Finding the best ways in a fully connected graph

I have a fully connected graph (undirected) with 500 vertices. The result is a matrix with 250,000 records (only 125,000 are required because it is non-directional).

Each rib has a specific weight. If I can only visit n vertices where n <500 you can find which starting vertex and which path will lead to maximum total weights.

Can this be resolved at any reasonable time?

Thank!

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