Tight analysis of the primal-dual method for edge-covering pliable set families

2025-04-04

AI Summary

This paper studies the minimum-cost edge cover problem on two structural families: γ-pliable set systems and <k-cut families. Using a primal-dual framework, we introduce a reverse-delete strategy and structural induction analysis, complemented by combinatorial constructions and tight instance design. Our contributions are twofold: (i) For γ-pliable set systems, we tighten the approximation ratio to 7—improving upon the prior upper bound of 10 and correcting the previously claimed lower bound of 8; (ii) For all cuts of size less than k in graphs, we establish a tight approximation ratio of 6—rectifying the erroneous prior claim of 5. Both results constitute the optimal achievable ratios for primal-dual algorithms on these families: our theoretical analysis is matched precisely by worst-case instances, thereby advancing the fundamental limits of approximation algorithms for edge cover under pliability constraints.

Technology Category

• Constraint Satisfaction and Optimization: Other Foundations of Constraint Satisfaction
• Search and Optimization: Combinatorial Optimization
• Reasoning under Uncertainty: Other Foundations of Reasoning under Uncertainty

Application Category

• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs
• Economics, Online Markets and Human Computation: Incentives in network design for Web infrastructures and ecosystems
• Web Mining and Content Analysis: Robustness and generalizability of Web mining methods

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 $gamma$-pliable set families, that have much weaker uncrossing properties. The approximation ratio $16$ was improved to $10$ by the author [WAOA 2025: 151-166]. Recently, Bansal [arXiv:2308.15714] stated approximation ratio $8$ for $gamma$-pliable families and an improved approximation ratio $5$ for an important particular case of the family of cuts of size $<k$ of a graph, but his proof has an error. 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. Our analysis is supplemented by examples showing that these approximation ratios are tight for the primal-dual algorithm.

Problem

• Improving approximation ratio for edge-covering pliable set families
• Analyzing primal-dual method for γ-pliable set families
• Providing tight approximation ratios for graph cut families

Innovation

• Improved primal-dual algorithm for pliable families
• Tight approximation ratio of 7 for γ-pliable
• Simple proof for 6-approximation on graph cuts