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 at least k nodes from a given set T of terminals.
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