Related papers: Unfairly Splitting Separable Necklaces
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,…
We resolve the computational complexity of two problems known as NECKLACE-SPLITTING and DISCRETE HAM SANDWICH, showing that they are PPA-complete. For NECKLACE SPLITTING, this result is specific to the important special case in which two…
The classic Ham-Sandwich theorem states that for any $d$ measurable sets in $\mathbb{R}^d$, there is a hyperplane that bisects them simultaneously. An extension by B\'ar\'any, Hubard, and Jer\'onimo [DCG 2008] states that if the sets are…
The classes PPA-$p$ have attracted attention lately, because they are the main candidates for capturing the complexity of Necklace Splitting with $p$ thieves, for prime $p$. However, these classes were not known to have complete problems of…
We prove several versions of N. Alon's "necklace-splitting theorem", subject to additional constraints, as illustrated by the following results. (1) The "almost equicardinal necklace-splitting theorem" claims that, without increasing the…
The famous Ham-Sandwich theorem states that any $d$ point sets in $\mathbb{R}^d$ can be simultaneously bisected by a single hyperplane. The $\alpha$-Ham-Sandwich theorem gives a sufficient condition for the existence of biased cuts, i.e.,…
We prove a common generalization of the Ham Sandwich theorem and Alon's Necklace Splitting theorem. Our main results show the existence of fair distributions of $m$ measures in $R^d$ among $r$ thieves using roughly $mr/d$ convex pieces,…
In the $\varepsilon$-Consensus-Halving problem, we are given $n$ probability measures $v_1, \dots, v_n$ on the interval $R = [0,1]$, and the goal is to partition $R$ into two parts $R^+$ and $R^-$ using at most $n$ cuts, so that $|v_i(R^+)…
We show that the computational problem CONSENSUS-HALVING is PPA-complete, the first PPA-completeness result for a problem whose definition does not involve an explicit circuit. We also show that an approximate version of this problem is…
We provide approximation algorithms for two problems, known as NECKLACE SPLITTING and $\epsilon$-CONSENSUS SPLITTING. In the problem $\epsilon$-CONSENSUS SPLITTING, there are $n$ non-atomic probability measures on the interval $[0, 1]$ and…
We address two variants of the classical necklace counting problem from enumerative combinatorics. In both cases, we fix a finite group $\mathcal{G}$ and a positive integer $n$. In the first variant, we count the ``identity-product…
A necklace splitting theorem of Goldberg and West asserts that any k-colored (continuous) necklace can be fairly split using at most k cuts. Motivated by the problem of Erd\H{o}s on strongly nonrepetitive sequences, Alon et al. proved that…
We study the consensus-halving problem of dividing an object into two portions, such that each of $n$ agents has equal valuation for the two portions. The $\epsilon$-approximate consensus-halving problem allows each agent to have an…
The necklace splitting problem is a classic problem in fair division with many applications, including data-informed fair hash maps. We extend necklace splitting to a dynamic setting, allowing for relocation, insertion, and deletion of…
The well-known "splitting necklace theorem" of Noga Alon says that each "necklace" having beads of n different colors can be fairly divided between k "thieves" by at most n(k-1) cuts. We demonstrate that Alon's result is a special case of a…
A necklace is an equivalence class of words of length $n$ over an alphabet under the cyclic shift (rotation) operation. As a classical object, there have been many algorithmic results for key operations on necklaces, including counting,…
A (continuous) necklace is simply an interval of the real line colored measurably with some number of colors. A well-known application of the Borsuk-Ulam theorem asserts that every $k$-colored necklace can be fairly split by at most $k$…
It is known that any open necklace with beads of $t$ types in which the number of beads of each type is divisible by $k$, can be partitioned by at most $(k-1)t$ cuts into intervals that can be distributed into $k$ collections, each…
This paper deals with two problems about splitting fairly a path with colored vertices, where "fairly" means that each part contains almost the same amount of vertices in each color. Our first result states that it is possible to remove one…
The well-known "necklace splitting theorem" of Alon asserts that every $k$-colored necklace can be fairly split into $q$ parts using at most $t$ cuts, provided $k(q-1)\leq t$. In a joint paper with Alon et al. we studied a kind of opposite…