The paper focuses on the Set Family Edge Cover problem, where the goal is to find a minimum-cost edge set that covers a given set family F. The author considers a class of set families called "pliable" families, which generalize the previously studied "uncrossable" families.
The key insights are:
The author shows that the primal-dual algorithm proposed by Williamson et al. (WGMV algorithm) achieves an approximation ratio of 10 for covering pliable set families, improving upon the previous ratio of 16 shown by Bansal et al.
The author introduces the concept of "pliable" and "γ-pliable" set families, which have weaker uncrossing properties compared to uncrossable families, but still allow for a constant-factor approximation.
The analysis relies on several structural lemmas about pliable families, including the existence of a laminar witness family and properties of "hollow chains" in the witness family.
The improved approximation ratio is achieved by a refined analysis of the WGMV algorithm, showing that the contribution of each "hollow chain" to the dual objective is at most 5, rather than the previous bound of 8.
The improved approximation ratio for covering pliable set families also leads to improved results for several variants of the Capacitated k-Edge Connected Spanning Subgraph problem.
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arxiv.org
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