Beyond matroids: Secretary Problem and Prophet Inequality with general constraints
Abstract
We study generalizations of the "Prophet Inequality" and "Secretary Problem", where the algorithm is restricted to an arbitrary downward-closed set system. For {0,1}-values, we give O(log n)-competitive algorithms for both problems. This is close to the \Omega(log n / loglog n) lower bound due to Babaioff, Immorlica, and Kleinberg. For general values, our results translate to O(log n log r)-competitive algorithms, where r is the cardinality of the largest feasible set. This resolves (up to the O(log r loglog n) factors) an open question posed to us by Bobby Kleinberg.
Cite
@article{arxiv.1604.00357,
title = {Beyond matroids: Secretary Problem and Prophet Inequality with general constraints},
author = {Aviad Rubinstein},
journal= {arXiv preprint arXiv:1604.00357},
year = {2024}
}
Comments
v2 includes a nice observation by Kesselheim, Singla, and Svensson: the prophet inequality bound can also be directly derived from the secretary algorithm (since the latter is "order-oblivious")!