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"The ACO algorithm is used to solve the problem more efficiently than a brute-force approach". That's rather misleading as it's not a problem at all to find a

"The ACO algorithm is used to solve the problem more efficiently than a brute-force approach". That's rather misleading as it's not a problem at all to find a solution (fe try the nearest neighbour heuristic). The problem with TSP is that you want to prove that your solution IS the optimum. If you're interested in the optimum, you need something like `branch-and-bound`. Anyway, I did the exact same visualization as part of my master's in 1996 (but using distributed memetic algorithms iso ACO)

jeremysalwen 7y ago
Obviously finding good tours with a limited computational budget is an interesting problem[1]. So, no, it's not misleading, as this method is more efficient at finding good tours than brute-force (as claimed), and is even better than the NN heuristic. [1] https://en.wikipedia.org/wiki/Travelling_salesman_problem#Heuristic_and_approximation_algorithms https://en.wikipedia.org/wiki/Travelling_salesman_problem#He...
- toolslive 7y ago
  at least you should specify which brute force approach you're comparing it with.
  - thaumasiotes 7y ago
    What do you imagine the different brute force approaches are? I would assume the brute force approach is to calculate the cost of every possible route and choose the one that had the lowest cost.
- 7thaccount 7y ago
  Branch-and-bound MIP isn't very computationally intensive. I run models with millions of variables and constraints on my laptop. The commercial solvers (fast and best) are very expensive though. So you're kind of right in a eay.
thaumasiotes 7y ago
> The problem with TSP is that you want to prove that your solution IS the optimum. This can't be right? The whole definition of an NP problem is that while the solution may or may not be hard to find, it's always easy to verify the correct solution if somebody gives it to you. As applied to "you want to prove that your solution is the optimum", that means there is a polynomial-time algorithm to do exactly that. That's what verifying the solution means. The problem with TSP is that it's hard to determine what the solution is, not that it's hard to determine that the correct solution really is correct.
- toolslive 7y ago
  with the TSP, any permutation of the list of cities is a correct solution. They just differ in length of the tour. An optimal solution S has len(S) = min {len(P) | P in Perm(cities)}
  - thaumasiotes 7y ago
    No, the correct solution is the one that achieves the minimum travel cost[1]. If any route at all was a correct solution, the problem wouldn't be NP-complete -- it would be the much easier problem of "determine whether a path exists between two nodes in a graph". [1] It's actually easier to analyze the version of the problem which asks you to find any solution of length less than some constant L.
    - toolslive 7y ago
      So your view is that the heuristic methods (NN, 2-opt,...) do not yield a solution?
      - thaumasiotes 7y ago
        They do not necessarily yield a correct solution. For example, under a heuristic approach, if you ask "is there a path of length < L?", you may get the answer "no" when the actual answer is "yes". You seem to be confused over the difference between "solution", in the sense of the answer to a well-defined mathematical problem, and "solution", in the sense of "what do I say to my boss?"

jeremysalwen 7y ago

Obviously finding good tours with a limited computational budget is an interesting problem[1]. So, no, it's not misleading, as this method is more efficient at finding good tours than brute-force (as claimed), and is even better than the NN heuristic. [1] https://en.wikipedia.org/wiki/Travelling_salesman_problem#Heuristic_and_approximation_algorithms https://en.wikipedia.org/wiki/Travelling_salesman_problem#He...

toolslive 7y ago
at least you should specify which brute force approach you're comparing it with.
- thaumasiotes 7y ago
  What do you imagine the different brute force approaches are? I would assume the brute force approach is to calculate the cost of every possible route and choose the one that had the lowest cost.
7thaccount 7y ago
Branch-and-bound MIP isn't very computationally intensive. I run models with millions of variables and constraints on my laptop. The commercial solvers (fast and best) are very expensive though. So you're kind of right in a eay.

thaumasiotes 7y ago

> The problem with TSP is that you want to prove that your solution IS the optimum. This can't be right? The whole definition of an NP problem is that while the solution may or may not be hard to find, it's always easy to verify the correct solution if somebody gives it to you. As applied to "you want to prove that your solution is the optimum", that means there is a polynomial-time algorithm to do exactly that. That's what verifying the solution means. The problem with TSP is that it's hard to determine what the solution is, not that it's hard to determine that the correct solution really is correct.

toolslive 7y ago
with the TSP, any permutation of the list of cities is a correct solution. They just differ in length of the tour. An optimal solution S has len(S) = min {len(P) | P in Perm(cities)}
- thaumasiotes 7y ago
  No, the correct solution is the one that achieves the minimum travel cost[1]. If any route at all was a correct solution, the problem wouldn't be NP-complete -- it would be the much easier problem of "determine whether a path exists between two nodes in a graph". [1] It's actually easier to analyze the version of the problem which asks you to find any solution of length less than some constant L.
  - toolslive 7y ago
    So your view is that the heuristic methods (NN, 2-opt,...) do not yield a solution?
    - thaumasiotes 7y ago
      They do not necessarily yield a correct solution. For example, under a heuristic approach, if you ask "is there a path of length < L?", you may get the answer "no" when the actual answer is "yes". You seem to be confused over the difference between "solution", in the sense of the answer to a well-defined mathematical problem, and "solution", in the sense of "what do I say to my boss?"