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Recent advances on the matroid secretary problem

Published: 03 June 2013 Publication History

Abstract

The matroid secretary problem was introduced by Babaioff, Immorlica, and Kleinberg in SODA 2007 as an online problem that was both mathematically interesting and had applications to online auctions. In this column I will introduce and motivate this problem, and give a survey of some of the exciting work that has been done on it over the past 6 years. While we have a much better understanding of matroid secretary now than we did in 2007, the main conjecture is still open: does there exist an O(1)-competitive algorithm.

References

[1]
Moshe Babaioff, Michael Dinitz, Anupam Gupta, Nicole Immorlica, and Kunal Talwar. Secretary problems: weights and discounts. In Proc. SODA, pages 1245--1254, 2009.
[2]
Mohammad Hossein Bateni, MohammadTaghi Hajiaghayi, and Morteza Zadimoghaddam. Submodular secretary problem and extensions. In Proc. APPROX/RANDOM, pages 39--52, 2010.
[3]
Moshe Babaioff, Nicole Immorlica, and Robert Kleinberg. Matroids, secretary problems, and online mechanisms. In Proc. SODA, pages 434--443, 2007.
[4]
Moshe Babaioff, Nicole Immorlica, David Kempe, and Robert Kleinberg. Online auctions and generalized secretary problems. SIGecom Exch., 7(2):7:1--7:11, June 2008.
[5]
Sourav Chakraborty and Oded Lachish. Improved competitive ratio for the matroid secretary problem. In Proc. SODA, pages 1702--1712, 2012.
[6]
Michael Dinitz and Guy Kortsarz. Matroid secretary for regular and decomposable matroids. In Proc. SODA, 2013.
[7]
Nedialko B. Dimitrov and C. Greg Plaxton. Competitive weighted matching in transversal matroids. In Proc. ICALP, pages 397--408, 2008.
[8]
E. B. Dynkin. The optimum choice of the instant for stopping a markov process. Soviet Math. Dokl, 4, 1963.
[9]
Jack Edmonds. Submodular functions, matroids, and certain polyhedra. In Combinatorial Structures and their Applications (Proc. Calgary Internat. Conf.), pages 69--87. Gordon and Breach, 1970.
[10]
Thomas S. Ferguson. Who solved the secretary problem? Statistical Science, 4(3):pp. 282--289, 1989.
[11]
Moran Feldman, Joseph Seffi Naor, and Roy Schwartz. Improved competitive ratios for submodular secretary problems. In Proc. APPROX/RANDOM, pages 218--229, 2011.
[12]
Freeman. The secretary problem and its extensions: A review. International Statistical Review, 51(2):pp. 189--206, 1983.
[13]
Anupam Gupta, Aaron Roth, Grant Schoenebeck, and Kunal Talwar. Constrained nonmonotone submodular maximization: offline and secretary algorithms. In Proc. WINE, pages 246--257, 2010.
[14]
Shayan Oveis Gharan and Jan Vondrák. On variants of the matroid secretary problem. In Proc. ESA, pages 335--346, 2011.
[15]
Sungjin Im and YajunWang. Secretary problems: laminar matroid and interval scheduling. In Proc. SODA, pages 1265--1274, 2011.
[16]
Patrick Jaillet, José Soto, and Rico Zenklusen. Advances on matroid secretary problems: Free order model and laminar case. In Proc. IPCO, pages 254--265, 2013.
[17]
Robert Kleinberg. A multiple-choice secretary algorithm with applications to online auctions. In Proc. SODA, pages 630--631, 2005.
[18]
Nitish Korula and Martin Pál. Algorithms for secretary problems on graphs and hypergraphs. In Proc. ICALP, pages 508--520, 2009.
[19]
Frederick Mosteller. Fifty Challenging Problems In Probability With Solutions. Dover Publications, 1987.
[20]
Tengyu Ma, Bo Tang, and Yajun Wang. The Simulated Greedy Algorithm for Several Submodular Matroid Secretary Problems. In Proc. STACS, pages 478--489, 2013.
[21]
J.G. Oxley. Matroid theory. Oxford Graduate Texts in Mathematics Series. Oxford University Press, Incorporated, 1992.
[22]
P. D. Seymour. Decomposition of regular matroids. J. Combin. Theory Ser. B, 28(3):305--359, 1980.
[23]
P. D. Seymour. Matroid minors. In R. L. Graham, M. Grötschel, and L. Lovász, editors, Handbook of combinatorics (vol. 1), pages 527--550. MIT Press, Cambridge, MA, USA, 1995.
[24]
José A. Soto. Matroid secretary problem in the random assignment model. In Proc. SODA, pages 1275--1284, 2011.
[25]
Hassler Whitney. On the abstract properties of linear dependence. American Journal of Mathematics, 57(3):pp. 509--533, 1935.

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Published In

cover image ACM SIGACT News
ACM SIGACT News  Volume 44, Issue 2
June 2013
142 pages
ISSN:0163-5700
DOI:10.1145/2491533
Issue’s Table of Contents

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Association for Computing Machinery

New York, NY, United States

Publication History

Published: 03 June 2013
Published in SIGACT Volume 44, Issue 2

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