Combinatorial Improvements of the 1-Tree Bound for the Traveling Salesman Problem

1-trees are used as relaxation for the symmetric Traveling Salesman Problem. We propose to strengthen this relaxation by adding additional constraints of the following type: select a set of nonadjacent vertices and require the 1-tree to contain exactly two edges adjacent to any of those vertices. This leads to the intersection of a graphical matroid with a partition matroid. We propose an algorithm for this problem (more efficient than general matroid intersection) and provide computational results to indicate the improvement over the general 1-tree relaxation.