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Let $G\left(\ V,\ E=V\times V\setminus\lbrace(v_i,v_i)\rbrace,\ \Omega: E\ni e_{ij}\mapsto\omega_{ij}\in\mathbb{R}\right)$ be a(n) (A)TSP instance.

Then

$$2*\ell(T_{\mathrm{opt}})\quad\ge\quad\sum_{v\in V}{\ \min_{(u,w)}\ \omega_{uv}+\omega_{vw}} $$


is a lower bound that can be calculated in $\Theta(n^2)$ time for TSPs time whereas for ATSPs I assume it might be $O(n^2\log(n))$ when utilizing adjacency lists that are sorted according to ascending edgeweights.


Questions:

  • is there already a name for the described lower bound?
  • are there online resources that discuss properties of that bound with regard to TSPs and ATSPs?


Remark: in practical TSP calculations one would of course take the maximum of other known lower bounds (most prominently the Held Karp bound) and the lower bound desribed above; that would guarantee that one gets the best of two worlds.

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