我们重新审视了盒子价值服从相关分布的经典潘多拉魔盒（PB）问题。近期工作 arXiv:1911.01632 针对该问题中一类受限策略——即按固定顺序访问盒子的策略——得到了常数近似算法。在本工作中，我们研究近似最优策略的复杂性，该策略可以根据迄今为止已看到的价值，自适应地选择接下来访问哪个盒子。我们的主要结果建立了 PB 与随机优化中被充分研究的 Uniform Decision Tree（UDT，统一决策树）问题以及 Min-Sum Set Cover（最小和集合覆盖）问题的一个变体 MSSC_f 之间保持近似性的等价关系。对于支撑集大小为 m 的分布，UDT 存在 log m 近似；尽管在多项式时间内获得常数因子近似是一个长期未决的问题，但常数因子近似可以在次指数时间内实现（arXiv:1906.11385）。我们的主要结果意味着 PB 和 MSSC_f 也具有相同性质。我们还研究了价值分布以 m 个乘积分布的混合这一更简洁形式给出的情形。该问题同样与 Optimal Decision Tree（最优决策树）的一个噪声变体相关，而后者难度显著更高。当每个盒子上的混合分量要么相同，要么在 TV 距离上相差 ε 时，我们给出了一个运行时间为 n^{\ ilde O(m^2/ε^2)} 的常数因子近似算法。
英文摘要：
We revisit the classic Pandora's Box (PB) problem under correlated distributions on the box values. Recent work of arXiv:1911.01632 obtained constant approximate algorithms for a restricted class of policies for the problem that visit boxes in a fixed order. In this work, we study the complexity of approximating the optimal policy which may adaptively choose which box to visit next based on the values seen so far.
Our main result establishes an approximation-preserving equivalence of PB to the well studied Uniform Decision Tree (UDT) problem from stochastic optimization and a variant of the Min-Sum Set Cover ($\text{MSSC}_f$) problem. For distributions of support $m$, UDT admits a $\log m$ approximation, and while a constant factor approximation in polynomial time is a long-standing open problem, constant factor approximations are achievable in subexponential time (arXiv:1906.11385). Our main result implies that the same properties hold for PB and $\text{MSSC}_f$.
We also study the case where the distribution over values is given more succinctly as a mixture of $m$ product distributions. This problem is again related to a noisy variant of the Optimal Decision Tree which is significantly more challenging. We give a constant-factor approximation that runs in time $n^{ \tilde O( m^2/\varepsilon^2 ) }$ when the mixture components on every box are either identical or separated in TV distance by $\varepsilon$.
Google Research（谷歌研究）；University of Maryland（马里兰大学）；Microsoft Research（微软研究院）；Harvard University（哈佛大学）
We consider online variations of the Pandora's box problem (Weitzman. 1979), a standard model for understanding issues related to the cost of acquiring information for decision-making. Our problem generalizes both the classic Pandora's box problem and the prophet inequality framework. Boxes are presented online, each with a random value and cost drew jointly from some known distribution. Pandora chooses online whether to open each box given its cost, and then chooses irrevocably whether to keep the revealed prize or pass on it. We aim for approximation algorithms against adversaries that can c…
Pandora's Box is a central problem in decision making under uncertainty that can model various real life scenarios. In this problem we are given $n$ boxes, each with a fixed opening cost, and an unknown value drawn from a known distribution, only revealed if we pay the opening cost. Our goal is to find a strategy for opening boxes to minimize the sum of the value selected and the opening cost paid. In this work we revisit Pandora's Box when the value distributions are correlated, first studied in Chawla et al. (arXiv:1911.01632). We show that the optimal algorithm for the independent case, giv…
Weitzman (1979) introduced the Pandora Box problem as a model for sequential search with inspection costs, and gave an elegant index-based policy that attains provably optimal expected payoff. In various scenarios, the searching agent may select an option without making a costly inspection. The variant of the Pandora box problem with non-obligatory inspection has attracted interest from both economics and algorithms researchers. Various simple algorithms have proved suboptimal, with the best known 0.8-approximation algorithm due to Guha et al. (2008). No hardness result for the problem was kno…
University of Southern California（南加州大学）；Northwestern University（西北大学）；Data Science Institute at the University of Chicago（芝加哥大学数据科学研究所）；University of Chicago（芝加哥大学）
The Pandora's Box problem and its extensions capture optimization problems with stochastic input where the algorithm can obtain instantiations of input random variables at some cost. To our knowledge, all previous work on this class of problems assumes that different random variables in the input are distributed independently. As such it does not capture many real-world settings. In this paper, we provide the first approximation algorithms for Pandora's Box-type problems with correlations. We assume that the algorithm has access to samples drawn from the joint distribution on input. Algorithms…
Martin Weitzman's "Pandora's problem" furnishes the mathematical basis for optimal search theory in economics. Nearly 40 years later, Laura Doval introduced a version of the problem in which the searcher is not obligated to pay the cost of inspecting an alternative's value before selecting it. Unlike the original Pandora's problem, the version with nonobligatory inspection cannot be solved optimally by any simple ranking-based policy, and it is unknown whether there exists any polynomial-time algorithm to compute the optimal policy. This motivates the study of approximately optimal policies th…
Weitzman introduced Pandora's box problem as a mathematical model of sequential search with inspection costs, in which a searcher is allowed to select a prize from one of $n$ alternatives. Several decades later, Doval introduced a close version of the problem, where the searcher does not need to incur the inspection cost of an alternative, and can select it uninspected. Unlike the original problem, the optimal solution to the nonobligatory inspection variant is proved to need adaptivity, and by recent work of [FLL22], finding the optimal solution is NP-hard. Our first main result is a structur…
Institute of Computer Science and Lamarr Institute for Machine Learning and Artificial Intelligence, University of Bonn, Germany（计算机科学研究所和拉马尔机器学习与人工智能研究所，波恩大学，德国）
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