Beyond matroids: Secretary Problem and Prophet Inequality with general constraints
Aviad Rubinstein
TL;DR
The paper develops a unified framework for online selection problems under general downward-closed constraints, addressing both Prophet Inequality and Secretary variants. It introduces a dynamic potential-based approach for the prophet and a two-phase, bucketed secretary algorithm, achieving $O(\log n)$-competitive ratios for 0-1 valuations and $O(\log n\cdot\log r)$ for general valuations, with $r$ the largest feasible set size. It proves near-tight lower bounds and discusses computational models, including oracle access, non-monotone feasibility, and order-oblivious reductions, highlighting the broad applicability of potential-function techniques in online combinatorial optimization. The results extend the matroid secretary/prophet literature to arbitrary downward-closed families, providing new guarantees and algorithmic tools with potential impact on online mechanism design and related online optimization problems.
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 Ω(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.