In pliable index coding (PICOD), a number of clients are connected via a noise-free broadcast channel to a server which has a list of messages. Each client has a unique subset of messages at the server as side-information and requests for any one message not in the side-information. A PICOD scheme of length $\ell$ is a set of $\ell$ encoded transmissions broadcast from the server such that all clients are satisfied. Finding the optimal (minimum) length of PICOD and designing PICOD schemes that have small length are the fundamental questions in PICOD. In this paper, we use a hypergraph-based approach to derive new achievability and converse results for PICOD. We present an algorithm which gives an achievable scheme for PICOD with length at most $\Delta(\mathcal{H})$, where $\Delta(\mathcal{H})$ is the maximum degree of any vertex in a hypergraph that represents the PICOD problem. We also give a lower bound for the optimal PICOD length using a new structural parameter associated with the PICOD hypergraph called the nesting number. Finally, we identify a class of problems for which our converse is tight, and also characterize the optimal PICOD lengths of problems with $\Delta(\mathcal{H})\in\{1,2,3\}$.

翻译：在可信赖的索引编码(PICOD)中,一些客户通过无噪音广播频道连接到一个服务器,该服务器有一份信息列表。每个客户在服务器上有一个独特的信息子集,作为侧信息,并请求任何非侧信息。一个长度为$\ell$(美元)的 PICOD 计划是一套由服务器播放的编码传输,使所有客户都满意。在代表 PICOD 问题的高空测量中,找到PICOD 和设计短长度的 PICOD 计划的最佳(最小)长度(最小)是PICOD 的基本问题。在本文中,我们使用基于高空的超文本方法为 PICOD 生成新的可接收性和反向结果。我们提出了一个算法,它为 PICOD 提供了可实现的方案,其长度最多为$\delta(mathcal{H}$( mathcalca{H} $(最小) 是代表 PICOD 问题的任何顶点的最大(最小) 度。我们用新的结构参数来设定最理想的 PICODD 的长度,最后是PICD 类的 和最接近的 。