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 $\varepsilon$.

翻译：我们重新审视经典潘多拉魔盒（PB）问题，并考虑盒子价值服从相关分布的情形。近期工作（arXiv:1911.01632）针对该问题中按固定顺序访问盒子的受限策略类，给出了常数近似算法。本文研究可能根据已观测价值自适应选择下一盒子的最优策略的近似复杂性。我们的主要结果建立了PB问题与随机优化中广泛研究的均匀决策树（UDT）问题及最小和集合覆盖变体（$\text{MSSC}_f$）之间的保近似等价性。对于支撑集大小为$m$的分布，UDT问题存在$\log m$近似算法，而多项式时间内常数因子近似是长期未解决问题，但亚指数时间内可实现常数因子近似（arXiv:1906.11385）。我们的主要结果表明PB与$\text{MSSC}_f$具有相同性质。我们还研究了当价值分布以$m$个乘积分布混合形式更简洁给出时的情形，该问题与更具挑战性的含噪最优决策树变体相关。我们给出一个常数因子近似算法，当每个盒子的混合分量在总变差距离上相同或间隔至少$\varepsilon$时，其运行时间为$n^{ \tilde O( m^2/\varepsilon^2 ) }$。