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

TL;DR

This paper demonstrates that the approximation ratio of 5 for covering small cuts using the primal-dual method is tight by providing an example where the ratio approaches 5.

Contribution

The paper constructs a specific example showing that the primal-dual algorithm's approximation ratio for small cuts coverage cannot be improved beyond 5.

Findings

The primal-dual algorithm's ratio is tight at 5.

An example is provided where the ratio approaches 5.

The result confirms the optimality of the known approximation bound.

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 $5$.