DOI: https://doi.org/10.1007/978-3-642-02930-1_42

  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-642-02929-5

  • Online ISBN: 978-3-642-02930-1

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