  This work develops algorithmic results for the classic cake-cutting problem
in which a divisible, heterogeneous resource (modeled as a cake) needs to be
partitioned among agents with distinct preferences. We focus on a standard
formulation of cake cutting wherein each agent must receive a contiguous piece
of the cake. While multiple hardness results exist in this setup for finding
fair/efficient cake divisions, we show that, if the value densities of the
agents satisfy the monotone likelihood ratio property (MLRP), then strong
algorithmic results hold for various notions of fairness and economic
efficiency.
  Addressing cake-cutting instances with MLRP, first we develop an algorithm
that finds cake divisions (with connected pieces) that are envy-free, up to an
arbitrary precision. The time complexity of our algorithm is polynomial in the
number of agents and the bit complexity of an underlying Lipschitz constant. We
obtain similar positive results for maximizing social (utilitarian) and
egalitarian welfare. In addition, we show that, under MLRP, the problem of
maximizing Nash social welfare admits a fully polynomial-time approximation
scheme (FPTAS).
  Many distribution families bear MLRP. In particular, this property holds if
all the value densities belong to any one of the following families: Gaussian
(with the same variance), linear, binomial, Poisson, and exponential
distributions. Furthermore, it is known that linear translations of any
log-concave function satisfy MLRP. Therefore, our results also hold when the
value densities of the agents are linear translations of the following
(log-concave) distributions: Laplace, gamma, beta, Subbotin, chi-square,
Dirichlet, and logistic. Hence, through MLRP, the current work obtains novel
cake-cutting algorithms for multiple distribution families.
