The N-Queens completion problem is NP-hard

Re:The N-Queens completion problem is NP-hard by mmarx on Monday September 04, 2017 @01:21PM (#15169561)

That's just a good heuristic, though, it does not guarantee finding a solution. Also note that NP is a class of decision problems, and finding a solution for the n-Queens problem is not.

None of the two decision problems that arise naturally from n-queens, deciding whether there exists solutions for a given board size, and verifying whether a candidate is indeed a solution, are NP-hard (solutions exist for n \not\in {2, 3}; verification can clearly be done in time quadratic in n). Counting solutions is #P-complete, though.

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Re:The N-Queens completion problem is NP-hard by thomasahle on Monday September 04, 2017 @04:46PM (#15170762)

> Counting solutions is #P-complete, though.

Are you sure? Where was this proven? It could easily be that http://oeis.org/A000170 had a polynomial time combinatorial formula.

Maybe some completion-counting problem could be shown to be #P-complete though.

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Re:The N-Queens completion problem is NP-hard by taeric on Monday September 04, 2017 @05:11PM (#15170879)

If you read the cites on that link, you'll see that the value for 26 was only added in 2016. They do not have a closed form formula, or they would have shown it.

I mean, they could just be slow revealing. I doubt it, though.

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Re:The N-Queens completion problem is NP-hard by thomasahle on Wednesday September 06, 2017 @08:22AM (#15183024)

Computational hardness doesn't depend on whether anybody knows an efficient algorithm. Just whether one exists.

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Re:The N-Queens completion problem is NP-hard by taeric on Wednesday September 06, 2017 @10:12AM (#15183802)

Fair enough. I was obviously reading the above as having a polynomial time as having an efficient time one. Where efficient was shorthand for "quick." Both leaps, I concede were misguided.

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