一种面向层级查询的统一算法
A Unifying Algorithm for Hierarchical Queries
AI总结：
本文提出一种统一算法，证明层级查询同时界定概率查询评估、Shapley 值计算和新提出的袋-集合最大化问题的易处理性边界，并基于 2-幺半群抽象代数结构为层级查询给出统一的多项式时间解法。
摘要：
层级查询类已知界定了以下两个关于无自连接布尔合取查询（SJF-BCQ）的广泛研究问题在易处理性与难处理性之间的二分边界：（i）在元组独立概率数据库上评估一个 SJF-BCQ；（ii）在一个使 SJF-BCQ 求值为真的数据库中计算某个事实的 Shapley 值。本文中，我们证明层级查询也界定了另一个自然算法问题在易处理性与难处理性之间的二分边界，我们称该问题为“袋-集合最大化”问题。与一个 SJF-BCQ $Q$ 相关联的袋-集合最大化问题要求：给定一个数据库 $\cal D$，在通过向 $\cal D$ 添加来自另一个给定数据库 $\cal D^r$ 的最多 $θ$ 个事实而得到的数据库 $\cal D'$ 上，找出 $Q$ 在袋语义下取得的最大值。对于非层级查询，我们证明袋-集合最大化问题是一个 NP 完全优化问题。更重要的是，对于层级查询，我们证明上述三个问题（概率查询评估、Shapley 值计算和袋-集合最大化）都接受一个统一的多项式时间算法，该算法运行在一个称为“2-幺半群”的抽象代数结构上。这三个问题各自需要针对手头问题定制不同的 2-幺半群实例化。
英文摘要：
The class of hierarchical queries is known to define the boundary of the dichotomy between tractability and intractability for the following two extensively studied problems about self-join free Boolean conjunctive queries (SJF-BCQ): (i) evaluating a SJF-BCQ on a tuple-independent probabilistic database; (ii) computing the Shapley value of a fact in a database on which a SJF-BCQ evaluates to true. Here, we establish that hierarchical queries define also the boundary of the dichotomy between tractability and intractability for a different natural algorithmic problem, which we call the "bag-set maximization" problem. The bag-set maximization problem associated with a SJF-BCQ $Q$ asks: given a database $\cal D$, find the biggest value that $Q$ takes under bag semantics on a database $\cal D'$ obtained from $\cal D$ by adding at most $θ$ facts from another given database $\cal D^r$. For non-hierarchical queries, we show that the bag-set maximization problem is an NP-complete optimization problem. More significantly, for hierarchical queries, we show that all three aforementioned problems (probabilistic query evaluation, Shapley value computation, and bag-set maximization) admit a single unifying polynomial-time algorithm that operates on an abstract algebraic structure, called a "2-monoid". Each of the three problems requires a different instantiation of the 2-monoid tailored for the problem at hand.
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