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What's the fundamental difference between the Knapsack problem and the travelling salesman (TSP) problem both of which are NP-hard, while the reality is that TSP could be solved much much faster?

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closed as unclear what you're asking by Noah Schweber, Stefan Kohl, Steven Sam, Chris Godsil, Timothy Chow Jul 25 '14 at 2:18

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It takes longer to pack for a trip than to take it? – The Masked Avenger Jul 24 '14 at 17:02
haha, this really makes me laugh! – user40780 Jul 24 '14 at 17:26
Studying why some theoretically hard problems seem to be practically easy is what gave rise to fields like parameterized complexity and average case complexity. I don't know what they have to say about these particular problems, though. – Dan Turetsky Jul 24 '14 at 19:56
Could you cite what is the reality behind the claim "that TSP could be solved much much faster"? I don't doubt the claim; I would just be interested to learn of what's behind it. Thanks. – Joseph O'Rourke Jul 24 '14 at 23:19
I agree with Joseph. To my knowledge, it is not the accepted view that the TSP can be solved "much much faster" than the knapsack problem. For example, theoretically, knapsack admits a FPTAS while (general) TSP is APX-complete (though Euclidean TSP admits a PTAS), so by that measure, knapsack is "easier." OTOH, if the question is about the solvability of "naturally occurring" instances, then user40780 needs to specify which classes of natural instances are being considered in each case. – Timothy Chow Jul 25 '14 at 2:25