**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](https://ieor.berkeley.edu/%7Ekaminsky/ieor251/notes/3-16-05.pdf) of the [Held-Karp lower bound](https://www2.seas.gwu.edu/%7Esimhaweb/champalg/tsp/tsp.html) 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](https://arxiv.org/abs/1501.01944),
so a clarification would be of help.



Relation of 1-trees to optimal tours
Just found the answer in the 1969 paper [THE TRAVELING-SALESMAN PROBLEM AND MINIMUM SPANNING TREES](https://www.cse.wustl.edu/%7Eychen/7102/Karp-TSP.pdf) 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.


