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or approximate optimal tour which we might get from any of the known methods. We now have an optimal tour which is say 2.5% of the optimal. If by our method we can improve or upgrade it to one step further to a shorter route then we can use inductive reasoning to establish that we can continue it till an optimal tour is reached. Now if we consider any hypothetical approximate optimal tour and apply our method of checking all the point on the existing net for being joined to their optimal branch, we can get a local optimal tour. Further our case of hypothetical or virtual segments will give us another local tour, which can be improved to a local optimal tour by our methods of segment to point and segment to segment check. Therefore, essentially our method, by the process of local improvements give a series
of local optimal tours which eventually lead to universal optimal tour . Basically our method has two kinds of improvements, one which is done on an existing network to get an local optimal tour. Further another improvement method gives a series of local optimal tours which lead to the ultimate optimal tour. The two improvements which we are discussing are segment to point check, segment to segment check and virtual segments check.
Therefore we have the optimal tour in the end. If for any reasons whatsoever we arrive at a junction when any of the corrections give no improvements then, indeed we have the optimal tour. This only means that all the points are joined to their optimal branches.
We can define these as interior local improvements and exterior local improvements. The interior local improvements are implemented when we compare all the points on a network without changing the basic configuration or network. This simply means that we cannot create a virtual segment in interior improvements. Hence in interior local improvements we can always reach at our starting point, if we wish. On the contrary in an external improvement we change the network via a virtual segment and any of the interior local improvements cannot take us back to our original network, without using a se