Extending the Primal-Dual 2-Approximation Algorithm Beyond Uncrossable Set Families

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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 languageEnglish
Title of host publicationInteger Programming and Combinatorial Optimization - 25th International Conference, IPCO 2024, Proceedings
EditorsJens Vygen, Jarosław Byrka
PublisherSpringer Science and Business Media Deutschland GmbH
Number of pages14
ISBN (Print)9783031598340
StatePublished - 2024
Event25th International Conference on Integer Programming and Combinatorial Optimization, IPCO 2024 - Wroclaw, Poland
Duration: 3 Jul 20245 Jul 2024

Publication series

NameLecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics)
Volume14679 LNCS
ISSN (Print)0302-9743
ISSN (Electronic)1611-3349


Conference25th International Conference on Integer Programming and Combinatorial Optimization, IPCO 2024

Bibliographical note

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


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