  Cake cutting is a classic model for studying fair division of a
heterogeneous, divisible resource among agents with individual preferences.
Addressing cake division under a typical requirement that each agent must
receive a connected piece of the cake, we develop approximation algorithms for
finding envy-free (fair) cake divisions. In particular, this work improves the
state-of-the-art additive approximation bound for this fundamental problem. Our
results hold for general cake division instances in which the agents'
valuations satisfy basic assumptions and are normalized (to have value $1$ for
the cake). Furthermore, the developed algorithms execute in polynomial time
under the standard Robertson-Webb query model.
  Prior work has shown that one can efficiently compute a cake division (with
connected pieces) in which the additive envy of any agent is at most $1/3$. An
efficient algorithm is also known for finding connected cake divisions that are
(almost) $1/2$-multiplicatively envy-free. Improving the additive approximation
guarantee and maintaining the multiplicative one, we develop a polynomial-time
algorithm that computes a connected cake division that is both
$\left(\frac{1}{4} +o(1) \right)$-additively envy-free and $\left(\frac{1}{2} -
o(1) \right)$-multiplicatively envy-free. Our algorithm is based on the ideas
of interval growing and envy-cycle-elimination.
  In addition, we study cake division instances in which the number of distinct
valuations across the agents is parametrically bounded. We show that such cake
division instances admit a fully polynomial-time approximation scheme for
connected envy-free cake division.
