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 classical Private Information Retrieval (PIR) and multi-message PIR (MPIR), and captures settings where the demand space is constrained by application-specific structure. Focusing on balanced ${\{0,1\}}$-linear schemes, we derive converse bounds on the maximum retrieval rate and minimum subpacketization level, and develop an optimization-based framework for constructing schemes for general structured demand families. Our results show that, for certain families, the PSSR rate converse bound can exceed the best-known MPIR rate upper bound; when this PSSR bound is achievable, MPIR rate-optimal schemes become suboptimal for those families. By exploiting demand structure, our PSSR schemes achieve higher retrieval rates for many families and never underperform the best-known balanced ${\{0,1\}}$-linear MPIR schemes. Our results also show that demand structure can reduce the required subpacketization even when the optimal rate is unchanged. 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（PSSR）”问题，其中用户从分布在N个非共谋服务器上的K条消息的数据库中检索D条消息，且需求受限为已知的结构化D子集族。该公式统一了经典的私有信息检索（PIR）和多消息PIR（MPIR），并刻画了需求空间受应用特定结构约束的场景。聚焦于平衡{0,1}-线性方案，我们推导了最大检索率与最小子分组化水平的对偶界，并开发了面向通用结构化需求族方案构建的优化框架。结果表明：对于特定需求族，PSSR率的对偶界可超越已知的MPIR率上界；当该PSSR界可达时，MPIR最优率方案对该族将退化为次优。通过利用需求结构，我们的PSSR方案对多数需求族实现了更高检索率，且从未劣于已知最优平衡{0,1}-线性MPIR方案。研究还表明，即使最优率保持不变，需求结构仍可降低所需子分组化水平。关于连续需求族的并行工作进一步展示了该框架的适用范围：通过获得具有显著更小子分组化且无域大小限制的最优率方案，改进基于MPIR的方案。