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 \(\textsf{PPA}\) -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 \(\textsf{UEOPL}\) [Chiu, Choudhary and Mulzer, ICALP’20]. We define the analogue of this well-separation 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. Since this version of necklace splitting reduces to \(\alpha \) -Ham Sandwich in a solution-preserving way it follows that instances of this version always have unique solutions. 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 \(2^{O(\ell \log \ell )}+O(m^2)\) . 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 \(\textsf{coNP}\) -complete [Bergold et al., SWAT’22].
