Unfairly Splitting Separable Necklaces

Abstract

The Necklace Splitting problem is a classical problem in combinatorics that has been intensively studied both from a combinatorial and a computational point of view. It is well-known that the Necklace Splitting problem reduces to the discrete Ham Sandwich problem. This reduction was crucial in the proof of PPA-completeness of the Ham Sandwich problem. Recently, Borzechowski, Schnider and Weber [ISAAC'23] introduced a variant of Necklace Splitting that similarly reduces to the $α$-Ham Sandwich problem, which lies in the complexity class UEOPL but is not known to be complete. To make this reduction work, the input necklace is guaranteed to be n-separable. They showed that these necklaces can be fairly split in polynomial time and thus this subproblem cannot be used to prove UEOPL-hardness for $α$-Ham Sandwich. We consider the more general unfair necklace splitting problem on n-separable necklaces, i.e., the problem of splitting these necklaces such that each thief gets a desired fraction of each type of jewels. This more general problem is the natural necklace-splitting-type version of $α$-Ham Sandwich, and its complexity status is one of the main open questions posed by Borzechowski, Schnider and Weber. We show that the unfair splitting problem is also polynomial-time solvable, and can thus also not be used to show UEOPL-hardness for $α$-Ham Sandwich.

Figures (14)

• Figure 1: Necklaces with 3 colours $a$, $b$ and $c$. The convex hulls of each component are shown.
• Figure 2: Walk graphs of the examples in \ref{['fig:exampleOfSeparability']}.
• Figure 3: The graphs $N_7$ to the left and $N_8$ to the right.
• Figure 4: Turning the walk graph to the label graph for $n=7$ (left) and $n=8$ (right). The coloured edges and vertices are added to the walk graph to obtain the label graph.
• Figure 5: Two traversals in the label graph for a vertex $v$. The arrows indicate the order in which the edges are traversed: in the first traversal $e_1$ and $e_2$ and in the second $e_3$ and $e_4$.

Theorems & Definitions (71)

• Definition 1: Necklace
• Definition 2: $\alpha$-Cut
• Definition 3: $\alpha$-Necklace-Splitting
• Theorem 4
• Theorem 5
• Definition 6: Separability
• Definition 7
• Lemma 8: nseparableNecklaces
• Lemma 9: $\alpha$-Ham-Sandwich Theorem, originalDiscreteAlphaHS
• Theorem 10