I have looked at rome2rio and few other sites, but it did not fit what I was looking for. I want an application or site, where in, if I feed it a list of cities it should tell me which city should be my starting point and what cities I need to traverse (with specific order) and what should be my final destination. I am mainly looking for travelling by road. The whole purpose is I need to make sure that I do not travel back and forth and waste time, I should travel optimally.
The problem that you state is known in academia as the Traveling Salesman Problem as pointed out by @ZachLipton.
Googling for that expression in combination with app/website/software gives a few good finds.
A nice online tool is Optimap which seems to use Google directions. Here is a random route along the US east coast. You can choose between a roundtrip and or ending up in a different city than where you started from.