## Abstract

We study the problem of finding a minimum weight connected subgraph spanning at least k vertices on planar, node-weighted graphs. We give a (4+ε) -approximation algorithm for this problem. We achieve this by utilizing the recent Lagrangian-multiplier preserving (LMP) primal-dual 3-approximation for the node-weighted prize-collecting Steiner tree problem by Byrka et al. (SWAT’16) and adopting an approach by Chudak et al. (Math. Prog. ’04) regarding Lagrangian relaxation for the edge-weighted variant. In particular, we improve the procedure of picking additional vertices (tree merging procedure) given by Sadeghian (2013) by taking a constant number of recursive steps and utilizing the limited guessing procedure of Arora and Karakostas (Math. Prog. ’06). More generally, our approach readily gives a(4/3.r+ε) -approximation on any graph class where the algorithm of Byrka et al. for the prize-collecting version gives an r-approximation. We argue that this can be interpreted as a generalization of an analogous result by Könemann et al. (Algorithmica ’11) for partial cover problems. Together with a lower bound construction by Mestre (STACS’08) for partial cover this implies that our bound is essentially best possible among algorithms that utilize an LMP algorithm for the Lagrangian relaxation as a black box. In addition to that, we argue by a more involved lower bound construction that even using the LMP algorithm by Byrka et al. in a non-black-box fashion could not beat the factor 4/3.r when the tree merging step relies only on the solutions output by the LMP algorithm.

Original language | English |
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Title of host publication | Approximation and Online Algorithms - 16th International Workshop, WAOA 2018, Revised Selected Papers |

Editors | Leah Epstein, Thomas Erlebach |

Publisher | Springer |

Pages | 87-101 |

Number of pages | 15 |

ISBN (Print) | 9783030046927 |

DOIs | |

Publication status | Published - 1 Jan 2018 |

MoE publication type | A4 Article in a conference publication |

Event | Workshop on Approximation and Online Algorithms - Helsinki, Finland Duration: 23 Aug 2018 → 24 Aug 2018 Conference number: 16 |

### Publication series

Name | Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics) |
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Volume | 11312 LNCS |

ISSN (Print) | 0302-9743 |

ISSN (Electronic) | 1611-3349 |

### Workshop

Workshop | Workshop on Approximation and Online Algorithms |
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Abbreviated title | WAOA |

Country/Territory | Finland |

City | Helsinki |

Period | 23/08/2018 → 24/08/2018 |