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Pandora's Box（PB）问题中盒子价值具有相关分布的情况。近期arXiv:1911.01632的工作针对该问题中按固定顺序访问盒子的受限策略类，获得了常数近似算法。本研究探讨了最优策略的近似复杂性——该策略可根据已观测到的价值自适应选择下一个待访问的盒子。我们的主要结果为PB问题与随机优化领域的经典Uniform Decision Tree（UDT）问题以及Min-Sum Set Cover（$\text{MSSC}_f$）问题的变体建立了保近似等价性。对于支持度为$m$的分布，UDT问题存在$\log m$近似算法，尽管多项式时间内实现常数因子近似仍是一个长期未决的公开问题，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和$\text{MSSC}_f$问题具有相同性质。我们还研究了价值分布以$m$个乘积分布混合形式更简洁呈现的情况。该问题同样与噪声变体的最优决策树相关，但其难度显著更高。当每个盒子的混合分量要么相同、要么在总变分距离上被$\varepsilon$分离时，我们给出一个运行时间为$n^{ \tilde O( m^2/\varepsilon^2 ) }$的常数因子近似算法。