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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Université de Montpellier
2021-11-01
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| Series: | Open Journal of Mathematical Optimization |
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| Online Access: | https://ojmo.centre-mersenne.org/articles/10.5802/ojmo.11/ |
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| author | Righini, Giovanni |
| author_facet | Righini, Giovanni |
| author_sort | Righini, Giovanni |
| collection | DOAJ |
| description | 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. |
| format | Article |
| id | doaj-art-49469975218b48ebb373392a6701ead4 |
| institution | Kabale University |
| issn | 2777-5860 |
| language | English |
| publishDate | 2021-11-01 |
| publisher | Université de Montpellier |
| record_format | Article |
| series | Open Journal of Mathematical Optimization |
| spelling | doaj-art-49469975218b48ebb373392a6701ead42025-02-07T14:02:31ZengUniversité de MontpellierOpen Journal of Mathematical Optimization2777-58602021-11-01211710.5802/ojmo.1110.5802/ojmo.11Efficient optimization of the Held–Karp lower boundRighini, Giovanni0University of Milan, Department of Computer Science via Celoria 18, Milano ItalyGiven 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.https://ojmo.centre-mersenne.org/articles/10.5802/ojmo.11/Traveling salesman problemMinimum spanning treeHeld–Karp lower boundUnion-Find data-structure. |
| spellingShingle | Righini, Giovanni Efficient optimization of the Held–Karp lower bound Open Journal of Mathematical Optimization Traveling salesman problem Minimum spanning tree Held–Karp lower bound Union-Find data-structure. |
| title | Efficient optimization of the Held–Karp lower bound |
| title_full | Efficient optimization of the Held–Karp lower bound |
| title_fullStr | Efficient optimization of the Held–Karp lower bound |
| title_full_unstemmed | Efficient optimization of the Held–Karp lower bound |
| title_short | Efficient optimization of the Held–Karp lower bound |
| title_sort | efficient optimization of the held karp lower bound |
| topic | Traveling salesman problem Minimum spanning tree Held–Karp lower bound Union-Find data-structure. |
| url | https://ojmo.centre-mersenne.org/articles/10.5802/ojmo.11/ |
| work_keys_str_mv | AT righinigiovanni efficientoptimizationoftheheldkarplowerbound |