Abstract
In the classical secretary problem an employer would like to choose the best candidate among n competing candidates that arrive in a random order. This basic concept of n elements arriving in a random order and irrevocable decisions made by an algorithm have been explored extensively over the years, and used for modeling the behavior of many processes. Our main contribution is a new linear programming technique that we introduce as a tool for obtaining and analyzing mechanisms for the secretary problem and its variants. The linear program is formulated using judiciously chosen variables and constraints and we show a one-to-one correspondence between mechanisms for the secretary problem and feasible solutions to the linear program. Capturing the set of mechanisms as a linear polytope holds the following immediate advantages.
-
Computing the optimal mechanism reduces to solving a linear program.
-
Proving an upper bound on the performance of any mechanism reduces to finding a feasible solution to the dual program.
-
Exploring variants of the problem is as simple as adding new constraints, or manipulating the objective function of the linear program.
We demonstrate these ideas by exploring some natural variants of the secretary problem. In particular, using our approach, we design optimal secretary mechanisms in which the probability of selecting a candidate at any position is equal. We refer to such mechanisms as incentive compatible and these mechanisms are motivated by the recent applications of secretary problems to online auctions. We also show a family of linear programs which characterize all mechanisms that are allowed to choose J candidates and gain profit from the K best candidates. We believe that linear programming based approach may be very helpful in the context of other variants of the secretary problem.
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Preview
Unable to display preview. Download preview PDF.
References
Awerbuch, B., Azar, Y., Meyerson, A.: Reducing Truth-Telling Online Mechanisms to Online Optimization. In: Proceedings of ACM Symposium on Theory of Computing, pp. 503–510 (2003)
Babaioff, M., Immorlica, N., Kempe, D., Kleinberg, R.: A Knapsack Secretary Problem with Applications. In: Charikar, M., Jansen, K., Reingold, O., Rolim, J.D.P. (eds.) RANDOM 2007 and APPROX 2007. LNCS, vol. 4627, pp. 16–28. Springer, Heidelberg (2007)
Babaioff, M., Immorlica, N., Kempe, D., Kleinberg, R.: Online Auctions and Generalized Secretary Problems. SIGecom Exchange 7, 1–11 (2008)
Babaioff, M., Immorlica, N., Kleinberg, R.: Matroids, Secretary Problems, and Online Mechanisms. In: Proceedings 18th ACM-SIAM Symposium on Discrete Algorithms (2007)
Babaioff, M., Dinitz, M., Gupta, A., Immorlica, N., Talwar, K.: Secretary problems: weights and discounts. In: SODA ’09: Proceedings of the Nineteenth Annual ACM -SIAM Symposium on Discrete Algorithms, pp. 1245–1254. Society for Industrial and Applied Mathematics, Philadelphia (2009)
Buchbinder, N., Singh, M., Jain, K.: Incentives in Online Auctions and Secretary Problems via Linear Programming (2009) (manuscript)
Buchbinder, N., Jain, K., Naor, J(S.): Online primal-dual algorithms for maximizing ad-auctions revenue. In: Arge, L., Hoffmann, M., Welzl, E. (eds.) ESA 2007. LNCS, vol. 4698, pp. 253–264. Springer, Heidelberg (2007)
Dynkin, E.B.: The Optimum Choice of the Instant for Stopping a Markov Process. Sov. Math. Dokl. 4 (1963)
Ferguson, T.S.: Who Solved the Secretary Problem? Statist. Sci. 4, 282–289 (1989)
Freeman, P.R.: The Secretary Problem and its Extensions: A Review. International Statistical Review 51, 189–206 (1983)
Gardner, M.: Mathematical Games. Scientific American, 150–153 (1960)
Goemans, M., Kleinberg, J.: An improved approximation ratio for the minimum latency problem. In: SODA ’96: Proceedings of the seventh annual ACM-SIAM symposium on Discrete algorithms, pp. 152–158 (1996)
Hajiaghayi, M.T., Kleinberg, R., Parkes, D.C.: Adaptive Limited-Supply Online Auctions. In: Proceedings of the 5th ACM Conference on Electronic Commerce (2004)
Jain, K., Mahdian, M., Markakis, E., Saberi, A., Vazirani, V.V.: Greedy facility location algorithms analyzed using dual fitting with factor-revealing lp. J. ACM 50(6), 795–824 (2003)
Kleinberg, R.: A Multiple-Choice Secretary Algorithm with Applications to Online Auctions. In: Proceedings of the Sixteenth Annual ACM-SIAM Symposium on Discrete algorithms (2005)
Korula, N., Pál, M.: Algorithms for secretary problems on graphs and hypergraphs. In: Albers, S., Marchetti-Spaccamela, A., Matias, Y., Nikoletseas, S., Thomas, W. (eds.) ICALP 2009. LNCS, vol. 5556, pp. 508–520. Springer, Heidelberg (2009)
Lavi, R., Nisan, N.: Competitive Analysis of Incentive Compatible On-line Auctions. In: Proceedings of 2nd ACM Conf. on Electronic Commerce, pp. 233–241 (2000)
Lindley, D.V.: Dynamic Programming and Decision Theory. Applied Statistics 10, 39–51 (1961)
Mehta, A., Saberi, A., Vazirani, U., Vazirani, V.: Adwords and generalized online matching. J. ACMÂ 54(5), 22 (2007)
Samuels, S.M.: Secretary Problems. In: Handbook of Sequential Analysis, vol. 118, pp. 381–405 (1991)
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2010 Springer-Verlag Berlin Heidelberg
About this paper
Cite this paper
Buchbinder, N., Jain, K., Singh, M. (2010). Secretary Problems via Linear Programming. In: Eisenbrand, F., Shepherd, F.B. (eds) Integer Programming and Combinatorial Optimization. IPCO 2010. Lecture Notes in Computer Science, vol 6080. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-13036-6_13
Download citation
DOI: https://doi.org/10.1007/978-3-642-13036-6_13
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-642-13035-9
Online ISBN: 978-3-642-13036-6
eBook Packages: Computer ScienceComputer Science (R0)