The secretary problem became one of the most prominent online selection problems due to its numerous applications in online mechanism design. The task is to select a maximum weight subset of elements subject to given constraints, where elements arrive one-by-one in random order, revealing a weight upon arrival. The decision whether to select an element has to be taken immediately after its arrival. The different applications that map to the secretary problem ask for different constraint families to be handled. The most prominent ones are matroid constraints, which both capture many relevant settings and admit strongly competitive secretary algorithms. However, dealing with more involved constraints proved to be much more difficult, and strong algorithms are known only for a few specific settings. In this paper, we present a general framework for dealing with the secretary problem over the intersection of several matroids. This framework allows us to combine and exploit the large set of matroid secretary algorithms known in the literature. As one consequence, we get constant-competitive secretary algorithms over the intersection of any constant number of matroids whose corresponding (single-)matroid secretary problems are currently known to have a constant-competitive algorithm. Moreover, we show that our results extend to submodular objectives.
|Number of pages||54|
|Journal||SIAM Journal on Computing|
|State||Published - 2022|
Bibliographical noteFunding Information:
\ast Received by the editors April 12, 2021; accepted for publication (in revised form) January 18, 2022; published electronically June 14, 2022. An extended abstract version of this work appeared in Proceedings of SODA, 2018. https://doi.org/10.1137/21M1411822 \bfF \bfu \bfn \bfd \bfi \bfn \bfg : The first author was supported by Israel Science Foundation grants 1357/16 and 459/20. The second author was supported by ERC Starting Grant 335288-OptApprox and by Swiss National Science Foundation project 200021-184656, ``Randomness in Problem Instances and Randomized Algorithms."" The third author received funding from Swiss National Science Foundation grant 200021 165866 and the European Research Council (ERC) under the European Union's Horizon 2020 research and innovation program (grant 817750) . \dagger Department of Mathematics and Computer Science, Open University of Israel. Current address: Department of Computer Science, University of Haifa, Israel (firstname.lastname@example.org). \ddagger School of Computer and Communication Sciences, EPFL, Lausanne, Switzerland (ola.svensson@ epfl.ch). \S Department of Mathematics, ETH Zurich, Zurich, Switzerland (email@example.com).
The first author was supported by Israel Science Foundation grants 1357/16 and 459/20. The second author was supported by ERC Starting Grant 335288-OptApprox and by Swiss National Science Foundation project 200021-184656, "Randomness in Problem Instances and Randomized Algorithms." The third author received funding from Swiss National Science Foundation grant 200021 165866 and the European Research Council (ERC) under the European Union's Horizon
© 2022 Moran Feldman, Ola Svensson, and Rico Zenklusen.
- matroid intersection
- matroid secretary problem
- online algorithms