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3/2-Approximation for the Forest Augmentation Problem

Ali Çivril

TL;DR

A $\frac{3}{2}$-approximation algorithm for the Forest Augmentation Problem (FAP) is described, which significantly improves upon the previous best ratio and proceeds toward the goal of a $\frac{3}{2}-approximation algorithm for the Weighted 2-ECSS.

Abstract

We describe a $\frac{3}{2}$-approximation algorithm for the Forest Augmentation Problem (\textsf{FAP}), which is a special case of the Weighted 2-Edge-Connected Spanning Subgraph Problem (\textsf{Weighted 2-ECSS}). This significantly improves upon the previous best ratio $1.9973$, and proceeds toward the goal of a $\frac{3}{2}$-approximation algorithm for \textsf{Weighted 2-ECSS}.

3/2-Approximation for the Forest Augmentation Problem

TL;DR

A -approximation algorithm for the Forest Augmentation Problem (FAP) is described, which significantly improves upon the previous best ratio and proceeds toward the goal of a $\frac{3}{2}-approximation algorithm for the Weighted 2-ECSS.

Abstract

We describe a -approximation algorithm for the Forest Augmentation Problem (\textsf{FAP}), which is a special case of the Weighted 2-Edge-Connected Spanning Subgraph Problem (\textsf{Weighted 2-ECSS}). This significantly improves upon the previous best ratio , and proceeds toward the goal of a -approximation algorithm for \textsf{Weighted 2-ECSS}.
Paper Structure (20 sections, 6 theorems, 3 equations, 9 figures, 2 algorithms)

This paper contains 20 sections, 6 theorems, 3 equations, 9 figures, 2 algorithms.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. The Algorithm for FAP
  4. The Intuition: The Primal-Dual Schema
  5. The Algorithm
  6. Proof of Theorem \ref{['main']}
  7. The Initial Assignment of the Dual Variables
  8. The General Cost Sharing Scheme
  9. The Cost Sharing for Special Segments
  10. The Cost Sharing for Other Segments
  11. The Cost Sharing for Strong Segments that are not Special Segments
  12. The Cost Sharing for Weak Segments
  13. Induction Base Case $k=1$
  14. Inductive Step: Increasing $k$ by One or Two
  15. Induction Base Case $k=2$
  16. ...and 5 more sections

Key Result

Theorem 1

There exists a polynomial-time $\frac{3}{2}$-approximation algorithm for FAP.

Figures (9)

  • Figure 1: An example of an improvement operation performed in the second step of the algorithm with a single reverse-delete operation
  • Figure 2: An example of an improvement operation performed in the second step of the algorithm with two reverse-delete operations
  • Figure 3: An example of an improvement operation performed in the second step of the algorithm with four reverse-delete operations
  • Figure 4: Illustrations for the proofs of Claim \ref{['special']} and Claim \ref{['opt-1']}, where the edges of an optimal solution are in solid lines
  • Figure 5: (a) The base case $k=1$; (b) Increasing the number of weak segments by one; (c) Increasing the number of weak segments by two: A possible set of positive dual variables are shown with dotted lines
  • ...and 4 more figures

Theorems & Definitions (17)

  • Theorem 1
  • Lemma 2
  • proof
  • Lemma 3
  • proof
  • Claim 4
  • proof
  • Claim 5
  • proof
  • Claim 6
  • ...and 7 more