I would like to know if there is any way to solve an NP-hard type problem, for example, the TSP, sum of subsets or knapsack problem, by using linear programming and not by brute force.

I ask this because I made a program which uses a linear programming algorithm and it solves them, but the execution time does not vary specifically from the input size, so I can't define its asymptotic behavior $O(n)$ yet.

For example, for the TSP with an input size $n=20$, depending on the cases and tested on an ordinary computer, it takes between $30$ seconds and $3$ minutes to solve the problem.

Greetings.