Partial-order analogue of the secretary problem: the binary tree case
Discrete Mathematics
Improved Algorithms and Analysis for Secretary Problems and Generalizations
SIAM Journal on Discrete Mathematics
A Ratio Inequality for Binary Trees and the Best Secretary
Combinatorics, Probability and 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
Matroids, secretary problems, and online mechanisms
SODA '07 Proceedings of the eighteenth annual ACM-SIAM symposium on Discrete algorithms
On a universal best choice algorithm for partially ordered sets
Random Structures & Algorithms
Online auctions and generalized secretary problems
ACM SIGecom Exchanges
Secretary problems: weights and discounts
SODA '09 Proceedings of the twentieth Annual ACM-SIAM Symposium on Discrete 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
Constrained non-monotone submodular maximization: offline and secretary algorithms
WINE'10 Proceedings of the 6th 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
Secretary problems via linear programming
IPCO'10 Proceedings of the 14th international conference on Integer Programming and Combinatorial Optimization
The best-choice problem for partially ordered objects
Operations Research Letters
Interviewing secretaries in parallel
Proceedings of the 13th ACM Conference on Electronic Commerce
Note: The best choice problem for a union of two linear orders with common maximum
Discrete Applied Mathematics
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The secretary problem lies at the core of mechanism design for online auctions. In this work we study the generalization of the classical secretary problem in a setting where there is only a partial order between the elements and the goal of the algorithm is to return one of the maximal elements of the poset. This is equivalent to the auction setting where the seller has a multidimensional objective function with only a partial order among the outcomes. We obtain an algorithm that succeeds with probability at least k-k/(k-1)((1 + log k1/(k-1))k - 1), where k is the number of maximal elements in the poset and is the only information about the poset that is known to the algorithm; the success probability approaches the classical bound of 1/e as k - 1. On the other hand, we prove an almost matching upper bound of k-1/(k-1) on the success probability of any algorithm for this problem; this upper bound holds even if the algorithm knows the complete structure of the poset.