186 research outputs found
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
The Outer Limits of Contention Resolution on Matroids and Connections to the Secretary Problem
Contention resolution schemes have proven to be a useful and unifying abstraction for a variety of constrained optimization problems, in both offline and online arrival models. Much of prior work restricts attention to product distributions for the input set of elements, and studies contention resolution for increasingly general packing constraints, both offline and online. In this paper, we instead focus on generalizing the input distribution, restricting attention to matroid constraints in both the offline and online random arrival models. In particular, we study contention resolution when the input set is arbitrarily distributed, and may exhibit positive and/or negative correlations between elements. We characterize the distributions for which offline contention resolution is possible, and establish some of their basic closure properties. Our characterization can be interpreted as a distributional generalization of the matroid covering theorem. For the online random arrival model, we show that contention resolution is intimately tied to the secretary problem via two results. First, we show that a competitive algorithm for the matroid secretary problem implies that online contention resolution is essentially as powerful as offline contention resolution for matroids, so long as the algorithm is given the input distribution. Second, we reduce the matroid secretary problem to the design of an online contention resolution scheme of a particular form
Combinatorial Stationary Prophet Inequalities
Numerous recent papers have studied the tension between thickening and
clearing a market in (uncertain, online) long-time horizon Markovian settings.
In particular, (Aouad and Sarita{\c{c}} EC'20, Collina et al. WINE'20, Kessel
et al. EC'22) studied what the latter referred to as the Stationary Prophet
Inequality Problem, due to its similarity to the classic finite-time horizon
prophet inequality problem. These works all consider unit-demand buyers.
Mirroring the long line of work on the classic prophet inequality problem
subject to combinatorial constraints, we initiate the study of the stationary
prophet inequality problem subject to combinatorially-constrained buyers.
Our results can be summarized succinctly as unearthing an algorithmic
connection between contention resolution schemes (CRS) and stationary prophet
inequalities. While the classic prophet inequality problem has a tight
connection to online CRS (Feldman et al. SODA'16, Lee and Singla ESA'18), we
show that for the stationary prophet inequality problem, offline CRS play a
similarly central role. We show that, up to small constant factors, the best
(ex-ante) competitive ratio achievable for the combinatorial prophet inequality
equals the best possible balancedness achievable by offline CRS for the same
combinatorial constraints
Simple and Optimal Online Contention Resolution Schemes for -Uniform Matroids
We provide a simple -selectable Online Contention
Resolution Scheme for -uniform matroids against a fixed-order adversary. If
and denote the set of selected elements and the set of realized
active elements among the first (respectively), our algorithm selects with
probability any active element such that . This implies
a prophet inequality against fixed-order
adversaries for -uniform matroids that is considerably simpler than previous
algorithms [Ala14, AKW14, JMZ22].
We also prove that no OCRS can be -selectable for -uniform matroids against an almighty adversary.
This guarantee is matched by the (known) simple greedy algorithm that accepts
every active element with probability
[HKS07].Comment: 26 pages, 15th Innovations in Theoretical Computer Science (ITCS
2024
Prophet Inequalities for Cost Minimization
Prophet inequalities for rewards maximization are fundamental to optimal
stopping theory with several applications to mechanism design and online
optimization. We study the cost minimization counterpart of the classical
prophet inequality, where one is facing a sequence of costs in an online manner and must stop at some point and take the last cost
seen. Given that the 's are independent, drawn from known distributions,
the goal is to devise a stopping strategy that minimizes the expected cost.
If the 's are not identically distributed, then no strategy can achieve
a bounded approximation if the arrival order is adversarial or random. This
leads us to consider the case where the 's are I.I.D.. For the I.I.D.
case, we give a complete characterization of the optimal stopping strategy, and
show that, if our distribution satisfies a mild condition, then the optimal
stopping strategy achieves a tight (distribution-dependent) constant-factor
approximation.
Our techniques provide a novel approach to analyze prophet inequalities,
utilizing the hazard rate of the distribution. We also show that when the
hazard rate is monotonically increasing (i.e. the distribution is MHR), this
constant is at most , and this is optimal for MHR distributions.
For the classical prophet inequality, single-threshold strategies can achieve
the optimal approximation factor. Motivated by this, we analyze
single-threshold strategies for the cost prophet inequality problem. We design
a threshold that achieves a
-factor approximation,
where the exponent in the logarithmic factor is a distribution-dependent
constant, and we show a matching lower bound.
We note that our results can be used to design approximately optimal posted
price-style mechanisms for procurement auctions which may be of independent
interest.Comment: 38 page
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Prophet Inequalities with Cancellation Costs
Most of the literature on online algorithms and sequential decision-making focuses on settings with “irrevocable decisions” where the algorithm’s decision upon arrival of the new input is set in stone and can never change in the future. One canonical example is the classic prophet inequality problem, where realizations of a sequence of independent random variables X1, X2,… with known distributions are drawn one by one and a decision maker decides when to stop and accept the arriving random variable, with the goal of maximizing the expected value of their pick. We consider “prophet inequalities with recourse” in the linear buyback cost setting, where after accepting a variable Xi, we can still discard Xi later and accept another variable Xj, at a buyback cost of f × Xi. The goal is to maximize the expected net reward, which is the value of the final accepted variable minus the total buyback cost. Our first main result is an optimal prophet inequality in the regime of f ≥ 1, where we prove that we can achieve an expected reward 1+f/1+2f times the expected offline optimum. The problem is still open for 0<f<1 and we give some partial results in this regime. In particular, as our second main result, we characterize the asymptotic behavior of the competitive ratio for small f and provide almost matching upper and lower bounds that show a factor of 1−Θ(flog(1/f)). Our results are obtained by two fundamentally different approaches: One is inspired by various proofs of the classical prophet inequality, while the second is based on combinatorial optimization techniques involving LP duality, flows, and cuts
Simple Mechanisms for Non-linear Agents
We consider agents with non-linear preferences given by private values and
private budgets. We quantify the extent to which posted pricing approximately
optimizes welfare and revenue for a single agent. We give a reduction framework
that extends the approximation of multi-agent pricing-based mechanisms from
linear utility to nonlinear utility. This reduction framework is broadly
applicable as Alaei et al. (2012) have shown that mechanisms for linear agents
can generally be interpreted as pricing-based mechanisms. We give example
applications of the framework to oblivious posted pricing (e.g., Chawla et al.,
2010), sequential posted pricing (e.g., Yan, 2011), and virtual surplus
maximization (Myerson, 1981)
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