  We study the fair allocation of a cake, which serves as a metaphor for a
divisible resource, under the requirement that each agent should receive a
contiguous piece of the cake. While it is known that no finite envy-free
algorithm exists in this setting, we exhibit efficient algorithms that produce
allocations with low envy among the agents. We then establish NP-hardness
results for various decision problems on the existence of envy-free
allocations, such as when we fix the ordering of the agents or constrain the
positions of certain cuts. In addition, we consider a discretized setting where
indivisible items lie on a line and show a number of hardness results extending
and strengthening those from prior work. Finally, we investigate connections
between approximate and exact envy-freeness, as well as between continuous and
discrete cake cutting.
