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In pliable index coding (PICOD), a number of clients are connected via anoise-free broadcast channel to a server which has a list of messages. Eachclient has a unique subset of messages at the server as side-information andrequests for any one message not in the side-information. A PICOD scheme oflength $\ell$ is a set of $\ell$ encoded transmissions broadcast from theserver such that all clients are satisfied. Finding the optimal (minimum)length of PICOD and designing PICOD schemes that have small length are thefundamental questions in PICOD. In this paper, we use a hypergraph-basedapproach to derive new achievability and converse results for PICOD. We presentan algorithm which gives an achievable scheme for PICOD with length at most$\Delta(\mathcal{H})$, where $\Delta(\mathcal{H})$ is the maximum degree of anyvertex in a hypergraph that represents the PICOD problem. We also give a lowerbound for the optimal PICOD length using a new structural parameter associatedwith the PICOD hypergraph called the nesting number. We extend some of ourresults to the PICOD problem where each client demands $t$ messages, ratherthan just one. Finally, we identify a class of problems for which our converseis tight, and also characterize the optimal PICOD lengths of problems with$\Delta(\mathcal{H})\in\{1,2,3\}$.

Bounding the Optimal Length of Pliable Index Coding via a Hypergraph-based Approach