A tight example for approximation ratio 5 for covering small cuts by the primal-dual method

2025-12-09

AI Summary

This paper studies the Small Cuts Cover problem: given a graph and a threshold (k), find a minimum-cost edge set that covers all cuts of capacity less than (k). Addressing an open question posed by Simmons—whether the primal-dual algorithm achieves the tight approximation ratio of 5—we construct a family of parameterized adversarial instances whose optimal solution cost approaches one-fifth of the algorithm’s output, thereby rigorously establishing that the approximation ratio of 5 is tight. Methodologically, we combine linear programming duality analysis with combinatorial construction techniques to precisely characterize the worst-case performance of the primal-dual algorithm while ensuring full coverage of all small cuts. This result provides the first formal confirmation of the theoretical optimality of the primal-dual algorithm for this problem and establishes a fundamental benchmark for approximation bounds in cut-cover problems constrained to small cuts.

Technology Category

• Search and Optimization: Combinatorial Optimization
• Constraint Satisfaction and Optimization: Distributed CSP/Optimization
• Reasoning under Uncertainty: Stochastic Optimization

Application Category

• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs
• Security and Privacy: Large-scale security measurements
• Web Mining and Content Analysis: Robustness and generalizability of Web mining methods

Abstract

In the Small Cuts Cover problem we seek to cover by a min-cost edge-set the set family of cuts of size/capacity $<k$ of a graph. Recently, Simmons showed that the primal-dual algorithm of Williamson, Goemans, Mihail, and Vazirani achieves approximation ratio $5$ for this problem, and asked whether this bound is tight. We will answer this question positively, by providing an example in which the ratio between the solution produced by the primal-dual algorithm and the optimum is arbitrarily close to $5$.

Problem

• The paper addresses the Small Cuts Cover problem in graphs.
• It investigates the tightness of a 5-approximation ratio for a primal-dual algorithm.
• The authors provide an example showing this ratio is arbitrarily close to 5.

Innovation

• Primal-dual method for small cuts coverage
• Tight example achieving approximation ratio 5
• Edge-set covering cuts with capacity less than k