6,951 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
Advances on Matroid Secretary Problems: Free Order Model and Laminar Case
The most well-known conjecture in the context of matroid secretary problems
claims the existence of a constant-factor approximation applicable to any
matroid. Whereas this conjecture remains open, modified forms of it were shown
to be true, when assuming that the assignment of weights to the secretaries is
not adversarial but uniformly random (Soto [SODA 2011], Oveis Gharan and
Vondr\'ak [ESA 2011]). However, so far, there was no variant of the matroid
secretary problem with adversarial weight assignment for which a
constant-factor approximation was found. We address this point by presenting a
9-approximation for the \emph{free order model}, a model suggested shortly
after the introduction of the matroid secretary problem, and for which no
constant-factor approximation was known so far. The free order model is a
relaxed version of the original matroid secretary problem, with the only
difference that one can choose the order in which secretaries are interviewed.
Furthermore, we consider the classical matroid secretary problem for the
special case of laminar matroids. Only recently, a constant-factor
approximation has been found for this case, using a clever but rather involved
method and analysis (Im and Wang, [SODA 2011]) that leads to a
16000/3-approximation. This is arguably the most involved special case of the
matroid secretary problem for which a constant-factor approximation is known.
We present a considerably simpler and stronger -approximation, based on reducing the problem to a matroid secretary
problem on a partition matroid
Simple Mechanisms for a Subadditive Buyer and Applications to Revenue Monotonicity
We study the revenue maximization problem of a seller with n heterogeneous
items for sale to a single buyer whose valuation function for sets of items is
unknown and drawn from some distribution D. We show that if D is a distribution
over subadditive valuations with independent items, then the better of pricing
each item separately or pricing only the grand bundle achieves a
constant-factor approximation to the revenue of the optimal mechanism. This
includes buyers who are k-demand, additive up to a matroid constraint, or
additive up to constraints of any downwards-closed set system (and whose values
for the individual items are sampled independently), as well as buyers who are
fractionally subadditive with item multipliers drawn independently. Our proof
makes use of the core-tail decomposition framework developed in prior work
showing similar results for the significantly simpler class of additive buyers
[LY13, BILW14].
In the second part of the paper, we develop a connection between
approximately optimal simple mechanisms and approximate revenue monotonicity
with respect to buyers' valuations. Revenue non-monotonicity is the phenomenon
that sometimes strictly increasing buyers' values for every set can strictly
decrease the revenue of the optimal mechanism [HR12]. Using our main result, we
derive a bound on how bad this degradation can be (and dub such a bound a proof
of approximate revenue monotonicity); we further show that better bounds on
approximate monotonicity imply a better analysis of our simple mechanisms.Comment: Updated title and body to version included in TEAC. Adapted Theorem
5.2 to accommodate \eta-BIC auctions (versus exactly BIC
Single-Sample Prophet Inequalities via Greedy-Ordered Selection
We study single-sample prophet inequalities (SSPIs), i.e., prophet inequalities where only a single sample from each prior distribution is available. Besides a direct, and optimal, SSPI for the basic single choice problem [Rubinstein et al., 2020], most existing SSPI results were obtained via an elegant, but inherently lossy reduction to order-oblivious secretary (OOS) policies [Azar et al., 2014]. Motivated by this discrepancy, we develop an intuitive and versatile greedy-based technique that yields SSPIs directly rather than through the reduction to OOSs. Our results can be seen as generalizing and unifying a number of existing results in the area of prophet and secretary problems. Our algorithms significantly improve on the competitive guarantees for a number of interesting scenarios (including general matching with edge arrivals, bipartite matching with vertex arrivals, and certain matroids), and capture new settings (such as budget additive combinatorial auctions). Complementing our algorithmic results, we also consider mechanism design variants. Finally, we analyze the power and limitations of different SSPI approaches by providing a partial converse to the reduction from SSPI to OOS given by Azar et al.</p
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