Correlation Clustering and Two-edge-connected Augmentation for Planar Graphs

Authors Philip N. Klein, Claire Mathieu, Hang Zhou



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Philip N. Klein
Claire Mathieu
Hang Zhou

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Philip N. Klein, Claire Mathieu, and Hang Zhou. Correlation Clustering and Two-edge-connected Augmentation for Planar Graphs. In 32nd International Symposium on Theoretical Aspects of Computer Science (STACS 2015). Leibniz International Proceedings in Informatics (LIPIcs), Volume 30, pp. 554-567, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2015)
https://doi.org/10.4230/LIPIcs.STACS.2015.554

Abstract

In correlation clustering, the input is a graph with edge-weights, where every edge is labelled either + or - according to similarity of its endpoints. The goal is to produce a partition of the vertices that disagrees with the edge labels as little as possible. In two-edge-connected augmentation, the input is a graph with edge-weights and a subset R of edges of the graph. The goal is to produce a minimum weight subset S of edges of the graph, such that for every edge in R, its endpoints are two-edge-connected in R\cup S. For planar graphs, we prove that correlation clustering reduces to two-edge-connected augmentation, and that both problems have a polynomial-time approximation scheme.
Keywords
  • correlation clustering
  • two-edge-connected augmentation
  • polynomial-time approximation scheme
  • planar graphs

