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I use FindShortestTour as explained here and it works nicely for a small number of cities. However when I try this with as little as 16 cities, the given tour isn't optimal anymore. I've tried using different Methods but none of them gave me the optimal tour (solutions within a 5% to 400% range of the optimal solution).

How can I force Mathematica to give me the optimal solution?

Edit:

This is my code (pretty much copy-paste from the link above as I'm very new to Mathematica)

dim = 16;
max = 100;

(* create symmetric matrix with random integers *)
d = RandomInteger[max, {dim, dim}];
d = Table[If[i > j, d[[i, j]], d[[j, i]]], {i, 1, Length[d[[1]]]}, {j, 1,Length[d[[1]]]}];
(d[[#, #]] = Infinity) & /@ Range[dim];
d // Grid

(* find tour *)
{len, tour} = FindShortestTour[Range[dim], DistanceFunction -> (d[[#1, #2]] &), 
  Method -> "TwoOpt"]

(* display tour *)
HighlightGraph[ WeightedAdjacencyGraph[d, DirectedEdges -> False, 
  GraphStyle -> "SmallNetwork", EdgeLabels -> "EdgeWeight"], 
 Style[UndirectedEdge[#1, #2], Thickness[.01], Red] & @@@ Partition[tour, 2, 1, 1]]
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Aerus
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  • "IntegerLinearProgramming" doesn't give you the shortest? – Rojo Oct 28 '13 at 15:32
  • @Rojo It's strangely enough one of the worst, it gave me 888 as distance whereas the optimal was 198. The best for this specific layout was TwoOpt giving me a solution of 205 but none of them ever seem to give me an optimal solution. (which I need to compare my algorithm against) – Aerus Oct 28 '13 at 15:35
  • @Aerus Perhaps you should post you code? – Lou Oct 28 '13 at 15:37
  • @Lou updated with code – Aerus Oct 28 '13 at 15:41
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    The fact that "IntegerLinearProgramming" fails in such spectacular fashion seems to indicate the second problem today that has shown up in certain integer programming library code. I guess I need to file some reports. – Daniel Lichtblau Oct 28 '13 at 16:17
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    Okay, this one at least will be fixed in a future release. – Daniel Lichtblau Oct 28 '13 at 17:23
  • The method "TwoOpt" is a local search combinatorial heuristic, it is fast but there is no guarantee of optimality of the solution. You can google "TSP 2-opt" for more info. For such methods the function should probably be renamed FindDecentlyShortTour (!) – A.G. Jan 13 '14 at 01:18
  • @DanielLichtblau This example now fails with a typo in the internal code. I filed a report. – Michael E2 Sep 27 '15 at 14:14
  • @MichaelE2 Sorry about that. Will fix. – Daniel Lichtblau Sep 28 '15 at 16:54

1 Answers1

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Daniel Lichtblau has confirmed that this is a problem with the integer linear programming code. Moreover, a bug report has been filed and the problem will be addressed in a future version.

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Oleksandr R.
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