# Relation of 1-trees to optimal tours

Question:

given a complete symmetric graph $$G(V,E)$$ with $$n$$ vertices and edges $$e_{ij}$$ having weight $$\omega_{ij}$$, does there always exists a vector of vertex potentials $$(\pi_1,\,\dots,\,\pi_n)$$ that, when added to the weight of adjacent edges, guarantee that the minimum 1-tree of the Held-Karp lower bound resembles the edge-set of optimal tour, i.e. of the lightest Hamilton cycle of $$G$$.

I have scrutinized some related websites but couldn't find an explicit statement about the existence of such a vector, but only estimates, so a clarification would be of help.

Just found the answer in the 1969 paper THE TRAVELING-SALESMAN PROBLEM AND MINIMUM SPANNING TREES by Held and Karp; in which it is shown that it is not always possible to find the otimal solution to the TSP instance by systematic variation of the vertex potentials $$\pi_i$$ and provide a necessary and sufficient condition under which it is possible, namely if a certain wellknown linear program has an optimal solution in integers.