Abstract
We consider situations in which a decision-maker with a fixed budget faces a sequence of options, each with a cost and a value, and must select a subset of them online so as to maximize the total value. Such situations arise in many contexts, e.g., hiring workers, scheduling jobs, and bidding in sponsored search auctions.
This problem, often called the online knapsack problem, is known to be inapproximable. Therefore, we make the enabling assumption that elements arrive in a random order. Hence our problem can be thought of as a weighted version of the classical secretary problem, which we call the knapsack secretary problem. Using the random-order assumption, we design a constant-competitive algorithm for arbitrary weights and values, as well as a e-competitive algorithm for the special case when all weights are equal (i.e., the multiple-choice secretary problem). In contrast to previous work on online knapsack problems, we do not assume any knowledge regarding the distribution of weights and values beyond the fact that the order is random.
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.
Similar content being viewed by others
References
Aggarwal, G., Hartline, J.: Knapsack auctions. In: SODA, pp. 1083–1092 (2006)
Babaioff, M., Immorlica, N., Kleinberg, R.: Matroids, secretary problems, and online mechanisms. In: SODA, pp. 434–443 (2007)
Borgs, C., Chayes, J., Etesami, O., Immorlica, N., Jain, K., Mahdian, M.: Dynamics of bid optimization in online advertisement auctions. In: Proceedings of the 16th International World Wide Web Conference, (to appear, 2007)
Buchbinder, N., Naor, J.: Online primal-dual algorithms for covering and packing problems. In: Brodal, G.S., Leonardi, S. (eds.) ESA 2005. LNCS, vol. 3669, Springer, Heidelberg (2005)
Buchbinder, N., Naor, J.: Improved bounds for online routing and packing via a primal-dual approach. In: FOCS (2006)
Chakrabarty, D., Zhou, Y., Lukose, R.: Budget constrained bidding in keyword auctions and online knapsack problems. In: WWW 2007, Workshop on Sponsored Search Auctions (2007)
Dean, B., Goemans, M., Vondrák, J.: Approximating the stochastic knapsack problem: The benefit of adaptivity. In: FOCS, pp. 208–217 (2004)
Dean, B., Goemans, M., Vondrák, J.: Adaptivity and approximation for stochastic packing problems. In: SODA, pp. 395–404 (2005)
Dynkin, E.B.: The optimum choice of the instant for stopping a Markov process. Sov. Math. Dokl. 4 (1963)
Feldman, J., Muthukrishnan, S., Pal, M., Stein, C.: Budget optimization in search-based advertising auctions. In: Proceedings of the 8th ACM Conference on Electronic Commerce (to appear)
Kleinberg, R.: A multiple-choice secretary problem with applications to online auctions. In: SODA, pp. 630–631 (2005)
Lueker, G.: Average-case analysis of off-line and on-line knapsack problems. In: SODA, pp. 179–188 (1995)
Marchetti-Spaccamela, A., Vercellis, C.: Stochastic on-line knapsack problems. Mathematical Programming 68, 73–104 (1995)
Motwani, R., Raghavan, P.: Randomized Algorithms. Cambridge University Press, Cambridge (1995)
Muthukrishnan, S., Pal, M., Svitkina, Z.: Stochastic models for budget optimization in search-based (manuscript, 2007)
Rusmevichientong, P., Williamson, D.P.: An adaptive algorithm for selecting profitable keywords for search-based advertising services. In: Proceedings of the 7th ACM Conference on Electronic Commerce, ACM Press, New York (2006)
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2007 Springer-Verlag Berlin Heidelberg
About this paper
Cite this paper
Babaioff, M., Immorlica, N., Kempe, D., Kleinberg, R. (2007). A Knapsack Secretary Problem with Applications. In: Charikar, M., Jansen, K., Reingold, O., Rolim, J.D.P. (eds) Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques. APPROX RANDOM 2007 2007. Lecture Notes in Computer Science, vol 4627. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-540-74208-1_2
Download citation
DOI: https://doi.org/10.1007/978-3-540-74208-1_2
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-540-74207-4
Online ISBN: 978-3-540-74208-1
eBook Packages: Computer ScienceComputer Science (R0)