Prophet Secretary for Combinatorial Auctions and Matroids

The secretary and the prophet inequality problems are central to the field of Stopping Theory. Recently, there has been a lot of work in generalizing these models to multiple items because of their applications in mechanism design. The most important of these generalizations are to matroids and to combinatorial auctions (extends bipartite matching). Kleinberg-Weinberg [KW12] and Feldman et al. [FGL15] show that for adversarial arrival order of random variables the optimal prophet inequalities give a 1/2-approximation. For many settings, however, it’s conceivable that the arrival order is chosen uniformly at random, akin to the secretary problem. For such a random arrival model, we improve upon the 1/2-approximation and obtain (1−1/e)approximation prophet inequalities for both matroids and combinatorial auctions. This also gives improvements to the results of Yan [Yan11] and Esfandiari et al. [EHLM17] who worked in the special cases where we can fully control the arrival order or when there is only a single item. Our techniques are threshold based. We convert our discrete problem into a continuous setting and then give a generic template on how to dynamically adjust these thresholds to lower bound the expected total welfare. Part of this work was done while the authors were visiting the Simons Institute for the Theory of Computing. Department of Computer Science, University of Maryland, College Park, MD 20742 USA. Email: {ehsani,hajiagha}@cs.umd.edu. Supported in part by NSF CAREER award CCF-1053605, NSF BIGDATA grant IIS-1546108, NSF AF:Medium grant CCF-1161365, DARPA GRAPHS/AFOSR grant FA9550-12-1-0423, and another DARPA SIMPLEX grant. Department of Computer Science, TU Dortmund, 44221 Dortmund, Germany. Email: [email protected]. Computer Science Department, Carnegie Mellon University, Pittsburgh, PA 15213, USA. Email: [email protected]. Supported in part by a CMU Presidential Fellowship and NSF awards CCF-1319811, CCF1536002, and CCF-1617790