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，其中用户从由 $N$ 个非共谋服务器复制的 $K$ 条消息的数据库中检索 $D$ 条消息，且需求被限制为已知的结构化 $D$ 子集族。该公式概括了经典私有信息检索（PIR）和多消息私有信息检索（MPIR），并涵盖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速率和最小子分组化水平的逆向界，并开发了一个基于优化的框架用于构建通用结构化需求族的方案。我们的结果表明，对于某些族，PSSR 速率逆向界可能超过已知的最佳 MPIR 速率上界；当此 PSSR 界可达时，MPIR 速率最优方案对这些族变得次优。通过利用需求结构，我们的 PSSR 方案为许多族实现了更高的检索速率，且性能从未低于已知的最佳平衡 ${\{0,1\}}$-线性 MPIR 方案。我们的结果还表明，即使最优速率不变，需求结构也能减少所需子分组化程度。我们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，通过生成具有显著更小子分组化且无域大小限制的速率最优方案，改进了基于 MPIR 的方案。