## Abstract

A set family F is uncrossable if A∩B,A∪B∈F or A\B,B\A∈F for any A,B∈F. A classic result of Williamson, Goemans, Mihail, and Vazirani [STOC 1993:708-717] states that the problem of covering an uncrossable set family by a min-cost edge set admits approximation ratio 2, by a primal-dual algorithm. They asked whether this result extends to a larger class of set families and combinatorial optimization problems. We define a new class of semi-uncrossable set families, when for any A,B∈F we have that A∩B∈F and one of A∪B,A\B,B\A is in F, or A\B,B\A∈F. We will show that the Williamson et al. algorithm extends to this new class of families and identify several “non-uncrossable” algorithmic problems that belong to this class. In particular, we will show that the union of an uncrossable family and a monotone family, or of an uncrossable family that has the disjointness property and a proper family, is a semi-uncrossable family, that in general is not uncrossable. For example, our result implies approximation ratio 2 for the problem of finding a min-cost subgraph H such that H contains a Steiner forest and every connected component of H contains zero or at least k nodes from a given set T of terminals.

Original language | English |
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Title of host publication | Integer Programming and Combinatorial Optimization - 25th International Conference, IPCO 2024, Proceedings |

Editors | Jens Vygen, Jarosław Byrka |

Publisher | Springer Science and Business Media Deutschland GmbH |

Pages | 351-364 |

Number of pages | 14 |

ISBN (Print) | 9783031598340 |

DOIs | |

State | Published - 2024 |

Event | 25th International Conference on Integer Programming and Combinatorial Optimization, IPCO 2024 - Wroclaw, Poland Duration: 3 Jul 2024 → 5 Jul 2024 |

### 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 | 14679 LNCS |

ISSN (Print) | 0302-9743 |

ISSN (Electronic) | 1611-3349 |

### Conference

Conference | 25th International Conference on Integer Programming and Combinatorial Optimization, IPCO 2024 |
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Country/Territory | Poland |

City | Wroclaw |

Period | 3/07/24 → 5/07/24 |

### Bibliographical note

Publisher Copyright:© The Author(s), under exclusive license to Springer Nature Switzerland AG 2024.