An Exact algorithm of searching for the largest size cluster in an integer sequence 2-clustering problem

Alexander Kel’manov, Sergey Khamidullin, Vladimir Khandeev, Artem Pyatkin

Research output: Chapter in Book/Report/Conference proceedingConference contributionResearchpeer-review

Abstract

A problem of partitioning a finite sequence of points in Euclidean space into two subsequences (clusters) maximizing the size of the first cluster subject to two constraints is considered. The first constraint deals with every two consecutive indices of elements of the first cluster: the difference between them is bounded from above and below by some constants. The second one restricts the value of a quadratic clustering function that is the sum of the intracluster sums over both clusters. The intracluster sum is the sum of squared distances between cluster elements and the cluster center. The center of the first cluster is unknown and determined as the centroid (i.e. as the mean value of its elements), while the center of the second one is zero. The strong NP-hardness of the problem is shown and an exact algorithm is suggested for the case of integer coordinates of input points. If the space dimension is bounded by some constant this algorithm runs in a pseudopolynomial time.

Original languageEnglish
Title of host publicationOptimization and Applications - 9th International Conference, OPTIMA 2018, Revised Selected Papers
EditorsYury Kochetov, Michael Khachay, Yury Evtushenko, Vlasta Malkova, Mikhail Posypkin, Milojica Jacimovic
PublisherSpringer-Verlag GmbH and Co. KG
Pages131-143
Number of pages13
ISBN (Print)9783030109332
DOIs
Publication statusPublished - 1 Jan 2019
Event9th International Conference on Optimization and Applications, OPTIMA 2018 - Petrovac, Montenegro
Duration: 1 Oct 20185 Oct 2018

Publication series

NameCommunications in Computer and Information Science
Volume974
ISSN (Print)1865-0929

Conference

Conference9th International Conference on Optimization and Applications, OPTIMA 2018
CountryMontenegro
CityPetrovac
Period01.10.201805.10.2018

Keywords

  • 2-partition
  • Euclidean space
  • Exact algorithm
  • Fixed space dimension
  • Integer coordinates
  • Longest subsequence
  • NP-hard problem
  • Pseudopolynomial running time
  • Quadratic variance
  • Sequence

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