A PTAS for the Cluster Editing Problem on Planar Graphs

Research output: Chapter in Book/Report/Conference proceedingChapterAcademic

Abstract

The goal of the cluster editing problem is to add or delete a minimum number of edges from a given graph, so that the resulting graph becomes a union of disjoint cliques. The cluster editing problem is closely related to correlation clustering and has applications, e.g. In image segmentation. For general graphs this problem is apx apx{\mathbb {apx}}-hard. In this paper we present an efficient polynomial time approximation scheme for the cluster editing problem on graphs embeddable in the plane with a few edge crossings. The running time of the algorithm is 2 o(? -1 log(? -1 )) n 2o(?-1log?(?-1))n{2^{o\left( \epsilon ^{-1} \log (\epsilon ^{-1})\right) }n} for planar graphs and 2 o(k 2 ? -1 log(k 2 ? -1 )) n 2o(k2?-1log?(k2?-1))n2^{o\left( k^2\epsilon ^{-1}\log \left( k^2\epsilon ^{-1}\right) \right) }n for planar graphs with at most k crossings.
Original languageEnglish
Title of host publicationApproximation and Online Algorithms
PublisherSpringer
Pages27-39
Volume10138
EditionLecture Notes in Computer Science
ISBN (Electronic)978-3-319-51741-4
ISBN (Print)978-3-319-51740-7
DOIs
Publication statusPublished - 7 Jan 2017

Publication series

SeriesLecture Notes in Computer Science
Volume10138

Keywords

  • Graph approximation
  • Correlation clustering
  • Cluster editing
  • PTAS
  • k-planarity
  • Microscopy cell segmentation

Cite this

Berger, A., Grigoriev, A., & Winokurow, A. (2017). A PTAS for the Cluster Editing Problem on Planar Graphs. In Approximation and Online Algorithms (Lecture Notes in Computer Science ed., Vol. 10138, pp. 27-39). Springer. Lecture Notes in Computer Science, Vol.. 10138 https://doi.org/10.1007/978-3-319-51741-4_3
Berger, Andre ; Grigoriev, Alexander ; Winokurow, Andrej. / A PTAS for the Cluster Editing Problem on Planar Graphs. Approximation and Online Algorithms. Vol. 10138 Lecture Notes in Computer Science. ed. Springer, 2017. pp. 27-39 (Lecture Notes in Computer Science, Vol. 10138).
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abstract = "The goal of the cluster editing problem is to add or delete a minimum number of edges from a given graph, so that the resulting graph becomes a union of disjoint cliques. The cluster editing problem is closely related to correlation clustering and has applications, e.g. In image segmentation. For general graphs this problem is apx apx{\mathbb {apx}}-hard. In this paper we present an efficient polynomial time approximation scheme for the cluster editing problem on graphs embeddable in the plane with a few edge crossings. The running time of the algorithm is 2 o(? -1 log(? -1 )) n 2o(?-1log?(?-1))n{2^{o\left( \epsilon ^{-1} \log (\epsilon ^{-1})\right) }n} for planar graphs and 2 o(k 2 ? -1 log(k 2 ? -1 )) n 2o(k2?-1log?(k2?-1))n2^{o\left( k^2\epsilon ^{-1}\log \left( k^2\epsilon ^{-1}\right) \right) }n for planar graphs with at most k crossings.",
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Berger, A, Grigoriev, A & Winokurow, A 2017, A PTAS for the Cluster Editing Problem on Planar Graphs. in Approximation and Online Algorithms. Lecture Notes in Computer Science edn, vol. 10138, Springer, Lecture Notes in Computer Science, vol. 10138, pp. 27-39. https://doi.org/10.1007/978-3-319-51741-4_3

A PTAS for the Cluster Editing Problem on Planar Graphs. / Berger, Andre; Grigoriev, Alexander; Winokurow, Andrej.

Approximation and Online Algorithms. Vol. 10138 Lecture Notes in Computer Science. ed. Springer, 2017. p. 27-39 (Lecture Notes in Computer Science, Vol. 10138).

Research output: Chapter in Book/Report/Conference proceedingChapterAcademic

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T1 - A PTAS for the Cluster Editing Problem on Planar Graphs

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N2 - The goal of the cluster editing problem is to add or delete a minimum number of edges from a given graph, so that the resulting graph becomes a union of disjoint cliques. The cluster editing problem is closely related to correlation clustering and has applications, e.g. In image segmentation. For general graphs this problem is apx apx{\mathbb {apx}}-hard. In this paper we present an efficient polynomial time approximation scheme for the cluster editing problem on graphs embeddable in the plane with a few edge crossings. The running time of the algorithm is 2 o(? -1 log(? -1 )) n 2o(?-1log?(?-1))n{2^{o\left( \epsilon ^{-1} \log (\epsilon ^{-1})\right) }n} for planar graphs and 2 o(k 2 ? -1 log(k 2 ? -1 )) n 2o(k2?-1log?(k2?-1))n2^{o\left( k^2\epsilon ^{-1}\log \left( k^2\epsilon ^{-1}\right) \right) }n for planar graphs with at most k crossings.

AB - The goal of the cluster editing problem is to add or delete a minimum number of edges from a given graph, so that the resulting graph becomes a union of disjoint cliques. The cluster editing problem is closely related to correlation clustering and has applications, e.g. In image segmentation. For general graphs this problem is apx apx{\mathbb {apx}}-hard. In this paper we present an efficient polynomial time approximation scheme for the cluster editing problem on graphs embeddable in the plane with a few edge crossings. The running time of the algorithm is 2 o(? -1 log(? -1 )) n 2o(?-1log?(?-1))n{2^{o\left( \epsilon ^{-1} \log (\epsilon ^{-1})\right) }n} for planar graphs and 2 o(k 2 ? -1 log(k 2 ? -1 )) n 2o(k2?-1log?(k2?-1))n2^{o\left( k^2\epsilon ^{-1}\log \left( k^2\epsilon ^{-1}\right) \right) }n for planar graphs with at most k crossings.

KW - Graph approximation

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KW - k-planarity

KW - Microscopy cell segmentation

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Berger A, Grigoriev A, Winokurow A. A PTAS for the Cluster Editing Problem on Planar Graphs. In Approximation and Online Algorithms. Lecture Notes in Computer Science ed. Vol. 10138. Springer. 2017. p. 27-39. (Lecture Notes in Computer Science, Vol. 10138). https://doi.org/10.1007/978-3-319-51741-4_3