A package query returns a package - a multiset of tuples - that maximizes or minimizes a linear objective function subject to linear constraints, thereby enabling in-database decision support. Prior work has established the equivalence of package queries to Integer Linear Programs (ILPs) and developed the SketchRefine algorithm for package query processing. While this algorithm was an important first step toward supporting prescriptive analytics scalably inside a relational database, it struggles when the data size grows beyond a few hundred million tuples or when the constraints become very tight. In this paper, we present Progressive Shading, a novel algorithm for processing package queries that can scale efficiently to billions of tuples and gracefully handle tight constraints. Progressive Shading solves a sequence of optimization problems over a hierarchy of relations, each resulting from an ever-finer partitioning of the original tuples into homogeneous groups until the original relation is obtained. This strategy avoids the premature discarding of high-quality tuples that can occur with SketchRefine. Our novel partitioning scheme, Dynamic Low Variance, can handle very large relations with multiple attributes and can dynamically adapt to both concentrated and spread-out sets of attribute values, provably outperforming traditional partitioning schemes such as KD-tree. We further optimize our system by replacing our off-the-shelf optimization software with customized ILP and LP solvers, called Dual Reducer and Parallel Dual Simplex respectively, that are highly accurate and orders of magnitude faster.

翻译：包查询返回一个包（即元组的多重集），该查询在满足线性约束条件下最大化或最小化线性目标函数，从而实现数据库内的决策支持。先前研究确立了包查询与整数线性规划（ILP）的等价性，并开发了用于包查询处理的SketchRefine算法。尽管该算法是迈向在关系数据库中可扩展地支持规范性分析的重要第一步，但当数据规模超过数亿个元组或约束条件变得非常严格时，其性能会显著下降。本文提出Progressive Shading算法，这是一种新颖的包查询处理方法，能够高效扩展至十亿级元组，并优雅地处理严格约束。Progressive Shading通过求解一系列基于关系层级结构的优化问题——其中每一层关系由原始元组按不断细化的同质分组划分得到，直至恢复原始关系——从而避免SketchRefine算法中优质元组被过早丢弃的问题。我们提出的新型划分方案——动态低方差（Dynamic Low Variance），能够处理包含多个属性的超大规模关系，并动态适应属性值集中与分散两种分布模式，在理论上优于KD树等传统划分方案。我们进一步通过定制化ILP和线性规划（LP）求解器（分别称为Dual Reducer和Parallel Dual Simplex）替代现成优化软件来优化系统，这些求解器具有高精度且速度快数个数量级。