Prophet Secretary for Matroids and XOS Valuations: A Separation at 1-1/e and a 0.6767 Guarantee for I.I.D. Weights

Abstract

Ehsani, Hajiaghayi, Kesselheim, and Singla (SODA 2018) give $(1-1/e)$-competitive algorithms for matroid prophet-secretary and prophet-secretary with XOS valuation functions. In this work we show that these two guarantees have different limits. For every matroid, we give a randomized $(1-1/e+\varepsilon_0)$-competitive algorithm, where $\varepsilon_0=10^{-6}$. The guarantee holds for independent nonnegative values with finite expected optimum. For explicitly given finite distributions with rational values and probabilities, the algorithm has polynomial worst-case running time in the matroid independence-oracle model. The oracle tests whether a queried set is independent. The algorithm makes $O(N^4)$ such queries in expectation and $O(N^4\log(N+2))$ in the worst case, where $N$ is the number of elements. We further consider the i.i.d. version of this problem and provide a $0.6767$ competitive ratio for every matroid with i.i.d.\ nonnegative weights from a known distribution. This guarantee even holds in fixed arrival order (a.k.a. i.i.d. matriod prophet inequalities), even when the algorithm learns the order only as the elements arrive. For combinatorial auctions with XOS valuations, however, we show that $1-1/e$ is the optimal competitive ratio. We give an explicit family of finite instances showing that no online algorithm can guarantee a larger competitive ratio than $1-1/e$, even when the buyers' valuations are i.i.d., every additive clause has coefficients in $\{0,1\}$, and computation is unrestricted. Thus the matroid improvement exploits structure absent from general allocation with XOS valuations.