Efficient splitting of necklaces

We provide efficient approximation algorithms for the Necklace Splitting problem. The input consists of a sequence of beads of n types and an integer k. The objective is to split the necklace, with a small number of cuts made between consecutive beads, and distribute the resulting intervals into k collections so that the discrepancy between the shares of any two collections, according to each type, is at most 1. We also consider an approximate version where each collection should contain at least a (1 - ∈)/k and at most a (1 + ∈)/k fraction of the beads of each type. It is known that there is always a solution making at most n(k - 1) cuts, and this number of cuts is optimal in general. The proof is topological and provides no efficient procedure for finding these cuts. It is also known that for k = 2, and some fixed positive ∈, finding a solution with n cuts is PPAD-hard. We describe an efficient algorithm that produces an ∈-approximate solution for k = 2 making n(2+log(1/∈)) cuts. This is an exponential improvement of a (1/∈)O(n) bound of Bhatt and Leighton from the 80s. We also present an online algorithm for the problem (in its natural online model), in which the number of cuts made to produce discrepancy at most 1 on each type is O(m2/3n), where m is the maximum number of beads of any type. Lastly, we establish a lower bound showing that for the online setup this is tight up to logarithmic factors. Similar results are obtained for k > 2.