Approximate solutions to the Travelling Salesperson Problem on semimetric graphs

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.