Abstract
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 (1ε)/k and at most a (1+ε)/k fraction of the beads of each type. It is known that there is always a solution making at most n(k1) 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 ε, finding a solution with n cuts is PPADhard.
We describe an efficient algorithm that produces an εapproximate solution for k = 2 making n (2+log (1/ε)) cuts. This is an exponential improvement of a (1/ε)^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 Õ(m^{2/3} n), 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.
BibTeX  Entry
@InProceedings{alon_et_al:LIPIcs.ICALP.2021.14,
author = {Alon, Noga and Graur, Andrei},
title = {{Efficient Splitting of Necklaces}},
booktitle = {48th International Colloquium on Automata, Languages, and Programming (ICALP 2021)},
pages = {14:114:17},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {9783959771955},
ISSN = {18688969},
year = {2021},
volume = {198},
editor = {Bansal, Nikhil and Merelli, Emanuela and Worrell, James},
publisher = {Schloss Dagstuhl  LeibnizZentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/opus/volltexte/2021/14083},
URN = {urn:nbn:de:0030drops140832},
doi = {10.4230/LIPIcs.ICALP.2021.14},
annote = {Keywords: necklace splitting, necklace halving, approximation algorithms, online algorithms, discrepancy}
}
Keywords: 

necklace splitting, necklace halving, approximation algorithms, online algorithms, discrepancy 
Collection: 

48th International Colloquium on Automata, Languages, and Programming (ICALP 2021) 
Issue Date: 

2021 
Date of publication: 

02.07.2021 