The Submodular Secretary Problem Goes Linear

During the last decade, the matroid secretary problem (MSP) became one of the most prominent classes of online selection problems. Partially linked to its numerous applications in mechanism design, substantial interest arose also in the study of nonlinear versions of MSP, with a focus on the submodular matroid secretary problem (SMSP). So far, O(1)-competitive algorithms have been obtained for SMSP over some basic matroid classes. This created some hope that, analogously to the matroid secretary conjecture, one may even obtain O(1)-competitive algorithms for SMSP over any matroid. However, up to now, most questions related to SMSP remained open, including whether SMSP may be substantially more difficult than MSP; and more generally, to what extend MSP and SMSP are related. Our goal is to address these points by presenting general black-box reductions from SMSP to MSP. In particular, we show that any O(1)-competitive algorithm for MSP, even restricted to a particular matroid class, can be transformed in a black-box way to an O(1)-competitive algorithm for SMSP over the same matroid class. This implies that the matroid secretary conjecture is equivalent to the same conjecture for SMSP. Hence, in this sense SMSP is not harder than MSP. Also, to find O(1)-competitive algorithms for SMSP over a particular matroid class, it suffices to consider MSP over the same matroid class. Using our reductions we obtain many first and improved O(1)-competitive algorithms for SMSP over various matroid classes by leveraging known algorithms for MSP. Moreover, our reductions imply an O(loglog(rank))-competitive algorithm for SMSP, thus, matching the currently best asymptotic algorithm for MSP, and substantially improving on the previously best O(log(rank))-competitive algorithm for SMSP.