Article

Improved competitive ratios for submodular secretary problems

Authors:

Joseph Seffi Naor,

Roy SchwartzAuthors Info & Claims

APPROX'11/RANDOM'11: Proceedings of the 14th international workshop and 15th international conference on Approximation, randomization, and combinatorial optimization: algorithms and techniques

Pages 218 - 229

Published: 17 August 2011 Publication History

Abstract

The Classical Secretary Problem was introduced during the 60's of the 20th century, nobody is sure exactly when. Since its introduction, many variants of the problem have been proposed and researched. In the classical secretary problem, and many of its variant, the input (which is a set of secretaries, or elements) arrives in a random order. In this paper we apply to the secretary problem a simple observation which states that the random order of the input can be generated by independently choosing a random continuous arrival time for each secretary. Surprisingly, this simple observation enables us to improve the competitive ratio of several known and studied variants of the secretary problem. In addition, in some cases the proofs we provide assuming random arrival times are shorter and simpler in comparison to existing proofs. In this work we consider three variants of the secretary problem, all of which have the same objective of maximizing the value of the chosen set of secretaries given a monotone submodular function f. In the first variant we are allowed to hire a set of secretaries only if it is an independent set of a given partition matroid. The second variant allows us to choose any set of up to k secretaries. In the last and third variant, we can hire any set of secretaries satisfying a given knapsack constraint.

References

[1]

Ajtai, M., Megiddo, N., Waarts, O.: Improved algorithms and analysis for secretary problems and generalizations. SIAM J. Discrete Math. 14(1), 1-27 (2001)

Digital Library

[2]

Babaioff, M., Dinitz, M., Gupta, A., Immorlica, N., Talwar, K.: Secretary problems: Weights and discounts. In: 20th ACM-SIAM Symposium on Discrete Algorithms, pp. 1245-1254. Society for Industrial and Applied Mathematics, Philadelphia (2009)

Digital Library

[3]

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)

Digital Library

[4]

Babaioff, M., Immorlica, N., Kleinberg, R.: Matroids, secretary problems, and online mechanisms. In: 18th ACM-SIAM Symposium on Discrete Algorithms, pp. 434-443. Society for Industrial and Applied Mathematics, Philadelphia (2007)

Digital Library

[5]

Bateni, M.H., Hajiaghayi, M.T., Zadimoghaddam, M.: Submodular secretary problem and extensions. In: Serna, M., Shaltiel, R., Jansen, K., Rolim, J. (eds.) APPROX 2010, LNCS, vol. 6302, pp. 39-52. Springer, Heidelberg (2010)

Digital Library

[6]

Buchbinder, N., Jain, K., Singh, M.: Secretary problems via linear programming. In: Eisenbrand, F., Shepherd, F.B. (eds.) IPCO 2010. LNCS, vol. 6080, pp. 163- 176. Springer, Heidelberg (2010)

Digital Library

[7]

Calinescu, G., Chekuri, C., Pál, M., Vondrák, J.: Maximizing a monotone submodular function subject to a matroid constraint. To appear in SIAM J. Comput.

Digital Library

[8]

Chekuri, C., Vondrák, J., Zenklusen, R.: Submodular function maximization via the multilinear relaxation and contention resolution schemes. In: 42nd ACM Symposium on Theory of Computer Science, pp. 783-792. ACM, New York (2011)

Digital Library

[9]

Dimitrov, N.B., Plaxton, C.G.: Competitive weighted matching in transversal matroids. In: Aceto, L., Damgård, I., Goldberg, L.A., Halldórsson, M.M., Ingólfsdóttir, A., Walukiewicz, I. (eds.) ICALP 2008, Part I. LNCS, vol. 5125, pp. 397-408. Springer, Heidelberg (2008)

Digital Library

[10]

Dynkin, E.B.: The optimum choice of the instant for stopping a markov process. Sov. Math. Dokl. 4, 627-629 (1963)

[11]

Feige, U.: On maximizing welfare when utility functions are subadditive. SIAM J. Comput. 39(1), 122-142 (2009)

Digital Library

[12]

Feige, U., Mirrokni, V.S., Vondrák, J.: Maximizing non-monotone submodular functions. In: 48th Annual IEEE Symposium on Foundations of Computer Science, pp. 461-471. IEEE Computer Society, Washington DC (2007)

Digital Library

[13]

Ferguson, T.S.: Who solved the secretary problem? Statistical Science 4(3), 282- 289 (1989)

[14]

Freeman, P.R.: The secretary problem and its extensions: A review. International Statistical Review 51(2), 189-206 (1983)

