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 (log n · log r)-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] .
downward closed feasibility constraints REF .
Beyond matroids: secretary problem and prophet inequality with general constraints
STOC 2016
Mathematics Computer Science