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What is the $NP$ problem whose status in $P$ or $NP$-complete is still unsettled, as of 2018?

This question is inspired by the following two recent breakthroughs:

  1. The work of Mulzer et. al on $NP$-completeness of min-weight triangulation.
  2. Recent quasi-polynomial algorithm of Babai for graph isomorphism. The link is to Helfgott's more recent (2017) paper.
John L.
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Thinh D. Nguyen
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3 Answers3

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In addition to Eugen's answer, the problem of determining whether an integer $N$ has a factor less or equal to $M$ is also not known to be in either P or NP-C.

DreamConspiracy
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One example is graph isomorphism.

Eugen
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IMHO, there are dozens of meaningful $NP$ problems which statuses have not been dichotomized.

In fact, each one of your rather long list of "Karp hardness of..." questions contains an $NP$ problem that has some such variant. As long as one delves deep enough into each of these, one would find some subtle variant for oneself.