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 $5$.

翻译：在小割覆盖问题中，我们试图通过最小代价的边集覆盖图中规模/容量$<k$的割所构成的集合族。最近，Simmons证明了Williamson、Goemans、Mihail和Vazirani的原始对偶算法对该问题能达到近似比$5$，并询问该界是否紧。我们将通过构造一个实例，使得原始对偶算法产生的解与最优解之比可任意接近$5$，从而肯定地回答该问题。