# Open Journal of Mathematical Optimization

Efficient optimization of the Held–Karp lower bound
Open Journal of Mathematical Optimization, Volume 2 (2021), article no. 9, 17 p.

Given a weighted undirected graph $G=\left(V,E\right)$, the Held–Karp lower bound for the Traveling Salesman Problem (TSP) is obtained by selecting an arbitrary vertex $\overline{p}\in V$, by computing a minimum cost tree spanning $V\setminus \left\{\overline{p}\right\}$ and adding two minimum cost edges adjacent to $\overline{p}$. In general, different selections of vertex $\overline{p}$ provide different lower bounds. In this paper it is shown that the selection of vertex $\overline{p}$ can be optimized, to obtain the largest possible Held–Karp lower bound, with the same worst-case computational time complexity required to compute a single minimum spanning tree. Although motivated by the optimization of the Held–Karp lower bound for the TSP, the algorithm solves a more general problem, allowing for the efficient pre-computation of alternative minimum spanning trees in weighted graphs where any vertex can be deleted.

Revised:
Accepted:
Published online:
DOI: 10.5802/ojmo.11
Keywords: Traveling salesman problem, Minimum spanning tree, Held–Karp lower bound, Union-Find data-structure.
Giovanni Righini 1

1 University of Milan, Department of Computer Science via Celoria 18, Milano Italy
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Giovanni Righini. Efficient optimization of the Held–Karp lower bound. Open Journal of Mathematical Optimization, Volume 2 (2021), article  no. 9, 17 p. doi : 10.5802/ojmo.11. https://ojmo.centre-mersenne.org/articles/10.5802/ojmo.11/

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