### Abstract

A base-monotone region with a base is a subset of the cells in a pixel grid such that if a cell is contained in the region then so are the ones on a shortest path from the cell to the base. The problem of decomposing a pixel grid into disjoint base-monotone regions was first studied in the context of image segmentation. It is known that for a given pixel grid and base-lines, one can compute in polynomial time a maximum-weight region that can be decomposed into disjoint base-monotone regions with respect to the given base-lines (Chun et al., 2012 [4]). We continue this line of research and show the NP-hardness of the problem of optimally locating k base-lines in a given n×n pixel grid. We then present an O(n^{3})-time 2-approximation algorithm for this problem. We also study two related problems, the k base-segment problem and the quad-decomposition problem, and present some complexity results for them.

Original language | English |
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Pages (from-to) | 71-84 |

Number of pages | 14 |

Journal | Theoretical Computer Science |

Volume | 555 |

Issue number | C |

DOIs | |

Publication status | Published - 2014 |

### All Science Journal Classification (ASJC) codes

- Theoretical Computer Science
- Computer Science(all)

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## Cite this

*Theoretical Computer Science*,

*555*(C), 71-84. https://doi.org/10.1016/j.tcs.2013.11.030