## Analyzing the Held-Karp TSP bound: A monotonicity property with application.(English)Zbl 0698.68050

Summary: In their 1971 paper on the traveling salesman problem and minimum spanning trees, Held and Karp showed that finding an optimally weighted 1-tree is equivalent to solving a linear program for the traveling salesman problem (TSP) with only node-degree constraints and subtour elimination constraints. We show that the Held-Karp 1-trees have a certain monotonicity property: given a particular instance of the symmetric TSP with triangle inequality, the cost of the minimum weighted 1-tree is monotonic with respect to the set of nodes included. As a consequence, we obtain an alternate proof of a result of Wolsey and show that linear programs with node-degree and subtour elimination constraints must have a cost at least (2/3)$${\mathcal O}{\mathcal P}{\mathcal T}$$ where $${\mathcal O}{\mathcal P}{\mathcal T}$$ is the cost of the optimum solution to the TSP instance.

### MSC:

 68Q25 Analysis of algorithms and problem complexity 90C35 Programming involving graphs or networks 05C05 Trees
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### References:

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