Online contention resolution schemes (OCRSs) were proposed by Feldman, Svensson, and Zenklusen as a generic technique to round a fractional solution in the matroid polytope in an online fashion. It has found applications in several stochastic combinatorial problems where there is a commitment constraint: on seeing the value of a stochastic element, the algorithm has to immediately and irrevocably decide whether to select it while always maintaining an independent set in the matroid. Although OCRSs immediately lead to prophet inequalities, these prophet inequalities are not optimal. Can we instead use prophet inequalities to design optimal OCRSs? We design the first optimal @math -OCRS for matroids by reducing the problem to designing a matroid prophet inequality where we compare to the stronger benchmark of an ex-ante relaxation. We also introduce and design optimal @math -random order CRSs for matroids, which are similar to OCRSs but the arrival is chosen uniformly at random.
Krengel and Sucheston gave the first tight @math -single item prophet inequality @cite_24 @cite_11 . The connection between multiple-choice prophet inequalities and mechanism design was recognized in @cite_6 ; they proved a prophet inequality for uniform matroids. This bound was later improved by Alaei @cite_1 using the , which is an OCRS in disguise. @cite_7 further developed the connection between prophet inequalities and mechanism design, and showed how to be @math -prophet inequality for general matroids in a variant where the algorithm may choose the element order. Yan @cite_0 improved this result to @math -competitive using the for submodular functions, first studied in @cite_16 @cite_4 . @cite_21 adapted correlation gaps to a polytope to design CRSs. Improved correlation gaps were presented in @cite_0 @cite_17 . The matroid prophet inequality was first explicitly formulated in @cite_18 . @cite_23 gave an alternate proof, and extended to Bernoulli submodular functions, using OCRSs. Finally, information theoretic @math -prophet inequalities are also known for general downward-closed constraints @cite_3 @cite_12 .
@cite_18 @cite_4 @cite_7 @cite_21 @cite_1 @cite_17 @cite_6 @cite_3 @cite_24 @cite_0 @cite_23 @cite_16 @cite_12 @cite_11
2012672170 2949499418 1584886343 2810576705
Consider a gambler who observes a sequence of independent, non-negative random numbers and is allowed to stop the sequence at any time, claiming a reward equal to the most recent observation. The famous prophet inequality of Krengel, Sucheston, and Garling asserts that a gambler who knows the distribution of each random variable can achieve at least half as much reward, in expectation, as a "prophet" who knows the sampled values of each random variable and can choose the largest one. We generalize this result to the setting in which the gambler and the prophet are allowed to make more than one selection, subject to a matroid constraint. We show that the gambler can still achieve at least half as much reward as the prophet; this result is the best possible, since it is known that the ratio cannot be improved even in the original prophet inequality, which corresponds to the special case of rank-one matroids. Generalizing the result still further, we show that under an intersection of @math matroid constraints, the prophet's reward exceeds the gambler's by a factor of at most @math , and this factor is also tight. Beyond their interest as theorems about pure online algoritms or optimal stopping rules, these results also have applications to mechanism design. Our results imply improved bounds on the ability of sequential posted-price mechanisms to approximate optimal mechanisms in both single-parameter and multi-parameter Bayesian settings. In particular, our results imply the first efficiently computable constant-factor approximations to the Bayesian optimal revenue in certain multi-parameter settings. Consider a gambler who observes a sequence of independent, non-negative random numbers and is allowed to stop the sequence at any time, claiming a reward equal to the most recent observation. The famous prophet inequality of Krengel, Sucheston, and Garling asserts that a gambler who knows the distribution of each random variable can achieve at least half as much reward, in expectation, as a "prophet" who knows the sampled values of each random variable and can choose the largest one. We generalize this result to the setting in which the gambler and the prophet are allowed to make more than one selection, subject to a matroid constraint. We show that the gambler can still achieve at least half as much reward as the prophet; this result is the best possible, since it is known that the ratio cannot be improved even in the original prophet inequality, which corresponds to the special case of rank-one matroids. Generalizing the result still further, we show that under an intersection of p matroid constraints, the prophet's reward exceeds the gambler's by a factor of at most O(p), and this factor is also tight. Beyond their interest as theorems about pure online algorithms or optimal stopping rules, these results also have applications to mechanism design. Our results imply improved bounds on the ability of sequential posted-price mechanisms to approximate Bayesian optimal mechanisms in both single-parameter and multi-parameter settings. In particular, our results imply the first efficiently computable constant-factor approximations to the Bayesian optimal revenue in certain multi-parameter settings. In the classical prophet inequality, a gambler observes a sequence of stochastic rewards V1, ..., Vn and must decide, for each reward Vi, whether to keep it and stop the game or to forfeit the reward forever and reveal the next value Vi. The gambler's goal is to obtain a constant fraction of the expected reward that the optimal offline algorithm would get. Recently, prophet inequalities have been generalized to settings where the gambler can choose k items, and, more generally, where he can choose any independent set in a matroid. However, all the existing algorithms require the gambler to know the distribution from which the rewards V1, ..., Vn are drawn. The assumption that the gambler knows the distribution from which V1, ..., Vn are drawn is very strong. Instead, we work with the much simpler assumption that the gambler only knows a few samples from this distribution. We construct the first single-sample prophet inequalities for many settings of interest, whose guarantees all match the best possible asymptotically, even with full knowledge of the distribution. Specifically, we provide a novel single-sample algorithm when the gambler can choose any k elements whose analysis is based on random walks with limited correlation. In addition, we provide a black-box method for converting specific types of solutions to the related secretary problem to single-sample prophet inequalities, and apply it to several existing algorithms. Finally, we provide a constant-sample prophet inequality for constant-degree bipartite matchings. In addition, we apply these results to design the first posted-price and multi-dimensional auction mechanisms with limited information in settings with asymmetric bidders. Connections between prophet inequalities and posted-price mechanisms are already known, but applying the existing framework requires knowledge of the underlying distributions, as well as the so-called "virtual values" even when the underlying prophet inequalities do not. We therefore provide an extension of this framework that bypasses virtual values altogether, allowing our mechanisms to take full advantage of the limited information required by our new prophet inequalities. Online contention resolution schemes (OCRSs) were proposed by Feldman, Svensson, and Zenklusen as a generic technique to round a fractional solution in the matroid polytope in an online fashion. It has found applications in several stochastic combinatorial problems where there is a commitment constraint: on seeing the value of a stochastic element, the algorithm has to immediately and irrevocably decide whether to select it while always maintaining an independent set in the matroid. Although OCRSs immediately lead to prophet inequalities, these prophet inequalities are not optimal. Can we instead use prophet inequalities to design optimal OCRSs? We design the first optimal @math -OCRS for matroids by reducing the problem to designing a matroid prophet inequality where we compare to the stronger benchmark of an ex-ante relaxation. We also introduce and design optimal @math -random order CRSs for matroids, which are similar to OCRSs but the arrival is chosen uniformly at random.