An FPT Algorithm for Splitting a Necklace Among Two Thieves
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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 α-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]∖ 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+ℓ)-separable necklaces with n types of jewels and m total jewels in time 2^O(ℓlogℓ)+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 α-Ham Sandwich through reduction from the 2-Thief-Necklace-Splitting problem cannot work. The second FPT algorithm tests (n-1+ℓ)-separability of a given necklace with n types of jewels in time 2^O(ℓ^2)· 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].
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