ISAAC 2015: Algorithms and Computation pp 129-139

# The Secretary Problem with a Choice Function

Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 9472)

## Abstract

In the classical secretary problem, a decision-maker is willing to hire the best secretary out of n applicants that arrive in a random order, and the goal is to maximize the probability of choosing the best applicant. In this paper, we introduce the secretary problem with a choice function. The choice function represents the preference of the decision-maker. In this problem, the decision-maker hires some applicants, and the goal is to maximize the probability of choosing the best set of applicants defined by the choice function. We see that the secretary problem with a path-independent choice function generalizes secretary version of the stable matching problem, the maximum weight bipartite matching problem, and the maximum weight base problem in a matroid. When the choice function is path-independent, we provide an algorithm that succeeds with probability at least $$1/e^k$$ where k is the maximum size of the choice, and prove that this is the best possible. Moreover, for the non-path-independent case, we prove that the success probability goes to arbitrary small for any algorithm even if the maximum size of the choice is 2.

## Notes

### Acknowledgement

The author thanks Tomomi Matsui and Keisuke Bando for valuable comments and suggestions. This work was supported by JSPS KAKENHI Grant Number 26887014.

### References

1. 1.
Aizerman, M., Malishevski, A.: General theory of best variants choice: some aspects. IEEE Trans. Autom. Control 26, 1030–1040 (1981)
2. 2.
Babaioff, M., Immorlica, N., Kempe, D., Kleinberg, R.D.: A knapsack secretary problem with applications. In: Charikar, M., Jansen, K., Reingold, O., Rolim, J.D.P. (eds.) APPROX and RANDOM 2007. LNCS, vol. 4627, pp. 16–28. Springer, Heidelberg (2007)
3. 3.
Babaioff, M., Immorlica, N., Kempe, D., Kleinberg, R.: Online auctions and generalized secretary problems. ACM SIGecom Exch. 7(2), 7:1–7:11 (2008)
4. 4.
Babaioff, M., Immorlica, N., Kleinberg, R.: Matroids, secretary problems, and online mechanisms. In: Proceedings of the Eighteenth Annual ACM-SIAM Symposium on Discrete Algorithms, pp. 434–443 (2007)Google Scholar
5. 5.
Buchbinder, N., Jain, K., Singh, M.: Secretary problems via linear programming. Math. Oper. Res. 39(1), 190–206 (2014)
6. 6.
Chan, T.H.H., Chen, F., Jiang, S.H.C.: Revealing optimal thresholds for generalized secretary problem via continuous LP: impacts on online $$k$$-item auction and bipartite $$k$$-matching with random arrival order. In: Proceedings of the Eighteenth Annual ACM-SIAM Symposium on Discrete Algorithms, pp. 1169–1188 (2015)Google Scholar
7. 7.
Dinitz, M.: Recent advances on the matroid secretary problem. SIGACT News 44(2), 126–142 (2013)
8. 8.
Dynkin, E.B.: The optimum choice of the instant for stopping a Markov process. Sov. Math. Dokl. 4, 627–629 (1963)
9. 9.
Feldman, M., Svensson, O., Zenklusen, R.: A simple O(log log(rank))-completitive algorithm for the matroid secretary problem. In: Proceedings of the Twenty-Sixth Annual ACM-SIAM Symposium on Discrete Algorithms, pp. 1189–1201 (2015)Google Scholar
10. 10.
Ferguson, T.S.: Who solved the secretary problem? Statist. Sci. 4(3), 282–289 (1989)
11. 11.
Freeman, P.R.: The secretary problem and its extensions: a review. Int. Statis. Rev. 51(2), 189–206 (1983)
12. 12.
Fujishige, S., Tamura, A.: A general two-sided matching market with discrete concave utility functions. Discrete Appl. Math. 154, 950–970 (2006)
13. 13.
Gale, D., Shapley, L.S.: College admissions and the stability of marriage. Am. Math. Mon. 69, 9–14 (1962)
14. 14.
Gale, D., Sotomayor, M.: Some remarks on the stable matching problem. Discrete Appl. Math. 11, 223–232 (1985)
15. 15.
Gilbert, J.P., Mosteller, F.: Recognizing the maximum of a sequence. J. Am. Statist. Assoc. 61, 35–73 (1966)
16. 16.
Gusfield, D., Irving, R.W.: The Stable Marriage Problem: Structure and Algorithms. MIT Press, Boston (1989)
17. 17.
Khuller, S., Mitchell, S.G., Vazirani, V.V.: On-line algorithms for weighted bipartite matching and stable marriages. Theoret. Comput. Sci. 127(2), 255–267 (1994)
18. 18.
Kleinberg, R.: A multiple-choice secretary algorithm with applications to online auctions. In: Proceedings of the Eighteenth Annual ACM-SIAM Symposium on Discrete Algorithms, pp. 630–631 (2005)Google Scholar
19. 19.
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, Part II. LNCS, vol. 5556, pp. 508–520. Springer, Heidelberg (2009)
20. 20.
Lachish, O.: O(log log rank) completitive-ratio for the matroid secretary problem. In: Proceedings of 55th Annual Symposium on Foundation of Compuster Science, pp. 326–335 (2014)Google Scholar
21. 21.
Lindley, D.V.: Dynamic programming and decision theory. Appl. Stat. 10, 39–52 (1961)
22. 22.
Manlove, D.F.: Algorithmics of Matching Under Preferences. World Scientific, Singapore (2013)
23. 23.
McVitie, D.G., Wilson, L.B.: The stable marriage problem. Commun. ACM 14(7), 486–490 (1971)
24. 24.
Murota, K.: Recent developments in discrete convex analysis. In: Cook, W.J., Lovász, L., Vygen, J. (eds.) Research Trends in Combinatorial Optimization, pp. 219–260. Springer, Heidelberg (2009)
25. 25.
Murota, K., Yokoi, Y.: On the lattice structure of stable allocations in a two-sided discrete-concave market. Math. Oper. Res. 40, 460–473 (2015)
26. 26.
Nikolaev, M.L.: On a generalization of the best choice problem. Theor. Probab. Its Appl. 22(1), 187–190 (1977)
27. 27.
Oxley, J.G.: Matroid Theory. Oxford University Press, New York (1992)
28. 28.
Roth, A.E., Sotomayor, M.: Two-Sided Matching: A Study in Game-Theoretic Modeling and Analysis. Cambridge University Press, Cambridge (1991)
29. 29.
Samuels, S.M.: Secretary problems. In: Ghosh, B.K., Sen, P.K. (eds.) Handbook of Sequential Analysis. Marcel Dekker, Boston (1991)Google Scholar
30. 30.
Tamaki, M.: Recognizing both the maximum and the second maximum of a sequence. J. Appl. Probab. 16(4), 803–812 (1979)
31. 31.
Vanderbei, R.J.: The optimal choice of a subset of a population. Math. Oper. Res. 5(4), 481–486 (1980)