Combinatorial Prophet Inequalities

We introduce a novel framework of Prophet Inequalities for combinatorial valuation functions. For a (non-monotone) submodular objective function over an arbitrary matroid feasibility constraint, we give an O(1)-competitive algorithm. For a monotone subadditive objective function over an arbitrary downward-closed feasibility constraint, we give an O(log n log r)-competitive algorithm (where r is the cardinality of the largest feasible subset). Inspired by the proof of our subadditive prophet inequality, we also obtain an O(log n · log r)-competitive algorithm for the Secretary Problem with a monotone subadditive objective function subject to an arbitrary downward-closed feasibility constraint. Even for the special case of a cardinality feasibility constraint, our algorithm circumvents an Ω( √ n) lower bound by Bateni, Hajiaghayi, and Zadimoghaddam [BHZ13] in a restricted query model. En route to our submodular prophet inequality, we prove a technical result of independent interest: we show a variant of the Correlation Gap Lemma [CCPV07, ADSY12] for non-monotone submodular functions.