Towards improving Christofides algorithm on fundamental classes by gluing convex combinations of tours

We present a new approach for gluing tours over certain tight, 3-edge cuts. Gluing cuts has been used in algorithms finding Hamilton cycles special graph classes and proving bounds 2-edge-connected subgraph problem, but not much was known this direction connected multigraphs. apply to the traveling salesman problem (TSP) case when objective function of subtour elimination relaxation is minimized by $$\theta $$ -cyclic point: $$x_e \in \{0,\theta , 1-\theta 1\}$$ where support subcubic each vertex incident at least one edge with x-value 1. Such points are sufficient resolve TSP general. For these points, we construct convex combination which can reduce usage edges 1 from $$\frac{3}{2}$$ Christofides algorithm $$\frac{3}{2}-\frac{\theta }{10}$$ while keeping fractional same as algorithm. A direct consequence result Uniform Cover Problem TSP: In $$\frac{2}{3}$$ -uniform \{0, \frac{2}{3}\}$$ give $$\frac{17}{12}$$ -approximation TSP. such lands us halfway between approximation ratios $$\frac{4}{3}$$ implied famous “four-thirds conjecture”.