Mittagsseminar (in cooperation with A. Steger, D. Steurer and B. Sudakov)

Date and Time: Thursday, November 02, 2023, 12:15 pm

Duration: 30 minutes

Location: CAB G51

Speaker: Simon Weber

(Comically bad) sandwiches and highly separable necklaces: Some FPT algorithms for necklace splitting

In the alpha-Ham Sandwich problem, one is given d point sets in R^d, and needs to find a hyperplane cutting off a certain number alpha_i of points from each of the point sets P_i. In general, this is of course not possible, but the assumption of well-separation guarantees a (unique!) solution. The computational complexity of finding this solution remains unknown, only UEOPL-membership is known. In this talk I discuss a natural approach to proving UEOPL-hardness of this problem: Like in the PPA-hardness proof of the classical Ham Sandwich problem, we wish to prove hardness by reducing from Necklace Splitting. To capture the well-separation assumption in the alpha-Ham Sandwich problem, we define a notion of separability in necklaces. However, we then show that necklace splitting on the highly separable necklaces corresponding to well-separated point sets is surprisingly easy: We give FPT algorithms (with separability as parameter) for both checking this separability and for solving necklace splitting. This is joint work with Michaela Borzechowski (FU Berlin) and Patrick Schnider.

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