Efficient optimization of the Held–Karp lower bound
Given a weighted undirected graph $G=(V,E)$, the Held–Karp lower bound for the Traveling Salesman Problem (TSP) is obtained by selecting an arbitrary vertex $\bar{p} \in V$, by computing a minimum cost tree spanning $V \backslash \lbrace \bar{p}\rbrace $ and adding two minimum cost edges adjacent to...
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| Main Author: | |
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| Format: | Article |
| Language: | English |
| Published: |
Université de Montpellier
2021-11-01
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| Series: | Open Journal of Mathematical Optimization |
| Subjects: | |
| Online Access: | https://ojmo.centre-mersenne.org/articles/10.5802/ojmo.11/ |
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| Summary: | Given a weighted undirected graph $G=(V,E)$, the Held–Karp lower bound for the Traveling Salesman Problem (TSP) is obtained by selecting an arbitrary vertex $\bar{p} \in V$, by computing a minimum cost tree spanning $V \backslash \lbrace \bar{p}\rbrace $ and adding two minimum cost edges adjacent to $\bar{p}$. In general, different selections of vertex $\bar{p}$ provide different lower bounds. In this paper it is shown that the selection of vertex $\bar{p}$ 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. |
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| ISSN: | 2777-5860 |