[15]

Gharan, S.O., Vondrák, J.: Submodular maximization by simulated annealing. In: 22nd ACM-SIAM Symposium on Discrete Algorithms, pp. 1096-1116 (2011)

Digital Library

[16]

Gilbert, J., Mosteller, F.: Recognizing the maximum of a sequence. In: Fienberg, S., Hoaglin, D. (eds.) Selected Papers of Frederick Mosteller. Springer Series in Statistics, pp. 355-398. Springer, New York (2006)

[17]

Gupta, A., Roth, A., Schoenebeck, G., Talwar, K.: Constrained nonmonotone submodular maximization: Offline and secretary algorithms. In: Saberi, A. (ed.) WINE 2010. LNCS, vol. 6484, pp. 246-257. Springer, Heidelberg (2010)

Digital Library

[18]

Im, S., Wang, Y.: Secretary problems: Laminar matroid and interval scheduling. In: 22nd ACM-SIAM Symposium on Discrete Algorithms, pp. 1096-1116 (2011)

Digital Library

[19]

Kleinberg, R.: A multiple-choice secretary algorithm with applications to online auctions. In: 16th ACM-SIAM Symposium on Discrete Algorithms, pp. 630-631. Society for Industrial and Applied Mathematics, Philadelphia (2005)

Digital Library

[20]

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)

Digital Library

[21]

Kulik, A., Shachnai, H., Tamir, T.: Maximizing submodular set functions subject to multiple linear constraints. In: 20th ACM-SIAM Symposium on Discrete Algorithms, pp. 545-554. Society for Industrial and Applied Mathematics, Philadelphia (2009)

Digital Library

[22]

Lee, J., Mirrokni, V.S., Nagarajan, V., Sviridenko, M.: Maximizing nonmonotone submodular functions under matroid or knapsack constraints. SIAM J. Discrete Mathematics 23(4), 2053-2078 (2010)

Digital Library

[23]

Lee, J., Sviridenko, M., Vondrák, J.: Submodular maximization over multiple matroids via generalized exchange properties. In: 12th International Workshop on Approximation Algorithms for Combinatorial Optimization Problems, pp. 244-257. Springer, Heidelberg (2009)

Digital Library

[24]

Lindley, D.V.: Dynamic programming and decision theory. Applied Statistics 10, 39-51 (1961)

[25]

Vondrák, J.: Symmetry and approximability of submodular maximization problems. In: 50th Annual IEEE Symposium on Foundations of Computer Science, pp. 651-670. IEEE Computer Society, Washington DC (2009)

Digital Library

Cited By

Feldman MSvensson OZenklusen RCzumaj A(2018)A framework for the secretary problem on the intersection of matroidsProceedings of the Twenty-Ninth Annual ACM-SIAM Symposium on Discrete Algorithms10.5555/3174304.3175318(735-752)Online publication date: 7-Jan-2018
https://dl.acm.org/doi/10.5555/3174304.3175318
Soto JTurkieltaub AVerdugo VCzumaj A(2018)Strong algorithms for the ordinal matroid secretary problemProceedings of the Twenty-Ninth Annual ACM-SIAM Symposium on Discrete Algorithms10.5555/3174304.3175317(715-734)Online publication date: 7-Jan-2018
https://dl.acm.org/doi/10.5555/3174304.3175317
Babaioff MImmorlica NKempe DKleinberg R(2018)Matroid Secretary ProblemsJournal of the ACM10.1145/321251265:6(1-26)Online publication date: 19-Nov-2018
https://dl.acm.org/doi/10.1145/3212512
Show More Cited By

Improved competitive ratios for submodular secretary problems
1. Theory of computation

Recommendations

Submodular secretary problem and extensions

Online auction is the essence of many modern markets, particularly networked markets, in which information about goods, agents, and outcomes is revealed over a period of time, and the agents must make irrevocable decisions without knowing future ...
Improved competitive ratio for the matroid secretary problem
SODA '12: Proceedings of the twenty-third annual ACM-SIAM symposium on Discrete algorithms

The Matroid Secretary Problem, introduced by Babaioff et al. (2007), is a generalization of the Classical Secretary Problem. In this problem, elements from a matroid are presented to an on-line algorithm in a random order. Each element has a weight ...
Improved Absolute Approximation Ratios for Two-Dimensional Packing Problems
APPROX '09 / RANDOM '09: Proceedings of the 12th International Workshop and 13th International Workshop on Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques

We consider the two-dimensional bin packing and strip packing problem, where a list of rectangles has to be packed into a minimal number of rectangular bins or a strip of minimal height, respectively. All packings have to be non-overlapping and ...

