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——在满足线性约束的前提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。尽管该算法作为在关系数据库内可扩展地支持规范分析的重要初步尝试，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，它便难以应对。本文提出渐进式着色（Progressive Shading）——一种新型包查询处理算法，能够高效扩展至数十亿个元组，并优雅处理严格约束。该算法通过求解关系层次结构上的一系列优化问题来实现：每一步通过将原始元组划分为越来越细的同质组，直至恢复原始关系。此策略避免了SketchRefine可能发生的高质量元组过早丢弃问题。我们提出的新型分区方案——动态低方差（Dynamic Low Variance）——能够处理含多属性的大规模关系，并动态适应属性值的集中与分散分布，在理论上优于KD-Tree等传统分区方案。通过采用定制的ILP与线性规划求解器（分别称为Dual Reducer和Parallel Dual Simplex）替代现成优化软件，我们进一步优化系统性能，这些求解器具有高精度且速度快数个数量级。