  We study the problem of fairly dividing a heterogeneous resource, commonly
known as cake cutting and chore division, in the presence of strategic agents.
While a number of results in this setting have been established in previous
works, they rely crucially on the free disposal assumption, meaning that the
mechanism is allowed to throw away part of the resource at no cost. In the
present work, we remove this assumption and focus on mechanisms that always
allocate the entire resource. We exhibit a truthful and envy-free mechanism for
cake cutting and chore division for two agents with piecewise uniform
valuations, and we complement our result by showing that such a mechanism does
not exist when certain additional constraints are imposed on the mechanisms.
Moreover, we provide bounds on the efficiency of mechanisms satisfying various
properties, and give truthful mechanisms for multiple agents with restricted
classes of valuations.
