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（元组的多重集），该包在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SketchRefine算法用于包查询处理。尽管该算法作为在关系数据库内可扩展支持规范分析的重要初步尝试，但当数据规模超过数亿元组或约束条件变得非常严格时，其性能会显著下降。本文提出渐进着色（Progressive Shading）算法，这是一种能高效扩展至十亿级元组并优雅处理严格约束的包查询处理方法。该算法通过在一组层级关系上求解一系列优化问题来实现——每个层级关系均由原始元组按同质性分组的逐步细化产生，直至还原为原始关系。这种策略避免了SketchRefine算法可能出现的过早丢弃高质量元组问题。我们提出的新型分区方案——动态低方差（Dynamic Low Variance）——能处理含多属性的超大型关系，并动态适应属性值的聚集与分散分布，理论上优于KD树等传统分区方案。通过定制化ILP与LP求解器（分别称为Dual Reducer和Parallel Dual Simplex）替换现成优化软件，我们进一步优化系统性能，这些求解器兼具高精度与数量级的性能提升。