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 Ω (log n/log log n) lower bound due to Babaioff, Immorlica, and Kleinberg. For general values, our results translate to O (logn · logr)-competitive algorithms, where r is the cardinality of the largest feasible set. This resolves (up to the O (log r · log log n) factors) an open question posed to us by Bobby Kleinberg [13].