Matroid Secretary Problems

We define a generalization of the classical secretary problem called the matroid secretary problem. In this problem, the elements of a matroid are presented to an online algorithm in uniformly random order. When an element arrives, the algorithm observes its value and must make an irrevocable decision whether or not to accept it. The accepted elements must form an independent set, and the objective is to maximize the combined value of these elements. We present an O(log k)-competitive algorithm for general matroids (where k is the rank of the matroid), and constant-competitive algorithms for several special cases including graphic matroids, truncated partition matroids, and bounded degree transversal matroids. We leave as an open question the existence of constant-competitive algorithms for general matroids. Our results have applications in welfare-maximizing online mechanism design for domains in which the sets of simultaneously satisfiable agents form a matroid.

[1]  Robert D. Kleinberg,et al.  Secretary Problems with Non-Uniform Arrival Order , 2015, STOC.

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

[3]  Mohammad Taghi Hajiaghayi,et al.  Adaptive limited-supply online auctions , 2004, EC '04.

[4]  Silvio Lattanzi,et al.  Hiring a secretary from a poset , 2011, EC '11.

[5]  Moran Feldman,et al.  The Submodular Secretary Problem Goes Linear , 2015, 2015 IEEE 56th Annual Symposium on Foundations of Computer Science.

[6]  Yoav Shoham,et al.  Truth revelation in approximately efficient combinatorial auctions , 2002, EC '99.

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

[8]  Ittai Abraham,et al.  Low-Distortion Inference of Latent Similarities from a Multiplex Social Network , 2012, SIAM J. Comput..

[9]  Oded Lachish,et al.  O(log log Rank) Competitive Ratio for the Matroid Secretary Problem , 2014, 2014 IEEE 55th Annual Symposium on Foundations of Computer Science.

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

[11]  Shai Vardi,et al.  The Returning Secretary , 2015, STACS.

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

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

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

[15]  Noam Nisan,et al.  Online ascending auctions for gradually expiring items , 2005, SODA '05.

[16]  Aviad Rubinstein,et al.  Beyond matroids: secretary problem and prophet inequality with general constraints , 2016, STOC.

[17]  Rajeev Motwani,et al.  Randomized Algorithms , 1995, SIGA.

[18]  Michael Dinitz,et al.  Matroid Secretary for Regular and Decomposable Matroids , 2012, SIAM J. Comput..

[19]  S. Matthew Weinberg,et al.  Matroid prophet inequalities , 2012, STOC '12.

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

[21]  Joseph Naor,et al.  Improved Competitive Ratios for Submodular Secretary Problems (Extended Abstract) , 2011, APPROX-RANDOM.

[22]  Moshe Tennenholtz,et al.  Interviewing secretaries in parallel , 2012, EC '12.

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

[24]  Tengyu Ma,et al.  The Simulated Greedy Algorithm for Several Submodular Matroid Secretary Problems , 2011, Theory of Computing Systems.

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

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

[27]  Michael Dinitz,et al.  Recent advances on the matroid secretary problem , 2013, SIGA.

[28]  Andreas Tönnis,et al.  Submodular Secretary Problems: Cardinality, Matching, and Linear Constraints , 2016, APPROX-RANDOM.

[29]  Alessandro Panconesi,et al.  Concentration of Measure for the Analysis of Randomized Algorithms , 2009 .

[30]  Ola Svensson,et al.  A Simple O(log log(rank))-Competitive Algorithm for the Matroid Secretary Problem , 2015, SODA.

[31]  Noam Nisan,et al.  Truthful approximation mechanisms for restricted combinatorial auctions , 2008, Games Econ. Behav..

[32]  David R. Karger,et al.  Random sampling and greedy sparsification for matroid optimization problems , 1998, Math. Program..

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

[34]  Mohit Singh,et al.  Secretary Problems via Linear Programming , 2010, IPCO.

[35]  Moshe Babaioff,et al.  Mechanism Design for Single-Value Domains , 2005, AAAI.

[36]  Berthold Vöcking,et al.  An Optimal Online Algorithm for Weighted Bipartite Matching and Extensions to Combinatorial Auctions , 2013, ESA.

[37]  Jan Vondrák,et al.  On Variants of the Matroid Secretary Problem , 2013, Algorithmica.

[38]  Nicole Immorlica,et al.  A Knapsack Secretary Problem with Applications , 2007, APPROX-RANDOM.

[39]  Nicole Immorlica,et al.  Secretary Problems with Competing Employers , 2006, WINE.

[40]  Shuchi Chawla,et al.  Secretary Problems with Convex Costs , 2011, ICALP.

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

[42]  Sahil Singla,et al.  Combinatorial Prophet Inequalities , 2016, SODA.