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We study the problem of hierarchical clustering on planar graphs. We formulate this in terms of an LP relaxation of ultrametric rounding. To solve this LP efficiently we introduce a dual cutting plane scheme that uses minimum cost perfect matching as a subroutine in order to efficiently explore the space of planar partitions. We apply our algorithm to the problem of hierarchical image segmentation.arXiv:1507.02407v3 fatcat:xjx22u5zp5cglmbaxsagra7mfy