  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. We extend some of our
results to the PICOD problem where each client demands $t$ messages, rather
than just one. Finally, we identify a class of problems for which our converse
is tight, and also characterize the optimal PICOD lengths of problems with
$\Delta(\mathcal{H})\in\{1,2,3\}$.
