我们研究经典先知不等式的最小化对应问题，常称为最小先知或成本先知不等式。与最大化设置不同，最小化设置更难，即使对于独立同分布变量也有指数下界。我们研究多选择松弛，目标是以常数竞争比最小化预期选择数。对于对抗顺序，常数竞争比需要近乎线性数量的选择；对于先知秘书模型，可实现常数竞争力且预期选择数指数级更小。我们给出精细分析并定义\(M\)，给出在先知秘书模型中以\(O(\min\{\ln \ln M, \ln n\})\)预期选择数实现常数竞争比的算法，证明其紧性，还考虑了算法和对手都选\(r\)个值并支付总和的多单元变体，将技术扩展到独立同分布变量的多单元变体以实现常数竞争比和少量预期选择数。
英文摘要：
We study the minimization counterpart of the classic prophet inequality, often termed the min prophet or cost prophet inequality. Unlike the maximization setting, where simple threshold algorithms achieve half of the prophet's value, the minimization setting is significantly harder, with an exponential lower bound even for i.i.d.\ variables.
We study a multi-choice relaxation in which the algorithm may select multiple variables and gets to choose the best amongst them (the minimum amongst those selected). Our goal is to minimize the expected number of selections while achieving a constant competitive ratio. For adversarial order, we show that a constant competitive ratio requires a nearly linear number of choices in expectation, ergo, $Ω(n/\ln n)$. In contrast, we show that for the prophet secretary model (random order) one can attain constant competitiveness while requiring only an exponentially smaller expected number of choices i.e. $O(\ln n)$. We give a refined analysis and define $M$ to be the ratio of the minimum expected value of any single variable to the expected minimum value of all variables (the prophet's value) and present an algorithm that achieves a constant competitive ratio with $O(\min\{\ln \ln M, \ln n\})$ choices in expectation for the prophet secretary. We show that this is tight up to low order log factors even for the special case of the i.i.d. model. We also show that if we insist on a deterministic bound on the number of choices then every constant competitive algorithm requires $n$ choices. This holds even in the i.i.d.\ setting
Finally, we consider a variant where both the algorithm and the adversary choose $r$ values and pay their sum, this is the minimization multi unit version. We extend our techniques to the multi-unit variant for i.i.d.\ variables, achieving a constant competitive ratio with a small expected number of choices.
Threshold Dynamics and Correlated Prophet Inequalities
Universidad de Chile（智利大学）；TNG Technology Consulting（TNG技术咨询）；Moroccan Center for Game Theory, Université Mohammed VI Polytechnique（摩洛哥博弈论中心，穆罕默德六世polytechnique大学）；Center for Mathematical Modeling & Department of Computer Science, University of Southern California（南加州大学数学建模中心与计算机科学系）；Department of Mathematics and Computer Science, University of Cologne（科隆大学数学与计算机科学系）；Institute for Mathematical and Computational Engineering, and Department of Industrial and Systems Engineering, Pontificia Universidad Católica de Chile（天主教智利大学数学与计算工程研究所及工业与系统工程系）
