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].

翻译：众所周知，双贼项链分割问题可归约为离散火腿三明治问题。事实上，这一归约对证明火腿三明治问题的PPA完全性至关重要[Filos-Ratsikas and Goldberg, STOC'19]。近期，一种称为$\alpha$-火腿三明治问题的变体被研究，其中点集被保证具有良好分离性[Steiger and Zhao, DCG'10]。该搜索问题的复杂性仍未知，但已知其属于UEOPL复杂性类[Chiu, Choudhary and Mulzer, ICALP'20]。我们在项链分割问题中定义了这种良好分离条件的对应概念：若对于$n$种宝石类型的任意子集$A$，存在至多$n$个分割点可将$A$与类型$[n]\setminus A$分离，则称该项链是$n$-可分离的。通过归约到火腿三明治问题可知，该版本项链分割问题具有唯一解。此外，我们提供了两个FPT算法：第一个FPT算法在时间复杂度$2^{O(\ell\log\ell)}+m^2$内，解决了具有$n$种宝石类型和$m$个总宝石的$(n-1+\ell)$-可分离项链上的双贼项链分割问题。特别地，这表明双贼项链分割问题在$n$-可分离项链上是多项式时间可解的。因此，试图通过双贼项链分割问题归约来证明$\alpha$-火腿三明治问题难解性的方法不可行。第二个FPT算法在时间复杂度$2^{O(\ell^2)}\cdot n^4$内，测试给定具有$n$种宝石类型项链的$(n-1+\ell)$-可分离性。特别地，尽管点集良好分离性测试是coNP完全的[Bergold et al., SWAT'22]，但$n$-可分离性可在多项式时间内完成测试。