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

Zeev Nutov

TL;DR

The paper analyzes the Small Cuts Cover problem and the tightness of the $5$-approximation bound achieved by the WGMV primal-dual algorithm. It presents a constructive construction by gluing p copies of a base instance along a shared axis to create a larger instance with common axis nodes, establishing a family of ${\cal F}$-cores that force the algorithm's behavior. In the glued instance, the primal-dual algorithm produces a costly red solution of cost $5p$ while a cheaper blue solution costs $p+2$, yielding a ratio $5p/(p+2)$ that tends to $5$ as $p$ grows. This positively answers Simmons' question about tightness for this algorithm class, while noting it does not prove an integrality gap of $5$ and relies on the specific iterative-primal-dual construction.

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 $5$.

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

TL;DR

The paper analyzes the Small Cuts Cover problem and the tightness of the -approximation bound achieved by the WGMV primal-dual algorithm. It presents a constructive construction by gluing p copies of a base instance along a shared axis to create a larger instance with common axis nodes, establishing a family of -cores that force the algorithm's behavior. In the glued instance, the primal-dual algorithm produces a costly red solution of cost while a cheaper blue solution costs , yielding a ratio that tends to as grows. This positively answers Simmons' question about tightness for this algorithm class, while noting it does not prove an integrality gap of and relies on the specific iterative-primal-dual construction.

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 of a graph. Recently, Simmons showed that the primal-dual algorithm of Williamson, Goemans, Mihail, and Vazirani achieves approximation ratio 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 .

Paper Structure

This paper contains 2 sections, 5 theorems, 6 equations, 2 figures.

Table of Contents

  1. Introduction
  2. The construction

Key Result

Theorem 1

For any integers $q,p,k$ such that $q \ge 1$, $p \ge 2$, and $k \ge 2pq+1$ there exists an instance of the Small Cuts Cover problem on $4p+3$ nodes such that the WGMV WGMV primal-dual algorithm computes a solution of cost $\frac{5p}{p+2} \cdot {\sf opt}$.

Figures (2)

  • Figure 1: The instance of Small cuts Cover$G,c,L$. The green edges are those of the input graph, and the number on a green edge is the multiplicity/capacity of the edge. The red and blue edges are links. The weight of a link is the number of non-white nodes it connects.The ${\cal F}$-cores are $\{t\}$, $\{r\}$ and $C$, and the cuts $A,X,Y$ in ${\cal F}^r \setminus {\cal F}^r_C$. The figure also shows two additional cuts in ${\cal F}^r_C$, but these cuts are not addressed in Lemma \ref{['l:cores']}.
  • Figure 2: Illustration of the construction. The green edges are those of the input graph, and the number on a green edge is the multiplicity/capacity of the edge. The red and blue edges are links. The weight of a link is the number of non-white nodes it connects. The case $p=2$; note that the cut $A_1 \cup A_2$ is also a small cut.The case $p=4$; only some links are shown.

Theorems & Definitions (5)

  • Theorem 1
  • Lemma 2
  • Lemma 3
  • Lemma 4
  • Lemma 5