We introduce the \emph{Private Structured-Subset Retrieval (PSSR)} problem, where a user retrieves $D$ messages from a database of $K$ messages replicated across $N$ non-colluding servers, and the demand is restricted to a known structured family of $D$-subsets. This formulation generalizes Multi-message Private Information Retrieval (MPIR) and captures settings where the demand space is constrained by application-specific structure. Focusing on balanced ${\{0,1\}}$-linear schemes, a class that includes several best-known MPIR schemes, we derive converse bounds on the maximum retrieval rate and minimum subpacketization level required to achieve any given rate. We also develop an optimization-based framework to construct schemes for general structured demand families, providing flexibility in optimizing the retrieval rate or the subpacketization level. When specialized to the full demand family, this framework recovers known balanced $\{0,1\}$-linear MPIR constructions; for more restricted demand families, it can exploit the demand structure to increase the retrieval rate, reduce the subpacketization level, or both. We demonstrate this through a structured-demand example in which the proposed PSSR scheme simultaneously achieves a higher rate and requires a smaller subpacketization than the best-known MPIR scheme for the same parameters $N$, $K$, and $D$. Our parallel work on contiguous-demand families further illustrates the scope of this framework by yielding rate-optimal schemes with substantially smaller subpacketization and no field-size restrictions, improving upon MPIR-based schemes.

翻译：我们提出私有结构化子集检索（Private Structured-Subset Retrieval, PSSR）问题：用户从复制在$N$个非共谋服务器上的$K$个消息数据库中检索$D$条消息，且需求仅限于已知结构化$D$子集族。该公式推广了多消息私有信息检索（MPIR），并刻画了需求空间受应用特定结构约束的场景。针对包含若干最优MPIR方案的平衡${\{0,1\}}$-线性方案类，我们推导了最大检索速率的上界，以及实现任意给定速率所需的最小子包化水平。我们同时开发了基于优化的框架，用于构造面向一般结构化需求族的方案，在优化检索速率或子包化水平方面提供灵活性。当特化至完整需求族时，该框架重现了已知的平衡$\{0,1\}$-线性MPIR构造；对于更受限的需求族，它能利用需求结构提升检索速率、降低子包化水平，或兼得两者。我们通过结构化需求示例证明：在相同参数$N$、$K$、$D$下，所提出的PSSR方案能同时实现比最优MPIR方案更高的速率和更小的子包化。我们在连续需求族上的并行工作进一步展示了该框架的应用范围——通过构造无域大小限制且子包化水平显著降低的速率最优方案，改进了基于MPIR的方案。