  It is well-known that the 2-Thief-Necklace-Splitting problem reduces to the
discrete Ham Sandwich problem. In fact, this reduction was crucial in the proof
of the PPA-completeness of the Ham Sandwich problem [Filos-Ratsikas and
Goldberg, STOC'19]. Recently, a variant of the Ham Sandwich problem called
$\alpha$-Ham Sandwich has been studied, in which the point sets are guaranteed
to be well-separated [Steiger and Zhao, DCG'10]. The complexity of this search
problem remains unknown, but it is known to lie in the complexity class UEOPL
[Chiu, Choudhary and Mulzer, ICALP'20]. We define the analogue of this
well-separability condition in the necklace splitting problem -- a necklace is
$n$-separable, if every subset $A$ of the $n$ types of jewels can be separated
from the types $[n]\setminus A$ by at most $n$ separator points. By the
reduction to the Ham Sandwich problem it follows that this version of necklace
splitting has a unique solution.
  We furthermore provide two FPT algorithms: The first FPT algorithm solves
2-Thief-Necklace-Splitting on $(n-1+\ell)$-separable necklaces with $n$ types
of jewels and $m$ total jewels in time $2^{O(\ell\log\ell)}+m^2$. In
particular, this shows that 2-Thief-Necklace-Splitting is polynomial-time
solvable on $n$-separable necklaces. Thus, attempts to show hardness of
$\alpha$-Ham Sandwich through reduction from the 2-Thief-Necklace-Splitting
problem cannot work. The second FPT algorithm tests $(n-1+\ell)$-separability
of a given necklace with $n$ types of jewels in time $2^{O(\ell^2)}\cdot n^4$.
In particular, $n$-separability can thus be tested in polynomial time, even
though testing well-separation of point sets is coNP-complete [Bergold et al.,
SWAT'22].
