Book contents
- Frontmatter
- Preface
- Contents
- 1 Introduction
- 2 Linear Programming Relaxations of the Symmetric TSP
- 3 Linear Programming Relaxations of the Asymmetric TSP
- 4 Duality, Cuts, and Uncrossing
- 5 Thin Trees and Random Trees
- 6 Asymmetric Graph TSP
- 7 Constant-Factor Approximation for the Asymmetric TSP
- 8 Algorithms for Subtour Cover
- 9 Asymmetric Path TSP
- 10 Parity Correction of Random Trees
- 11 Proving the Main Payment Theorem for Hierarchies
- 12 Removable Pairings
- 13 Ear-Decompositions, Matchings, and Matroids
- 14 Symmetric Path TSP and T-Tours
- 15 Best-of-Many Christofides and Variants
- 16 Path TSP by Dynamic Programming
- 17 Further Results, Related Problems
- 18 State of the Art, Open Problems
- Bibliography
- Index
12 - Removable Pairings
Published online by Cambridge University Press: 14 November 2024
Book contents
- Frontmatter
- Preface
- Contents
- 1 Introduction
- 2 Linear Programming Relaxations of the Symmetric TSP
- 3 Linear Programming Relaxations of the Asymmetric TSP
- 4 Duality, Cuts, and Uncrossing
- 5 Thin Trees and Random Trees
- 6 Asymmetric Graph TSP
- 7 Constant-Factor Approximation for the Asymmetric TSP
- 8 Algorithms for Subtour Cover
- 9 Asymmetric Path TSP
- 10 Parity Correction of Random Trees
- 11 Proving the Main Payment Theorem for Hierarchies
- 12 Removable Pairings
- 13 Ear-Decompositions, Matchings, and Matroids
- 14 Symmetric Path TSP and T-Tours
- 15 Best-of-Many Christofides and Variants
- 16 Path TSP by Dynamic Programming
- 17 Further Results, Related Problems
- 18 State of the Art, Open Problems
- Bibliography
- Index
Summary
So far, all algorithms for Symmetric TSP began with a spanning tree and then added edges to make the graph Eulerian. In contrast, Mömke and Svensson suggested to begin with a 2-connected graph; then we may also delete some edges for making it Eulerian, and this may be cheaper overall. They introduced the notion of removable pairings, which allow to control that we maintain connectivity when deleting edges.
This idea led to a substantial improvement and is still used for the best algorithm for Graph TSP that we know today (cf. Chapter 12). It also yields the ratio 4/3 for the special case of subcubic graphs.
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- Approximation Algorithms for Traveling Salesman Problems , pp. 265 - 279Publisher: Cambridge University PressPrint publication year: 2024