Finding low cost TSP and 2-matching solutions using certain half-integer subtour vertices

Given the complete graph Kn = (V,E) on n nodes with edge costs c ∈ R, the symmetric traveling salesman problem (henceforth TSP) is to find a Hamilton cycle (or tour) in Kn of minimum cost. This problem is known to be NP-hard, even in the case where the costs satisfy the triangle inequality, i.e. when cij + cjk ≥ cik for all i, j, k ∈ V (see [5]). When the costs satisfy the triangle inequality, we say that this instance is a metric TSP. For any edge set F ⊆ E and x ∈ R, let x(F ) denote the sum ∑