The Submodular Secretary Problem Goes Linear

Abstract

During the last decade, the matroid secretary problem (MSP) became one of the most prominent classes of online selection problems. The interest in MSP is twofold: on the one hand, there are many interesting applications of MSP, and on the other hand, there is strong hope that MSP admits $O(1)$-competitive algorithms, which is the claim of the well-known matroid secretary conjecture. Partially linked to its numerous applications in online auctions, substantial interest arose also in the study of nonlinear versions of MSP, with a focus on the submodular MSP (SMSP). The fact that submodularity captures the property of diminishing returns, a very natural property for valuation functions, is a key reason for the interest in 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 questi...