Towards Improving Christofides Algorithm for Half-Integer TSP

We study the traveling salesman problem (TSP) in the case when the objective function of the subtour linear programming relaxation is minimized by a half-cycle point: x_e in {0,1/2,1} where the half-edges form a 2-factor and the 1-edges form a perfect matching. Such points are sufficient to resolve half-integer TSP in general and they have been conjectured to demonstrate the largest integrality gap for the subtour relaxation. For half-cycle points, the best-known approximation guarantee is 3/2 due to Christofides' famous algorithm. Proving an integrality gap of alpha for the subtour relaxation is equivalent to showing that alpha x can be written as a convex combination of tours, where x is any feasible solution for this relaxation. To beat Christofides' bound, our goal is to show that (3/2 - epsilon)x can be written as a convex combination of tours for some positive constant epsilon. Let y_e = 3/2-epsilon when x_e = 1 and y_e = 3/4 when x_e = 1/2. As a first step towards this goal, our main result is to show that y can be written as a convex combination of tours. In other words, we show that we can save on 1-edges, which has several applications. Among them, it gives an alternative algorithm for the recently studied uniform cover problem. Our main new technique is a procedure to glue tours over proper 3-edge cuts that are tight with respect to x, thus reducing the problem to a base case in which such cuts do not occur.