How to prove that this crossword puzzle is NP-complete?

  • We have an instance of crosswords in a square grid $G$ ​​of size $m \times m$.

  • We have a set of black boxes $N$.

  • We have a dictionary of words $D \subseteq\Sigma^{\ast}$, where $\Sigma$ is a finite alphabet.

The question is whether there is a way to fill in the white boxes ($G\setminus N$) so that all the horizontal and vertical words belong to $D$. Here we suppose $|\Sigma|\geq 3$.

Also note that the reduction should not create a dictionary that is too large.

  • $\begingroup$ You can reduce some version of Planar SAT to your problem. It would require some work. $\endgroup$ Feb 8, 2020 at 5:10
  • $\begingroup$ I think it's possible to use 3-SAT here, but how to do it? $\endgroup$
    – tala
    Feb 11, 2020 at 22:12
  • $\begingroup$ Why do you think it’s possible? Somebody told you? $\endgroup$ Feb 11, 2020 at 22:14
  • $\begingroup$ No, I just know that all NP-complete problems get reduced to 3-SAT $\endgroup$
    – tala
    Feb 11, 2020 at 22:19
  • $\begingroup$ Right, but sometimes there isn’t any simple direct reduction. $\endgroup$ Feb 11, 2020 at 22:21


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