Analysis of time complexity of travelling salesman problem
When solving TSP using dynamic programming you get something akin to the following:

TSP(graph, start, target) {
  if start == target {
    return 0;
  }

  min = infinity;
  for neighbor in neighbors(graph, start, target) {
    tour_length = TSP(remove(graph, start), neighbor, target)
                  + distance(graph, start, neighbor);
    if tour_length &lt; min {
      min = tour_length;
    }
  }
  return min;
}


(neighbors only returns target as a viable neighbor if no other choice is available)

This algorithm recursively finds the shortest tour starting from each neighbor and returns the minimum of those. If you do all this with dynamic programming you can calculate the amount of work you're doing like so:
There are $n$ possible start vertices and $2^n$ possible subgraphs. So this function will be called on at most $n\cdot2^n$ distinct arguments (the target never changes). Each call performs at most $O(n)$ work (there are at most $n$ neighbors). Hence the total work you're doing is $O(n^2\,2^n)$.