Comments

Information & Contributors

Information

Published In

cover image Guide Proceedings

APPROX'11/RANDOM'11: Proceedings of the 14th international workshop and 15th international conference on Approximation, randomization, and combinatorial optimization: algorithms and techniques

August 2011

699 pages

ISBN:9783642229343

Editors:
Leslie Ann Goldberg
University of Liverpool, Department of Computer Science, Liverpool, UK
,
Klaus Jansen
University of Kiel, Department of Computer Science, Kiel, Germany
,
R. Ravi
Carnegie Mellon University, Pittsburgh, PA
,
José D. P. Rolim
University of Geneva, Carouge, Switzerland

Publisher

Springer-Verlag

Berlin, Heidelberg

Publication History

Published: 17 August 2011

Qualifiers

Article

Contributors

Other Metrics

View Article Metrics

Bibliometrics & Citations

Bibliometrics

Article Metrics

17
Total Citations
View Citations
0
Total Downloads

Downloads (Last 12 months)0
Downloads (Last 6 weeks)0

Reflects downloads up to 11 Feb 2025

Other Metrics

View Author Metrics

Citations

Cited By

Feldman MSvensson OZenklusen RCzumaj A(2018)A framework for the secretary problem on the intersection of matroidsProceedings of the Twenty-Ninth Annual ACM-SIAM Symposium on Discrete Algorithms10.5555/3174304.3175318(735-752)Online publication date: 7-Jan-2018
https://dl.acm.org/doi/10.5555/3174304.3175318
Soto JTurkieltaub AVerdugo VCzumaj A(2018)Strong algorithms for the ordinal matroid secretary problemProceedings of the Twenty-Ninth Annual ACM-SIAM Symposium on Discrete Algorithms10.5555/3174304.3175317(715-734)Online publication date: 7-Jan-2018
https://dl.acm.org/doi/10.5555/3174304.3175317
Babaioff MImmorlica NKempe DKleinberg R(2018)Matroid Secretary ProblemsJournal of the ACM10.1145/321251265:6(1-26)Online publication date: 19-Nov-2018
https://dl.acm.org/doi/10.1145/3212512
Rubinstein ASingla SKlein P(2017)Combinatorial prophet inequalitiesProceedings of the Twenty-Eighth Annual ACM-SIAM Symposium on Discrete Algorithms10.5555/3039686.3039796(1671-1687)Online publication date: 16-Jan-2017
https://dl.acm.org/doi/10.5555/3039686.3039796
Feldman MIzsak RKlein P(2017)Building a good teamProceedings of the Twenty-Eighth Annual ACM-SIAM Symposium on Discrete Algorithms10.5555/3039686.3039795(1651-1670)Online publication date: 16-Jan-2017
https://dl.acm.org/doi/10.5555/3039686.3039795
Fujii KKashima H(2016)Budgeted stream-based active learning via adaptive submodular maximizationProceedings of the 30th International Conference on Neural Information Processing Systems10.5555/3157096.3157154(514-522)Online publication date: 5-Dec-2016
https://dl.acm.org/doi/10.5555/3157096.3157154
Fraigniaud PHalldórsson MPatt-Shamir BRawitz DRosén A(2016)Shrinking Maxima, Decreasing CostsAlgorithmica10.1007/s00453-015-9995-874:4(1205-1223)Online publication date: 1-Apr-2016
https://dl.acm.org/doi/10.1007/s00453-015-9995-8
Ma TTang BWang Y(2016)The Simulated Greedy Algorithm for Several Submodular Matroid Secretary ProblemsTheory of Computing Systems10.1007/s00224-015-9642-458:4(681-706)Online publication date: 1-May-2016
https://dl.acm.org/doi/10.1007/s00224-015-9642-4
Buchbinder NFeldman MSchwartz RIndyk P(2015)Online submodular maximization with preemptionProceedings of the twenty-sixth annual ACM-SIAM symposium on Discrete algorithms10.5555/2722129.2722209(1202-1216)Online publication date: 4-Jan-2015
https://dl.acm.org/doi/10.5555/2722129.2722209
Feldman MSvensson OZenklusen RIndyk P(2015)A simple O(log log(rank))-competitive algorithm for the matroid secretary problemProceedings of the twenty-sixth annual ACM-SIAM symposium on Discrete algorithms10.5555/2722129.2722208(1189-1201)Online publication date: 4-Jan-2015
https://dl.acm.org/doi/10.5555/2722129.2722208
Show More Cited By

View Options

View options

Figures

Tables

Media

View Table of Conten