43 research outputs found
Optimal Single-Choice Prophet Inequalities from Samples
We study the single-choice Prophet Inequality problem when the gambler is
given access to samples. We show that the optimal competitive ratio of
can be achieved with a single sample from each distribution. When the
distributions are identical, we show that for any constant ,
samples from the distribution suffice to achieve the optimal competitive
ratio () within , resolving an open problem of
Correa, D\"utting, Fischer, and Schewior.Comment: Appears in Innovations in Theoretical Computer Science (ITCS) 202
Learning Reserve Prices in Second-Price Auctions
This paper proves the tight sample complexity of Second-Price Auction with
Anonymous Reserve, up to a logarithmic factor, for all value distribution
families that have been considered in the literature. Compared to Myerson
Auction, whose sample complexity was settled very recently in (Guo, Huang and
Zhang, STOC 2019), Anonymous Reserve requires much fewer samples for learning.
We follow a similar framework as the Guo-Huang-Zhang work, but replace their
information theoretical argument with a direct proof
Optimal Online Contention Resolution Schemes via Ex-Ante Prophet Inequalities
Online contention resolution schemes (OCRSs) were proposed by Feldman, Svensson, and Zenklusen [Moran Feldman et al., 2016] 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 1/2-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 (1-1/e)-random order CRSs for matroids, which are similar to OCRSs but the arrival order is chosen uniformly at random
Beating Greedy For Approximating Reserve Prices in Multi-Unit VCG Auctions
We study the problem of finding personalized reserve prices for unit-demand
buyers in multi-unit eager VCG auctions with correlated buyers. The input to
this problem is a dataset of submitted bids of buyers in a set of auctions.
The goal is to find a vector of reserve prices, one for each buyer, that
maximizes the total revenue across all auctions.
Roughgarden and Wang (2016) showed that this problem is APX-hard but admits a
greedy -approximation algorithm. Later, Derakhshan, Golrezai, and
Paes Leme (2019) gave an LP-based algorithm achieving a -approximation
for the (important) special case of the problem with a single-item, thereby
beating greedy. We show in this paper that the algorithm of Derakhshan et al.
in fact does not beat greedy for the general multi-item problem. This raises
the question of whether or not the general problem admits a
better-than- approximation.
In this paper, we answer this question in the affirmative and provide a
polynomial-time algorithm with a significantly better approximation-factor of
. Our solution is based on a novel linear programming formulation, for
which we propose two different rounding schemes. We prove that the best of
these two and the no-reserve case (all-zero vector) is a -approximation