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Tight Analysis of the Primal-Dual Method for Edge-Covering Pliable Set Families

Author Zeev Nutov

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ACM Subject Classification
  • Theory of computation → Design and analysis of algorithms
Keywords
  • primal dual method
  • pliable set family
  • approximation algorithms

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Abstract

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 with a reverse delete phase. Bansal, Cheriyan, Grout, and Ibrahimpur [ICALP 2023: 15:1–15:19] showed that this algorithm achieves approximation ratio 16 for a larger class of so called γ-pliable set families, that have much weaker uncrossing properties. The approximation ratio 16 was improved to 10 in [Z. Nutov, 2025]. Recently, Bansal [I. Bansal, 2024] obtained approximation ratio 8 for γ-pliable families and also considered an important particular case of the family of cuts of size < k of a graph H. We will improve the approximation ratio to 7 for the former case and give a simple proof of approximation ratio 6 for the latter case. Furthermore, if H is λ-edge-connected then we will show a slightly better approximation ratio 6 - 1/(β+1), where β = ⌊(k-1)/(⌈(λ+1)/2⌉)⌋. Our analysis is supplemented by examples indicating that these approximation ratios are asymptotically tight for the primal-dual algorithm.

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Zeev Nutov. Tight Analysis of the Primal-Dual Method for Edge-Covering Pliable Set Families. In 50th International Symposium on Mathematical Foundations of Computer Science (MFCS 2025). Leibniz International Proceedings in Informatics (LIPIcs), Volume 345, pp. 82:1-82:14, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2025) https://doi.org/10.4230/LIPIcs.MFCS.2025.82

Author Details

Zeev Nutov
  • The Open University of Israel, Ra'anana, Israel

References

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