Approximate solutions to the Travelling Salesperson Problem on semimetric graphs

TL;DR

This paper extends Christofides' approximation algorithm to all complete weighted graphs with positive weights by utilizing the relaxed polygonal inequality, introducing a $rac{3 ext{ extgamma}}{2}$-approximation for the TSP on semimetric graphs.

Contribution

It generalizes the Christofides algorithm to semimetric graphs using the relaxed polygonal inequality, achieving a new approximation ratio.

Findings

Achieves a $rac{3 ext{ extgamma}}{2}$-approximation for TSP on semimetric graphs.

Proves every finite graph admits a $ ext{ extgamma}$-polygon structure.

Extends applicability of approximation techniques to broader graph classes.

Abstract

With the aid of the relaxed polygonal inequality (introduced by Fagin et al.) we strive to extend the applicability of Christofides approximation technique to the scope of all complete finite weighted graphs with positive weights. First section acquaints the Reader with the class of semimetric graphs and proves that every finite graph admits $\gamma$-polygon structure. Sections 2 and 3 establish the necessary notions from the graph and optimization theory to tackle the Traveling Salesperson Problem. In section 4 the minimal spanning tree method is introduced, while section 5 focuses on the analysis of this method through the lens of $\gamma$-polygon graphs. The final section of the paper adjusts the technique of Christofides by obtaining $\frac{3\gamma}{2}-$approximation for the TSP.