We design new minimal-subpacketization schemes for information-theoretic private information retrieval on graph-based replicated databases. In graph-based replication, the system consists of $K$ files replicated across $N$ servers according to a graph with $N$ vertices and $K$ edges. The client wants to retrieve one desired file, while keeping the index of the desired file private from each server via a query-response protocol. We seek PIR protocols that have (a) high rate, which is the ratio of the file-size to the total download cost, and (b) low subpacketization, which acts as a constraint on the size of the files for executing the protocol. We report two new schemes which have unit-subpacketization (which is minimal): (i) for a special class of graphs known as star graphs, and (ii) for general graphs. Our star-graph scheme has a better rate than previously known schemes with low subpacketization for general star graphs. Our scheme for general graphs uses a decomposition of the graph via independent sets. This scheme achieves a rate lower than prior schemes for the complete graph, however it can achieve higher rates than known for some specific graph classes. An extension of our scheme to the case of multigraphs achieves a higher rate than previous schemes for the complete multi-graph.

翻译：我们为基于图复制的数据库上的信息论私有信息检索设计了新的最小化子分组方案。在基于图的复制系统中，$K$ 个文件根据一个具有 $N$ 个顶点和 $K$ 条边的图，复制存储在 $N$ 个服务器上。客户端希望通过查询-响应协议检索一个所需文件，同时向每个服务器隐藏所需文件的索引。我们寻求满足以下条件的PIR协议：(a) 高码率，即文件大小与总下载开销的比值；(b) 低子分组数，这是对协议执行时文件大小的约束。我们报告了两种具有单位子分组（即最小子分组）的新方案：(i) 针对一类特殊的图（称为星图），以及 (ii) 针对一般图。我们的星图方案在码率上优于先前已知的、针对一般星图的低子分组方案。我们针对一般图的方案通过独立集对图进行分解。该方案在完全图上的码率低于先前的方案，但对于某些特定图类，它可以达到比已知方案更高的码率。我们将该方案扩展到多重图的情况，在完全多重图上实现了比先前方案更高的码率。