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I have some trouble trying to summarize the worst-case ratio of these heuristics for the metric (this means that it satisfies the triangle inequality) traveling salesman problem:

  • Nearest neighbor
  • Nearest insertion
  • Cheapest insertion
  • Farthest insertion

Nearest neighbor:

Here it says that the NN has a w-C ratio of

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This one, page 8, same as this one says that it is

enter image description here

Which changes a lot.

Insertion algorithms: Pretty match everyone agrees that the w-c ratio for cheapest and nearest insertion is <= 2 (always just for instances satisfying the triangle inequality) but coming to the farthest insertion every source is different:

here:

enter image description here (forgot to change NN to FI)

While here It is

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And here there is also a different one:

enter image description here

Regarding the FI, I think that it depends on the starting sub-tour. But in the NN, that ceil or floor bracket changes a lot the results, and since they all come from good sources, I can't figure out the right one.

Can someone summerize the actual known worst-case ratio for these algorithms?

Makyen
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Spinnaker
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  • Floor versus ceiling is not a material difference for most purposes. The differences for farthest are more worrying, though. Unfortunately, the only way to resolve this is to go spelunking in the TSP literature. – David Eisenstat Aug 31 '15 at 18:17
  • If I had to guess, the floor vs. ceiling thing is a typo, and the FI bounds quoted were proved in separate papers, since they're not actually in conflict. – David Eisenstat Aug 31 '15 at 18:31

1 Answers1

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NN: The correct bound uses ceiling, not floor (at least as proved in the original paper by Rosenkrantz et al. -- here, if you have access). I don't think there's a more recent bound that uses floor.

FI: Rosenkrantz et al. prove that the first bound applies to any insertion heuristic, including NN. Moreover, that bound is better than the other two (except for very small n). So I would use that bound. Note, however, that log really means log_2 in that formula. (I'm not sure where the other two bounds came from.)

One other note: It is known that there is no fixed worst-case bound for NN. It is not known whether there is a fixed worst-case bound for FI.

LarrySnyder610
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