Efficient splitting of necklaces

Noga Alon, Andrei Graur

Research output: Chapter in Book/Report/Conference proceedingConference contribution

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 (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.

Original languageEnglish (US)
Title of host publication48th International Colloquium on Automata, Languages, and Programming, ICALP 2021
EditorsNikhil Bansal, Emanuela Merelli, James Worrell
PublisherSchloss Dagstuhl- Leibniz-Zentrum fur Informatik GmbH, Dagstuhl Publishing
ISBN (Electronic)9783959771955
DOIs
StatePublished - Jul 1 2021
Event48th International Colloquium on Automata, Languages, and Programming, ICALP 2021 - Virtual, Glasgow, United Kingdom
Duration: Jul 12 2021Jul 16 2021

Publication series

NameLeibniz International Proceedings in Informatics, LIPIcs
Volume198
ISSN (Print)1868-8969

Conference

Conference48th International Colloquium on Automata, Languages, and Programming, ICALP 2021
Country/TerritoryUnited Kingdom
CityVirtual, Glasgow
Period7/12/217/16/21

All Science Journal Classification (ASJC) codes

  • Software

Keywords

  • Approximation algorithms
  • Discrepancy
  • Necklace halving
  • Necklace splitting
  • Online algorithms

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