Recent advances on the matroid secretary problem

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.

[1]  Morteza Zadimoghaddam,et al.  Submodular secretary problem and extensions , 2013, TALG.

[2]  Michael Dinitz,et al.  Secretary problems: weights and discounts , 2009, SODA.

[3]  Robert D. Kleinberg.  A multiple-choice secretary algorithm with applications to online auctions , 2005, SODA '05.

[4]  José A. Soto,et al.  Matroid secretary problem in the random assignment model , 2010, SODA '11.

[5]  Tengyu Ma,et al.  The Simulated Greedy Algorithm for Several Submodular Matroid Secretary Problems , 2013, STACS.

[6]  Martin Pál,et al.  Algorithms for Secretary Problems on Graphs and Hypergraphs , 2008, ICALP.

[7]  C. Greg Plaxton,et al.  Competitive Weighted Matching in Transversal Matroids , 2008, ICALP.

[8]  Paul D. Seymour,et al.  Decomposition of regular matroids , 1980, J. Comb. Theory, Ser. B.

[9]  Thomas S. Ferguson,et al.  Who Solved the Secretary Problem , 1989 .

[10]  Frederick Mosteller,et al.  Fifty Challenging Problems in Probability with Solutions , 1987 .

[11]  Sourav Chakraborty,et al.  Improved competitive ratio for the matroid secretary problem , 2012, SODA.

[12]  Aaron Roth,et al.  Constrained Non-monotone Submodular Maximization: Offline and Secretary Algorithms , 2010, WINE.

[13]  Nicole Immorlica,et al.  Matroids, secretary problems, and online mechanisms , 2007, SODA '07.

[14]  R. Graham,et al.  Handbook of Combinatorics , 1995 .

[15]  J. Oxley.  Matroid Theory (Oxford Graduate Texts in Mathematics) , 2006 .

[16]  Roy Schwartz,et al.  Improved competitive ratios for submodular secretary problems , 2011 .

[17]  Nicole Immorlica,et al.  Online auctions and generalized secretary problems , 2008, SECO.

[18]  Jack Edmonds,et al.  Submodular Functions, Matroids, and Certain Polyhedra , 2001, Combinatorial Optimization.

[19]  Patrick Jaillet,et al.  Advances on Matroid Secretary Problems: Free Order Model and Laminar Case , 2012, IPCO.

[20]  Jan Vondrák,et al.  On Variants of the Matroid Secretary Problem , 2011, ESA.

[21]  P. D. Seymour,et al.  Matroid minors , 1996 .

[22]  Yajun Wang,et al.  Secretary problems: laminar matroid and interval scheduling , 2011, SODA '11.

[23]  P. Freeman.  The Secretary Problem and its Extensions: A Review , 1983 .

[24]  H. Whitney.  On the Abstract Properties of Linear Dependence , 1935 .