  In the $\varepsilon$-Consensus-Halving problem, a fundamental problem in fair
division, there are $n$ agents with valuations over the interval $[0,1]$, and
the goal is to divide the interval into pieces and assign a label "$+$" or
"$-$" to each piece, such that every agent values the total amount of "$+$" and
the total amount of "$-$" almost equally. The problem was recently proven by
Filos-Ratsikas and Goldberg [2019] to be the first "natural" complete problem
for the computational class PPA, answering a decade-old open question.
  In this paper, we examine the extent to which the problem becomes easy to
solve, if one restricts the class of valuation functions. To this end, we
provide the following contributions. First, we obtain a strengthening of the
PPA-hardness result of [Filos-Ratsikas and Goldberg, 2019], to the case when
agents have piecewise uniform valuations with only two blocks. We obtain this
result via a new reduction, which is in fact conceptually much simpler than the
corresponding one in [Filos-Ratsikas and Goldberg, 2019]. Then, we consider the
case of single-block (uniform) valuations and provide a parameterized
polynomial time algorithm for solving $\varepsilon$-Consensus-Halving for any
$\varepsilon$, as well as a polynomial-time algorithm for $\varepsilon=1/2$.
Finally, an important application of our new techniques is the first hardness
result for a generalization of Consensus-Halving, the Consensus-$1/k$-Division
problem [Simmons and Su, 2003]. In particular, we prove that
$\varepsilon$-Consensus-$1/3$-Division is PPAD-hard.
