Packing Returning Secretaries
We study online secretary problems with returns in combinatorial packing domains with n candidates that arrive sequentially over time in random order. The goal is to accept a feasible packing of candidates of maximum total value. In the first variant, each candidate arrives exactly twice. All 2n arrivals occur in random order. We propose a simple 0.5-competitive algorithm that can be combined with arbitrary approximation algorithms for the packing domain, even when the total value of candidates is a subadditive function. For bipartite matching, we obtain an algorithm with competitive ratio at least 0.5721 - o(1) for growing n, and an algorithm with ratio at least 0.5459 for all n >= 1. We extend all algorithms and ratios to k >= 2 arrivals per candidate.
In the second variant, there is a pool of undecided candidates. In each round, a random candidate from the pool arrives. Upon arrival a candidate can be either decided (accept/reject) or postponed (returned into the pool). We mainly focus on minimizing the expected number of postponements when computing an optimal solution. An expected number of Theta(n log n) is always sufficient. For matroids, we show that the expected number can be reduced to O(r log (n/r)), where r <=n/2 is the minimum of the ranks of matroid and dual matroid. For bipartite matching, we show a bound of O(r log n), where r is the size of the optimum matching. For general packing, we show a lower bound of Omega(n log log n), even when the size of the optimum is r = Theta(log n).
Secretary Problem
Coupon Collector Problem
Matroids
Theory of computation~Online algorithms
65:1-65:12
Regular Paper
Supported by a grant from the German-Israeli Foundation for Scientific Research and Development (GIF).
https://arxiv.org/abs/1810.11216
Martin
Hoefer
Martin Hoefer
Goethe University Frankfurt/Main, Germany
Lisa
Wilhelmi
Lisa Wilhelmi
Goethe University Frankfurt/Main, Germany
10.4230/LIPIcs.ISAAC.2018.65
Moshe Babaioff, Nicole Immorlica, David Kempe, and Robert Kleinberg. A Knapsack Secretary Problem with Applications. In Proc. 10th Workshop Approximation Algorithms for Combinatorial Optimization Problems (APPROX), pages 16-28, 2007.
Moshe Babaioff, Nicole Immorlica, David Kempe, and Robert Kleinberg. Online Auctions and Generalized Secretary Problems. SIGecom Exchanges, 7(2), 2008.
Moshe Babaioff, Nicole Immorlica, and Robert Kleinberg. Matroids, secretary problems, and online mechanisms. In Proc. 18th Symp. Discrete Algorithms (SODA), pages 434-443, 2007.
Arnon Boneh and Micha Hofri. The coupon-collector problem revisited - a survey of engineering problems and computational methods. Comm. Stat. Stoch. Models, 13(1):39-66, 1997.
Ning Chen, Martin Hoefer, Marvin Künnemann, Chengyu Lin, and Peihan Miao. Secretary Markets with Local Information. In Proc. 42nd Intl. Coll. Automata, Languages &Programming (ICALP), volume 2, pages 552-563, 2015.
Michael Dinitz. Recent advances on the matroid secretary problem. SIGACT News, 44(2):126-142, 2013.
Uriel Feige. On Maximizing Welfare When Utility Functions Are Subadditive. SIAM J. Comput., 39(1):122-142, 2009.
Moran Feldman, Ola Svensson, and Rico Zenklusen. A Simple O(log log(rank))-Competitive Algorithm for the Matroid Secretary Problem. Math. Oper. Res., 43(2):638-650, 2018.
Moran Feldman and Rico Zenklusen. The Submodular Secretary Problem Goes Linear. SIAM J. Comput., 47(2):330-366, 2018.
Thomas Ferguson. Who Solved the Secretary Problem? Statistical Science, 4(3):282-289, 1989.
Amos Fiat, Ilia Gorelik, Haim Kaplan, and Slava Novgorodov. The Temp Secretary Problem. In Proc. 23rd European Symp. Algorithms (ESA), pages 631-642, 2015.
Philippe Flajolet, Danièle Gardy, and Loÿs Thimonier. Birthday Paradox, Coupon Collectors, Caching Algorithms and Self-Organizing Search. Disc. Appl. Math., 39(3):207-229, 1992.
Oliver Göbel, Martin Hoefer, Thomas Kesselheim, Thomas Schleiden, and Berthold Vöcking. Online Independent Set Beyond the Worst-Case: Secretaries, Prophets and Periods. In Proc. 41st Intl. Coll. Automata, Languages &Programming (ICALP), volume 2, pages 508-519, 2014.
MohammadTaghi Hajiaghayi, Robert Kleinberg, and David Parkes. Adaptive limited-supply online auctions. In Proc. 5th Conf. Electronic Commerce (EC), pages 71-80, 2004.
Martin Hoefer and Bojana Kodric. Combinatorial Secretary Problems with Ordinal Information. In Proc. 44th Intl. Coll. Automata, Languages &Programming (ICALP), pages 133:1-133:14, 2017.
Thomas Kesselheim, Klaus Radke, Andreas Tönnis, and Berthold Vöcking. An Optimal Online Algorithm for Weighted Bipartite Matching and Extensions to Combinatorial Auctions. In Proc. 21st European Symp. Algorithms (ESA), pages 589-600, 2013.
Thomas Kesselheim, Klaus Radke, Andreas Tönnis, and Berthold Vöcking. Primal Beats Dual on Online Packing LPs in the Random-Order Model. In Proc. 46th Symp. Theory of Comput. (STOC), pages 303-312, 2014.
Thomas Kesselheim and Andreas Tönnis. Submodular Secretary Problems: Cardinality, Matching, and Linear Constraints. In Proc. 20th Workshop Approximation Algorithms for Combinatorial Optimization Problems (APPROX), pages 16:1-16:22, 2017.
John Kobza, Sheldon Jacobson, and Diane Vaughan. A Survey of the Coupon Collector’s Problem with Random Sample Sizes. Methodol. Comput. Appl. Probab., 9:573-584, 2007.
Nitish Korula and Martin Pál. Algorithms for Secretary Problems on Graphs and Hypergraphs. In Proc. 36th Intl. Coll. Automata, Languages &Programming (ICALP), pages 508-520, 2009.
Oded Lachish. O(log log Rank)-Competitive Ratio for the Matroid Secretary Problem. In Proc. 55th Symp. Foundations of Comput. Sci. (FOCS), pages 326-335, 2014.
Marco Molinaro and R. Ravi. Geometry of Online Packing Linear Programs. Math. Oper. Res., 39(1):46-59, 2014.
Aviad Rubinstein. Beyond matroids: secretary problem and prophet inequality with general constraints. In Proc. 48th Symp. Theory of Comput. (STOC), pages 324-332, 2016.
José Soto, Abner Turkieltaub, and Victor Verdugo. Strong Algorithms for the Ordinal Matroid Secretary Problem. In Proc. 29th Symp. Discrete Algorithms (SODA), pages 715-734, 2018.
Shai Vardi. The Returning Secretary. In Proc. 32nd Symp. Theoret. Aspects of Comput. Sci. (STACS), pages 716-729, 2015. Full version CoRR abs/1404.0614. URL: http://arxiv.org/abs/1404.0614.
http://arxiv.org/abs/1404.0614
Martin Hoefer and Lisa Wilhelmi
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