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（即元组的多重集合），在线性约束下最大化或最小化线性目标函数，从而支持数据库内决策分析。先前研究已证明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。尽管该算法是将规范分析可扩展地集成到关系数据库中的重要里程碑，但当数据规模超过数亿个元组或约束条件变得极为严格时，其性能会显著下降。本文提出渐进式阴影（Progressive Shading）算法——一种新型包查询处理方法，可高效扩展至数十亿级元组，且能优雅处理严格约束。该算法通过求解基于关系层次结构的优化问题序列实现：每次迭代对原始元组进行更细粒度的同构分组，直至恢复原始关系。这一策略避免了SketchRefine中可能过早丢弃高质量元组的问题。我们提出的动态低方差（Dynamic Low Variance）分区方案可处理包含多属性的大型关系，并能动态适应属性值的集中与离散分布，理论上优于KD树等传统分区方案。此外，我们通过定制化ILP与LP求解器（分别称之为Dual Reducer和Parallel Dual Simplex）替代现成优化软件，在保持高精度的同时实现数量级的速度提升。