Proceedings of the twenty-first annual symposium on Principles of distributed computing
Adaptive limited-supply online auctions
EC '04 Proceedings of the 5th ACM conference on Electronic commerce
A multiple-choice secretary algorithm with applications to online auctions
SODA '05 Proceedings of the sixteenth annual ACM-SIAM symposium on Discrete algorithms
Competitive Analysis of Aggregate Max in Windowed Streaming
ICALP '09 Proceedings of the 36th International Colloquium on Automata, Languages and Programming: Part I
Auctions with severely bounded communication
Journal of Artificial Intelligence Research
Optimal algorithms for k-search with application in option pricing
ESA'07 Proceedings of the 15th annual European conference on Algorithms
Submodular secretary problem and extensions
APPROX/RANDOM'10 Proceedings of the 13th international conference on Approximation, and 14 the International conference on Randomization, and combinatorial optimization: algorithms and techniques
Delay-tolerant delivery of quality information in ad hoc networks
Journal of Parallel and Distributed Computing
Hiring a secretary from a poset
Proceedings of the 12th ACM conference on Electronic commerce
Improved competitive ratios for submodular secretary problems
APPROX'11/RANDOM'11 Proceedings of the 14th international workshop and 15th international conference on Approximation, randomization, and combinatorial optimization: algorithms and techniques
Secretary problems with competing employers
WINE'06 Proceedings of the Second international conference on Internet and Network Economics
Matroid secretary problem in the random assignment model
Proceedings of the twenty-second annual ACM-SIAM symposium on Discrete Algorithms
Optimal, quality-aware scheduling of data consumption in mobile ad hoc networks
Journal of Parallel and Distributed Computing
Submodular secretary problem and extensions
ACM Transactions on Algorithms (TALG)
Hi-index | 0.00 |
In the classical secretary problem, n objects from an ordered set arrive in random order, and one has to accept k of them so that the final decision about each object is made only on the basis of its rank relative to the ones already seen. Variants of the problem depend on the goal: either maximize the probability of accepting the best k objects, or minimize the expectation of the sum of the ranks (or powers of ranks) of the accepted objects. The problem and its generalizations are at the core of tasks with a large data set, in which it may be impractical to backtrack and select previous choices.Optimal algorithms for the special case of k = 1 are well known. Partial solutions for the first variant with general k are also known. In contrast, an explicit solution for the second variant with general k has not been known. It seems that the fact that the expected sum of powers of the ranks of selected items is bounded as n tends to infinity has been known to follow from standard results. We derive our results by obtaining explicit algorithms. For each $z \geq 1$, the resulting expected sum of the zth powers of the ranks of the selected objects is at most $k^{z + 1}/(z + 1) + C(z) \cdot k^{z + 0.5}\log k$, where log k \equiv \max\{1, \log_2 k\}$, whereas the best possible value at all is kz + 1/(z + 1) + O(kz). Our methods are very intuitive and apply to some generalizations. We also derive a lower bound on the trade-off between the probability of selecting the best object and the expected rank of the selected object.