ANALYZING THE HELD KARP TSP BOUND - A MONOTONICITY PROPERTY WITH APPLICATION

In their 1971 paper on the travelling 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. In this paper 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 3OPT where OPT is the cost of the optimum solution to the TSP instance. © 1990.