  We provide approximation algorithms for two problems, known as NECKLACE
SPLITTING and $\epsilon$-CONSENSUS SPLITTING. In the problem
$\epsilon$-CONSENSUS SPLITTING, there are $n$ non-atomic probability measures
on the interval $[0, 1]$ and $k$ agents. The goal is to divide the interval,
via at most $n (k-1)$ cuts, into pieces and distribute them to the $k$ agents
in an approximately equitable way, so that the discrepancy between the shares
of any two agents, according to each measure, is at most $2 \epsilon / k$. It
is known that this is possible even for $\epsilon = 0$. NECKLACE SPLITTING is a
discrete version of $\epsilon$-CONSENSUS SPLITTING. For $k = 2$ and some
absolute positive constant $\epsilon$, both of these problems are PPAD-hard.
  We consider two types of approximation. The first provides every agent a
positive amount of measure of each type under the constraint of making at most
$n (k - 1)$ cuts. The second obtains an approximately equitable split with as
few cuts as possible. Apart from the offline model, we consider the online
model as well, where the interval (or necklace) is presented as a stream, and
decisions about cutting and distributing must be made on the spot.
  For the first type of approximation, we describe an efficient algorithm that
gives every agent at least $\frac{1}{nk}$ of each measure and works even
online. For the second type of approximation, we provide an efficient online
algorithm that makes $\text{poly}(n, k, \epsilon)$ cuts and an offline
algorithm making $O(nk \log \frac{k}{\epsilon})$ cuts. We also establish lower
bounds for the number of cuts required in the online model for both problems
even for $k=2$ agents, showing that the number of cuts in our online algorithm
is optimal up to a logarithmic factor.
