Assume that I have a number of sites (locations) and the time it takes to travel from each site to each site is pre-computed

Example Data - Site to Site Pre-Calculated Times in minutes

From Start Site
To A 20
To B 15
To C 15

From Site A
To B 10
To C 15

Site B
To A 10
To C 20

Site C
To A 15
To B 20

4 Sites is fairly simple, but what if the site set was say 1000 sites?

Given a large site set What would the best approach be to quickly find the optimal routes from the start site while visiting every other site just once?

Route Solutions from Start Site for 3 sites from a start site

1 A(20) B(10) C(20) = 50
2 A(20) C(15) B(20) = 55

3 B(15) A(10) C(15) = 40
4 B(15) C(20) A(15) = 50

5 C(15) A(15) B(10) = 40
6 C(15) B(20) A(10) = 45