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References

  1. Nir Ailon, Moses Charikar, and Alantha Newman. Aggregating inconsistent information: Ranking and clustering. Journal of the ACM, 55(5), 2008. Google Scholar
  2. Nir Ailon and Edo Liberty. Correlation clustering revisited: The true cost of error minimization problems. In International Colloquium on Automata, Languages and Programming, pages 24-36. Springer, 2009. Google Scholar
  3. Sharon Alpert, Meirav Galun, Ronen Basri, and Achi Brandt. Image segmentation by probabilistic bottom-up aggregation and cue integration. In Computer Vision and Pattern Recognition, pages 1-8. IEEE, 2007. Google Scholar
  4. Amir Alush and Jacob Goldberger. Ensemble segmentation using efficient integer linear programming. Pattern Analysis and Machine Intelligence, 34(10):1966-1977, 2012. Google Scholar
  5. Amir Alush and Jacob Goldberger. Break and conquer: Efficient correlation clustering for image segmentation. In Similarity-Based Pattern Recognition, volume 7953, pages 134-147. Springer, 2013. Google Scholar
  6. Bjoern Andres, Jörg H. Kappes, Thorsten Beier, Ullrich Kothe, and Fred A. Hamprecht. Probabilistic image segmentation with closedness constraints. In International Conference on Computer Vision, pages 2611-2618. IEEE, 2011. Google Scholar
  7. Yoram Bachrach, Pushmeet Kohli, Vladimir Kolmogorov, and Morteza Zadimoghaddam. Optimal coalition structure generation in cooperative graph games. In Conference on Artificial Intelligence, 2013. Google Scholar
  8. Nikhil Bansal, Avrim Blum, and Shuchi Chawla. Correlation clustering. Machine Learning, 56(1-3):89-113, 2004. Google Scholar
  9. MohammadHossein Bateni, MohammadTaghi Hajiaghayi, and Dániel Marx. Approximation schemes for Steiner forest on planar graphs and graphs of bounded treewidth. Journal of the ACM, 58(5):21, 2011. Google Scholar
  10. Amir Ben-Dor, Ron Shamir, and Zohar Yakhini. Clustering gene expression patterns. Journal of Computational Biology, 6(3/4):281-297, 1999. Google Scholar
  11. A. Berger and M. Grigni. Minimum weight 2-edge-connected spanning subgraphs in planar graphs. In International Colloquium on Automata, Languages and Programming, volume 4596, pages 90-101, 2007. Google Scholar
  12. Glencora Borradaile, Erik D. Demaine, and Siamak Tazari. Polynomial-time approximation schemes for subset-connectivity problems in bounded-genus graphs. Algorithmica, 68(2):287-311, 2014. Google Scholar
  13. Glencora Borradaile and Philip N. Klein. The two-edge connectivity survivable network problem in planar graphs. International Colloquium on Automata, Languages and Programming, pages 485-501, 2008. Google Scholar
  14. Glencora Borradaile, Philip N. Klein, and Claire Mathieu. An O(n log n) approximation scheme for Steiner tree in planar graphs. ACM Transactions on Algorithms, 5(3):31, 2009. Google Scholar
  15. Sebastian Böcker and Jan Baumbach. Cluster editing. In Paola Bonizzoni, Vasco Brattka, and Benedikt Löwe, editors, The Nature of Computation. Logic, Algorithms, Applications, volume 7921, pages 33-44. Springer, 2013. Google Scholar
  16. Moses Charikar, Venkatesan Guruswami, and Anthony Wirth. Clustering with qualitative information. Journal of Computer and System Sciences, 71(3):360-383, 2005. Google Scholar
  17. Erik D. Demaine, Dotan Emanuel, Amos Fiat, and Nicole Immorlica. Correlation clustering in general weighted graphs. Theoretical Computer Science, 361(2):172-187, 2006. Google Scholar
  18. Frederic Dorn, Eelko Penninkx, Hans L. Bodlaender, and Fedor V. Fomin. Efficient exact algorithms on planar graphs: Exploiting sphere cut decompositions. Algorithmica, 58(3):790-810, 2010. Google Scholar
  19. Kapali P. Eswaran and R. Endre Tarjan. Augmentation problems. SIAM Journal on Computing, 5(4):653-665, 1976. Google Scholar
  20. Guy Even, Jon Feldman, Guy Kortsarz, and Zeev Nutov. A 1.8 approximation algorithm for augmenting edge-connectivity of a graph from 1 to 2. ACM Transactions on Algorithms, 5(2):21:1-21:17, 2009. Google Scholar
  21. Guy Even, Guy Kortsarz, and Zeev Nutov. A 1.5-approximation algorithm for augmenting edge-connectivity of a graph from 1 to 2. Information Processing Letters, 111(6):296-300, 2011. Google Scholar
  22. Aristides Gionis, Heikki Mannila, and Panayiotis Tsaparas. Clustering aggregation. ACM Transactions on Knowledge Discovery from Data, 1(1), 2007. Google Scholar
  23. Ioannis Giotis and Venkatesan Guruswami. Correlation clustering with a fixed number of clusters. In Symposium on Discrete algorithm, pages 1167-1176. ACM, 2006. Google Scholar
  24. Kamal Jain. A factor 2 approximation algorithm for the generalized Steiner network problem. Combinatorica, 21(1):39-60, 2001. Google Scholar
  25. Samir Khuller and Uzi Vishkin. Biconnectivity approximations and graph carvings. Journal of the ACM, 41(2):214-235, 1994. Google Scholar
  26. Sungwoong Kim, Sebastian Nowozin, Pushmeet Kohli, and Chang Dong Yoo. Higher-order correlation clustering for image segmentation. In Advances in Neural Information Processing Systems, pages 1530-1538, 2011. Google Scholar
  27. Philip N. Klein and Shay Mozes. Optimization algorithms for planar graphs. In preparation, manuscript at http://planarity.org. Google Scholar
  28. Philip N. Klein and R. Ravi. When cycles collapse: A general approximation technique for constrained two-connectivity problems. In Integer Programming and Combinatorial Optimization, pages 39-55, 1993. Google Scholar
  29. Guy Kortsarz and Zeev Nutov. Approximating minimum cost connectivity problems. Approximation Algorithms and Metahueristics, 2007. Google Scholar
  30. David R. Martin, Charless C. Fowlkes, and Jitendra Malik. Learning to detect natural image boundaries using local brightness, color, and texture cues. Pattern Analysis and Machine Intelligence, 26(5):530-549, 2004. Google Scholar
  31. Claire Mathieu and Warren Schudy. Correlation clustering with noisy input. In Symposium on Discrete Algorithms, pages 712-728, 2010. Google Scholar
  32. Hiroshi Nagamochi. An approximation for finding a smallest 2-edge-connected subgraph containing a specified spanning tree. Discrete Applied Mathematics, 126(1):83 - 113, 2003. Google Scholar
  33. R Ravi. Approximation algorithms for Steiner augmentations for two-connectivity. Technical Report CS-92-21, Department of Computer Science, Brown University, 1992. Google Scholar
  34. Mauricio Resende and Panos Pardalos. Handbook of optimization in telecommunications. Springer, 2008. Google Scholar
  35. Chaitanya Swamy. Correlation clustering: maximizing agreements via semidefinite programming. In Symposium on Discrete Algorithms, pages 526-527. SIAM, 2004. Google Scholar
  36. Julian Yarkony, Alexander Ihler, and Charless C Fowlkes. Fast planar correlation clustering for image segmentation. In European Conference on Computer Vision, volume 7577, pages 568-581. Springer, 2012. Google Scholar
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